10 49: Difference between revisions
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{{Knot Presentations}} |
{{Knot Presentations}} |
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<center><table border=1 cellpadding=10><tr align=center valign=top> |
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<td> |
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[[Braid Representatives|Minimum Braid Representative]]: |
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<table cellspacing=0 cellpadding=0 border=0> |
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<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr> |
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</table> |
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[[Invariants from Braid Theory|Length]] is 11, width is 4. |
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[[Invariants from Braid Theory|Braid index]] is 4. |
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</td> |
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<td> |
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[[Lightly Documented Features|A Morse Link Presentation]]: |
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[[Image:{{PAGENAME}}_ML.gif]] |
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</td> |
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</tr></table></center> |
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{{3D Invariants}} |
{{3D Invariants}} |
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{{4D Invariants}} |
{{4D Invariants}} |
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{{Polynomial Invariants}} |
{{Polynomial Invariants}} |
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=== "Similar" Knots (within the Atlas) === |
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Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
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{...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{...} |
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{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
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<tr align=center><td>-27</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>-27</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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</table>}} |
</table>}} |
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{{Display Coloured Jones|J2=<math> q^{-6} -2 q^{-7} + q^{-8} +7 q^{-9} -10 q^{-10} -3 q^{-11} +24 q^{-12} -19 q^{-13} -18 q^{-14} +46 q^{-15} -20 q^{-16} -41 q^{-17} +65 q^{-18} -14 q^{-19} -62 q^{-20} +72 q^{-21} -3 q^{-22} -69 q^{-23} +63 q^{-24} +6 q^{-25} -56 q^{-26} +41 q^{-27} +7 q^{-28} -32 q^{-29} +20 q^{-30} +4 q^{-31} -13 q^{-32} +7 q^{-33} + q^{-34} -3 q^{-35} + q^{-36} </math>|J3=<math> q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} +2 q^{-13} -10 q^{-14} -3 q^{-15} +17 q^{-16} +16 q^{-17} -30 q^{-18} -28 q^{-19} +29 q^{-20} +62 q^{-21} -35 q^{-22} -83 q^{-23} +11 q^{-24} +121 q^{-25} +6 q^{-26} -133 q^{-27} -55 q^{-28} +158 q^{-29} +83 q^{-30} -146 q^{-31} -138 q^{-32} +150 q^{-33} +165 q^{-34} -125 q^{-35} -209 q^{-36} +111 q^{-37} +232 q^{-38} -85 q^{-39} -250 q^{-40} +57 q^{-41} +259 q^{-42} -33 q^{-43} -246 q^{-44} +3 q^{-45} +228 q^{-46} +10 q^{-47} -183 q^{-48} -30 q^{-49} +150 q^{-50} +23 q^{-51} -100 q^{-52} -25 q^{-53} +72 q^{-54} +11 q^{-55} -43 q^{-56} -7 q^{-57} +30 q^{-58} - q^{-59} -18 q^{-60} + q^{-61} +13 q^{-62} -2 q^{-63} -8 q^{-64} +2 q^{-65} +3 q^{-66} + q^{-67} -3 q^{-68} + q^{-69} </math>|J4=<math> q^{-12} -2 q^{-13} + q^{-14} +3 q^{-15} -2 q^{-16} +2 q^{-17} -11 q^{-18} +3 q^{-19} +19 q^{-20} +6 q^{-22} -50 q^{-23} -15 q^{-24} +55 q^{-25} +36 q^{-26} +52 q^{-27} -121 q^{-28} -100 q^{-29} +50 q^{-30} +92 q^{-31} +212 q^{-32} -137 q^{-33} -229 q^{-34} -75 q^{-35} +50 q^{-36} +439 q^{-37} -3 q^{-38} -246 q^{-39} -266 q^{-40} -195 q^{-41} +558 q^{-42} +219 q^{-43} -41 q^{-44} -346 q^{-45} -565 q^{-46} +449 q^{-47} +352 q^{-48} +322 q^{-49} -215 q^{-50} -892 q^{-51} +178 q^{-52} +312 q^{-53} +683 q^{-54} +57 q^{-55} -1086 q^{-56} -123 q^{-57} +165 q^{-58} +957 q^{-59} +347 q^{-60} -1163 q^{-61} -385 q^{-62} -12 q^{-63} +1125 q^{-64} +604 q^{-65} -1120 q^{-66} -589 q^{-67} -216 q^{-68} +1146 q^{-69} +802 q^{-70} -909 q^{-71} -658 q^{-72} -437 q^{-73} +937 q^{-74} +864 q^{-75} -548 q^{-76} -516 q^{-77} -569 q^{-78} +553 q^{-79} +707 q^{-80} -210 q^{-81} -223 q^{-82} -503 q^{-83} +195 q^{-84} +407 q^{-85} -44 q^{-86} +22 q^{-87} -304 q^{-88} +20 q^{-89} +151 q^{-90} -27 q^{-91} +105 q^{-92} -125 q^{-93} -8 q^{-94} +31 q^{-95} -42 q^{-96} +76 q^{-97} -35 q^{-98} +5 q^{-99} +3 q^{-100} -34 q^{-101} +31 q^{-102} -8 q^{-103} +7 q^{-104} +2 q^{-105} -14 q^{-106} +7 q^{-107} -2 q^{-108} +3 q^{-109} + q^{-110} -3 q^{-111} + q^{-112} </math>|J5=<math> q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} + q^{-21} -5 q^{-22} +4 q^{-23} +16 q^{-24} +3 q^{-25} -16 q^{-26} -15 q^{-27} -26 q^{-28} +9 q^{-29} +59 q^{-30} +60 q^{-31} -9 q^{-32} -75 q^{-33} -128 q^{-34} -62 q^{-35} +111 q^{-36} +220 q^{-37} +154 q^{-38} -51 q^{-39} -318 q^{-40} -338 q^{-41} -45 q^{-42} +336 q^{-43} +505 q^{-44} +313 q^{-45} -272 q^{-46} -682 q^{-47} -541 q^{-48} +13 q^{-49} +648 q^{-50} +884 q^{-51} +332 q^{-52} -515 q^{-53} -962 q^{-54} -759 q^{-55} +44 q^{-56} +967 q^{-57} +1110 q^{-58} +444 q^{-59} -541 q^{-60} -1290 q^{-61} -1129 q^{-62} +5 q^{-63} +1184 q^{-64} +1603 q^{-65} +881 q^{-66} -792 q^{-67} -2058 q^{-68} -1669 q^{-69} +111 q^{-70} +2105 q^{-71} +2624 q^{-72} +735 q^{-73} -2083 q^{-74} -3268 q^{-75} -1679 q^{-76} +1673 q^{-77} +3936 q^{-78} +2621 q^{-79} -1305 q^{-80} -4281 q^{-81} -3476 q^{-82} +726 q^{-83} +4614 q^{-84} +4245 q^{-85} -263 q^{-86} -4766 q^{-87} -4888 q^{-88} -246 q^{-89} +4898 q^{-90} +5453 q^{-91} +696 q^{-92} -4953 q^{-93} -5942 q^{-94} -1145 q^{-95} +4912 q^{-96} +6344 q^{-97} +1666 q^{-98} -4761 q^{-99} -6647 q^{-100} -2184 q^{-101} +4356 q^{-102} +6783 q^{-103} +2788 q^{-104} -3794 q^{-105} -6656 q^{-106} -3288 q^{-107} +2924 q^{-108} +6252 q^{-109} +3736 q^{-110} -2038 q^{-111} -5513 q^{-112} -3845 q^{-113} +993 q^{-114} +4554 q^{-115} +3790 q^{-116} -202 q^{-117} -3461 q^{-118} -3336 q^{-119} -493 q^{-120} +2381 q^{-121} +2807 q^{-122} +813 q^{-123} -1446 q^{-124} -2102 q^{-125} -967 q^{-126} +736 q^{-127} +1485 q^{-128} +874 q^{-129} -256 q^{-130} -919 q^{-131} -739 q^{-132} -3 q^{-133} +535 q^{-134} +512 q^{-135} +127 q^{-136} -245 q^{-137} -353 q^{-138} -151 q^{-139} +105 q^{-140} +196 q^{-141} +124 q^{-142} -12 q^{-143} -100 q^{-144} -95 q^{-145} -17 q^{-146} +51 q^{-147} +50 q^{-148} +18 q^{-149} -6 q^{-150} -30 q^{-151} -24 q^{-152} +9 q^{-153} +13 q^{-154} +2 q^{-155} +9 q^{-156} -4 q^{-157} -10 q^{-158} + q^{-159} +3 q^{-160} -2 q^{-161} +3 q^{-162} + q^{-163} -3 q^{-164} + q^{-165} </math>|J6=<math> q^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +7 q^{-25} -4 q^{-26} + q^{-27} +18 q^{-28} -6 q^{-29} -14 q^{-30} -26 q^{-31} +11 q^{-32} -3 q^{-33} +16 q^{-34} +80 q^{-35} +18 q^{-36} -35 q^{-37} -124 q^{-38} -53 q^{-39} -70 q^{-40} +31 q^{-41} +273 q^{-42} +215 q^{-43} +92 q^{-44} -255 q^{-45} -282 q^{-46} -451 q^{-47} -242 q^{-48} +426 q^{-49} +679 q^{-50} +722 q^{-51} +56 q^{-52} -289 q^{-53} -1139 q^{-54} -1213 q^{-55} -192 q^{-56} +726 q^{-57} +1596 q^{-58} +1172 q^{-59} +883 q^{-60} -1033 q^{-61} -2196 q^{-62} -1764 q^{-63} -709 q^{-64} +1146 q^{-65} +1823 q^{-66} +2971 q^{-67} +931 q^{-68} -1169 q^{-69} -2337 q^{-70} -2724 q^{-71} -1479 q^{-72} -319 q^{-73} +3247 q^{-74} +3080 q^{-75} +2287 q^{-76} +569 q^{-77} -1923 q^{-78} -3765 q^{-79} -5031 q^{-80} -741 q^{-81} +1534 q^{-82} +4841 q^{-83} +6038 q^{-84} +3853 q^{-85} -1567 q^{-86} -8292 q^{-87} -7378 q^{-88} -5412 q^{-89} +2308 q^{-90} +9640 q^{-91} +12321 q^{-92} +6273 q^{-93} -6117 q^{-94} -12041 q^{-95} -14969 q^{-96} -5916 q^{-97} +7818 q^{-98} +18862 q^{-99} +16640 q^{-100} +1555 q^{-101} -11654 q^{-102} -22733 q^{-103} -16559 q^{-104} +992 q^{-105} +20875 q^{-106} +25452 q^{-107} +11417 q^{-108} -6958 q^{-109} -26535 q^{-110} -25864 q^{-111} -7635 q^{-112} +19197 q^{-113} +30886 q^{-114} +20112 q^{-115} -894 q^{-116} -27214 q^{-117} -32251 q^{-118} -15125 q^{-119} +16299 q^{-120} +33658 q^{-121} +26376 q^{-122} +4206 q^{-123} -26760 q^{-124} -36359 q^{-125} -20636 q^{-126} +13767 q^{-127} +35282 q^{-128} +30977 q^{-129} +8172 q^{-130} -26057 q^{-131} -39398 q^{-132} -25252 q^{-133} +10985 q^{-134} +35995 q^{-135} +35014 q^{-136} +12603 q^{-137} -23783 q^{-138} -41087 q^{-139} -30043 q^{-140} +5916 q^{-141} +33877 q^{-142} +37709 q^{-143} +18451 q^{-144} -17672 q^{-145} -38986 q^{-146} -33663 q^{-147} -1981 q^{-148} +26646 q^{-149} +36104 q^{-150} +23614 q^{-151} -7840 q^{-152} -30942 q^{-153} -32690 q^{-154} -9664 q^{-155} +15236 q^{-156} +28304 q^{-157} +24224 q^{-158} +1736 q^{-159} -18758 q^{-160} -25605 q^{-161} -12863 q^{-162} +4383 q^{-163} +16862 q^{-164} +19110 q^{-165} +6667 q^{-166} -7621 q^{-167} -15495 q^{-168} -10696 q^{-169} -1703 q^{-170} +6957 q^{-171} +11498 q^{-172} +6459 q^{-173} -1231 q^{-174} -7100 q^{-175} -6233 q^{-176} -2976 q^{-177} +1467 q^{-178} +5371 q^{-179} +4009 q^{-180} +812 q^{-181} -2451 q^{-182} -2655 q^{-183} -2075 q^{-184} -390 q^{-185} +2030 q^{-186} +1869 q^{-187} +854 q^{-188} -626 q^{-189} -813 q^{-190} -1033 q^{-191} -604 q^{-192} +648 q^{-193} +695 q^{-194} +490 q^{-195} -95 q^{-196} -125 q^{-197} -408 q^{-198} -396 q^{-199} +171 q^{-200} +195 q^{-201} +211 q^{-202} +6 q^{-203} +49 q^{-204} -122 q^{-205} -184 q^{-206} +37 q^{-207} +30 q^{-208} +68 q^{-209} +2 q^{-210} +48 q^{-211} -24 q^{-212} -62 q^{-213} +10 q^{-214} -3 q^{-215} +17 q^{-216} -5 q^{-217} +19 q^{-218} -2 q^{-219} -16 q^{-220} +5 q^{-221} -3 q^{-222} +3 q^{-223} -2 q^{-224} +3 q^{-225} + q^{-226} -3 q^{-227} + q^{-228} </math>|J7=Not Available}} |
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{{Computer Talk Header}} |
{{Computer Talk Header}} |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
</tr> |
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<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August |
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 20, 16, 1], |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 20, 16, 1], |
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X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], |
X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], |
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X[19, 12, 20, 13], X[13, 6, 14, 7], X[7, 2, 8, 3]]</nowiki></pre></td></tr> |
X[19, 12, 20, 13], X[13, 6, 14, 7], X[7, 2, 8, 3]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 7, -6, 8, -9, 3, -4, 5, -7, |
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6, -8, 4]</nowiki></pre></td></tr> |
6, -8, 4]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 8, 14, 2, 16, 18, 6, 20, 10, 12]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {-1, -1, -1, -1, 2, -1, -3, -2, -2, -2, -3}]</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {-1, -1, -1, -1, 2, -1, -3, -2, -2, -2, -3}]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 49]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BraidIndex[Knot[10, 49]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 49]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_49_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 49]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 3, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 49]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 8 12 2 3 |
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-13 + -- - -- + -- + 12 t - 8 t + 3 t |
-13 + -- - -- + -- + 12 t - 8 t + 3 t |
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3 2 t |
3 2 t |
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t t</nowiki></pre></td></tr> |
t t</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 49]][z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 49]][z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 |
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1 + 7 z + 10 z + 3 z</nowiki></pre></td></tr> |
1 + 7 z + 10 z + 3 z</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 49]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{59, -6}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 49]], KnotSignature[Knot[10, 49]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{59, -6}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 49]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -13 3 5 8 9 10 9 6 5 2 -3 |
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q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q |
q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q |
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12 11 10 9 8 7 6 5 4 |
12 11 10 9 8 7 6 5 4 |
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q q q q q q q q q</nowiki></pre></td></tr> |
q q q q q q q q q</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 49]}</nowiki></pre></td></tr> |
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<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 49]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 49]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -40 -38 -36 3 2 -28 2 3 3 2 2 |
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q + q - q - --- - --- - q - --- + --- + --- + --- + --- - |
q + q - q - --- - --- - q - --- + --- + --- + --- + --- - |
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32 30 26 24 20 18 14 |
32 30 26 24 20 18 14 |
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| Line 92: | Line 147: | ||
-12 -10 |
-12 -10 |
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q + q</nowiki></pre></td></tr> |
q + q</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 49]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[10, 49]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6 8 10 12 6 2 8 2 10 2 12 2 |
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a + 5 a - 7 a + 2 a + 4 a z + 12 a z - 10 a z + a z + |
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6 4 8 4 10 4 6 6 8 6 |
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4 a z + 9 a z - 3 a z + a z + 2 a z</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 49]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6 8 10 12 9 11 15 6 2 |
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-a + 5 a + 7 a + 2 a - 9 a z - 10 a z + a z + 4 a z - |
-a + 5 a + 7 a + 2 a - 9 a z - 10 a z + a z + 4 a z - |
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| Line 113: | Line 176: | ||
10 8 12 8 9 9 11 9 |
10 8 12 8 9 9 11 9 |
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6 a z + 3 a z + a z + a z</nowiki></pre></td></tr> |
6 a z + 3 a z + a z + a z</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 49]], Vassiliev[3][Knot[10, 49]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 49]], Vassiliev[3][Knot[10, 49]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{7, -16}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -7 -5 1 2 1 3 2 5 |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 49]][q, t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -7 -5 1 2 1 3 2 5 |
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q + q + ------- + ------ + ------ + ------ + ------ + ------ + |
q + q + ------- + ------ + ------ + ------ + ------ + ------ + |
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27 10 25 9 23 9 23 8 21 8 21 7 |
27 10 25 9 23 9 23 8 21 8 21 7 |
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| Line 130: | Line 195: | ||
13 3 11 3 11 2 9 2 7 |
13 3 11 3 11 2 9 2 7 |
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q t q t q t q t q t</nowiki></pre></td></tr> |
q t q t q t q t q t</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[10, 49], 2][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -36 3 -34 7 13 4 20 32 7 41 56 |
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q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + |
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35 33 32 31 30 29 28 27 26 |
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q q q q q q q q q |
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6 63 69 3 72 62 14 65 41 20 46 |
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--- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - |
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25 24 23 22 21 20 19 18 17 16 15 |
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q q q q q q q q q q q |
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18 19 24 3 10 7 -8 2 -6 |
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--- - --- + --- - --- - --- + -- + q - -- + q |
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14 13 12 11 10 9 7 |
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q q q q q q q</nowiki></pre></td></tr> |
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</table> |
</table> |
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See/edit the [[Rolfsen_Splice_Template]]. |
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[[Category:Knot Page]] |
[[Category:Knot Page]] |
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Revision as of 18:02, 29 August 2005
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Visit 10 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 49's page at Knotilus! Visit 10 49's page at the original Knot Atlas! |
Knot presentations
| Planar diagram presentation | X1425 X3849 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X7283 |
| Gauss code | -1, 10, -2, 1, -3, 9, -10, 2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4 |
| Dowker-Thistlethwaite code | 4 8 14 2 16 18 6 20 10 12 |
| Conway Notation | [41,21,2] |
Length is 11, width is 4. Braid index is 4. |
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Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +3 t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^6+10 z^4+7 z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 59, -6 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -10 q^{-8} +9 q^{-9} -8 q^{-10} +5 q^{-11} -3 q^{-12} + q^{-13} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^{12}+2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-7 a^{10}+2 z^6 a^8+9 z^4 a^8+12 z^2 a^8+5 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6+a^6} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-4 z^3 a^{15}+z a^{15}+4 z^6 a^{14}-4 z^4 a^{14}+4 z^7 a^{13}-4 z^5 a^{13}+z^3 a^{13}+3 z^8 a^{12}-3 z^6 a^{12}+2 z^4 a^{12}-2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+5 z^7 a^{11}-19 z^5 a^{11}+24 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+26 z^4 a^{10}-20 z^2 a^{10}+7 a^{10}+z^9 a^9+3 z^7 a^9-18 z^5 a^9+22 z^3 a^9-9 z a^9+3 z^8 a^8-11 z^6 a^8+15 z^4 a^8-13 z^2 a^8+5 a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6} |
| The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}+q^{38}-q^{36}-3 q^{32}-2 q^{30}-q^{28}-2 q^{26}+3 q^{24}+3 q^{20}+2 q^{18}+2 q^{14}-q^{12}+q^{10}} |
| The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{210}-2 q^{208}+4 q^{206}-6 q^{204}+4 q^{202}-2 q^{200}-4 q^{198}+12 q^{196}-18 q^{194}+22 q^{192}-20 q^{190}+9 q^{188}+5 q^{186}-22 q^{184}+38 q^{182}-43 q^{180}+41 q^{178}-26 q^{176}+2 q^{174}+26 q^{172}-46 q^{170}+60 q^{168}-54 q^{166}+33 q^{164}-30 q^{160}+49 q^{158}-44 q^{156}+24 q^{154}+8 q^{152}-36 q^{150}+40 q^{148}-26 q^{146}-13 q^{144}+53 q^{142}-80 q^{140}+72 q^{138}-36 q^{136}-23 q^{134}+71 q^{132}-105 q^{130}+97 q^{128}-66 q^{126}+8 q^{124}+41 q^{122}-78 q^{120}+84 q^{118}-61 q^{116}+18 q^{114}+20 q^{112}-48 q^{110}+48 q^{108}-26 q^{106}-6 q^{104}+43 q^{102}-56 q^{100}+48 q^{98}-10 q^{96}-32 q^{94}+72 q^{92}-79 q^{90}+62 q^{88}-20 q^{86}-22 q^{84}+58 q^{82}-66 q^{80}+58 q^{78}-29 q^{76}+q^{74}+20 q^{72}-30 q^{70}+27 q^{68}-17 q^{66}+9 q^{64}+q^{62}-4 q^{60}+5 q^{58}-4 q^{56}+3 q^{54}-q^{52}+q^{50}} |
Further Quantum Invariants
Further quantum knot invariants for 10_49.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{27}-2 q^{25}+2 q^{23}-3 q^{21}+q^{19}-q^{17}-q^{15}+3 q^{13}-q^{11}+3 q^9-q^7+q^5} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{74}-2 q^{72}-q^{70}+5 q^{68}-5 q^{66}-2 q^{64}+11 q^{62}-8 q^{60}-5 q^{58}+16 q^{56}-8 q^{54}-9 q^{52}+13 q^{50}-9 q^{46}+7 q^{42}-4 q^{40}-11 q^{38}+10 q^{36}+4 q^{34}-15 q^{32}+8 q^{30}+9 q^{28}-13 q^{26}+2 q^{24}+11 q^{22}-6 q^{20}-2 q^{18}+6 q^{16}-q^{12}+q^{10}} |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{141}-2 q^{139}-q^{137}+2 q^{135}+3 q^{133}-2 q^{131}-5 q^{129}+5 q^{127}+4 q^{125}-6 q^{123}-5 q^{121}+12 q^{119}+4 q^{117}-21 q^{115}-9 q^{113}+33 q^{111}+15 q^{109}-42 q^{107}-30 q^{105}+48 q^{103}+43 q^{101}-40 q^{99}-53 q^{97}+25 q^{95}+58 q^{93}-5 q^{91}-48 q^{89}-17 q^{87}+37 q^{85}+33 q^{83}-19 q^{81}-46 q^{79}+8 q^{77}+49 q^{75}+9 q^{73}-58 q^{71}-19 q^{69}+52 q^{67}+31 q^{65}-51 q^{63}-43 q^{61}+40 q^{59}+53 q^{57}-24 q^{55}-61 q^{53}+5 q^{51}+55 q^{49}+14 q^{47}-45 q^{45}-27 q^{43}+28 q^{41}+33 q^{39}-13 q^{37}-25 q^{35}+20 q^{31}+6 q^{29}-8 q^{27}-4 q^{25}+4 q^{23}+3 q^{21}-q^{17}+q^{15}} |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{228}-2 q^{226}-q^{224}+2 q^{222}+6 q^{218}-5 q^{216}-4 q^{214}-6 q^{210}+18 q^{208}-2 q^{206}-q^{204}-3 q^{202}-30 q^{200}+15 q^{198}+7 q^{196}+35 q^{194}+22 q^{192}-68 q^{190}-39 q^{188}-24 q^{186}+96 q^{184}+124 q^{182}-55 q^{180}-138 q^{178}-155 q^{176}+101 q^{174}+276 q^{172}+77 q^{170}-168 q^{168}-334 q^{166}-34 q^{164}+324 q^{162}+258 q^{160}-35 q^{158}-373 q^{156}-216 q^{154}+168 q^{152}+300 q^{150}+158 q^{148}-203 q^{146}-265 q^{144}-56 q^{142}+165 q^{140}+234 q^{138}+23 q^{136}-175 q^{134}-196 q^{132}+8 q^{130}+212 q^{128}+169 q^{126}-88 q^{124}-256 q^{122}-79 q^{120}+183 q^{118}+260 q^{116}-30 q^{114}-304 q^{112}-157 q^{110}+144 q^{108}+338 q^{106}+66 q^{104}-295 q^{102}-255 q^{100}+16 q^{98}+343 q^{96}+212 q^{94}-151 q^{92}-284 q^{90}-175 q^{88}+195 q^{86}+275 q^{84}+70 q^{82}-152 q^{80}-271 q^{78}-26 q^{76}+165 q^{74}+182 q^{72}+48 q^{70}-179 q^{68}-137 q^{66}-12 q^{64}+117 q^{62}+133 q^{60}-27 q^{58}-83 q^{56}-78 q^{54}+7 q^{52}+78 q^{50}+32 q^{48}-4 q^{46}-40 q^{44}-22 q^{42}+17 q^{40}+13 q^{38}+11 q^{36}-5 q^{34}-7 q^{32}+2 q^{30}+q^{28}+3 q^{26}-q^{22}+q^{20}} |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{335}-2 q^{333}-q^{331}+2 q^{329}+3 q^{325}+3 q^{323}-4 q^{321}-9 q^{319}-3 q^{317}+q^{315}+11 q^{313}+19 q^{311}+5 q^{309}-21 q^{307}-36 q^{305}-20 q^{303}+17 q^{301}+59 q^{299}+66 q^{297}+q^{295}-93 q^{293}-123 q^{291}-49 q^{289}+96 q^{287}+218 q^{285}+162 q^{283}-91 q^{281}-324 q^{279}-321 q^{277}-5 q^{275}+425 q^{273}+573 q^{271}+187 q^{269}-487 q^{267}-870 q^{265}-508 q^{263}+442 q^{261}+1181 q^{259}+953 q^{257}-230 q^{255}-1420 q^{253}-1481 q^{251}-159 q^{249}+1486 q^{247}+1960 q^{245}+726 q^{243}-1289 q^{241}-2304 q^{239}-1321 q^{237}+852 q^{235}+2338 q^{233}+1829 q^{231}-223 q^{229}-2059 q^{227}-2113 q^{225}-415 q^{223}+1516 q^{221}+2073 q^{219}+930 q^{217}-826 q^{215}-1774 q^{213}-1243 q^{211}+189 q^{209}+1293 q^{207}+1302 q^{205}+335 q^{203}-787 q^{201}-1226 q^{199}-670 q^{197}+369 q^{195}+1074 q^{193}+882 q^{191}-88 q^{189}-979 q^{187}-993 q^{185}-94 q^{183}+960 q^{181}+1147 q^{179}+188 q^{177}-1020 q^{175}-1304 q^{173}-332 q^{171}+1080 q^{169}+1565 q^{167}+523 q^{165}-1101 q^{163}-1779 q^{161}-832 q^{159}+965 q^{157}+1978 q^{155}+1200 q^{153}-686 q^{151}-1998 q^{149}-1566 q^{147}+224 q^{145}+1823 q^{143}+1848 q^{141}+321 q^{139}-1422 q^{137}-1924 q^{135}-851 q^{133}+823 q^{131}+1752 q^{129}+1254 q^{127}-168 q^{125}-1327 q^{123}-1401 q^{121}-439 q^{119}+734 q^{117}+1265 q^{115}+844 q^{113}-115 q^{111}-893 q^{109}-976 q^{107}-372 q^{105}+400 q^{103}+821 q^{101}+654 q^{99}+50 q^{97}-521 q^{95}-664 q^{93}-341 q^{91}+155 q^{89}+499 q^{87}+453 q^{85}+89 q^{83}-262 q^{81}-378 q^{79}-222 q^{77}+54 q^{75}+244 q^{73}+220 q^{71}+57 q^{69}-103 q^{67}-155 q^{65}-84 q^{63}+18 q^{61}+78 q^{59}+71 q^{57}+14 q^{55}-29 q^{53}-34 q^{51}-13 q^{49}+3 q^{47}+17 q^{45}+12 q^{43}-q^{41}-4 q^{39}-q^{37}-q^{35}+q^{33}+3 q^{31}-q^{27}+q^{25}} |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}+q^{38}-q^{36}-3 q^{32}-2 q^{30}-q^{28}-2 q^{26}+3 q^{24}+3 q^{20}+2 q^{18}+2 q^{14}-q^{12}+q^{10}} |
| 1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{108}-4 q^{106}+10 q^{104}-20 q^{102}+34 q^{100}-54 q^{98}+78 q^{96}-104 q^{94}+130 q^{92}-156 q^{90}+180 q^{88}-196 q^{86}+207 q^{84}-204 q^{82}+184 q^{80}-144 q^{78}+75 q^{76}+12 q^{74}-118 q^{72}+238 q^{70}-343 q^{68}+440 q^{66}-492 q^{64}+518 q^{62}-497 q^{60}+434 q^{58}-350 q^{56}+220 q^{54}-107 q^{52}-32 q^{50}+132 q^{48}-226 q^{46}+277 q^{44}-292 q^{42}+280 q^{40}-234 q^{38}+193 q^{36}-130 q^{34}+92 q^{32}-48 q^{30}+31 q^{28}-12 q^{26}+6 q^{24}-2 q^{22}+q^{20}} |
| 2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{100}+q^{98}-2 q^{94}-3 q^{92}-2 q^{90}-3 q^{88}+q^{86}+3 q^{84}+2 q^{82}+3 q^{80}+7 q^{78}+7 q^{76}-2 q^{74}-2 q^{72}+6 q^{70}-10 q^{66}-5 q^{64}+q^{62}-8 q^{60}-9 q^{58}-q^{56}-5 q^{52}+q^{50}+7 q^{48}-3 q^{46}+8 q^{42}+3 q^{40}-5 q^{38}+3 q^{36}+8 q^{34}+q^{32}-3 q^{30}+3 q^{28}+4 q^{26}-q^{24}-q^{22}+q^{20}} |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-2 q^{86}+4 q^{82}-6 q^{80}-q^{78}+9 q^{76}-8 q^{74}-4 q^{72}+14 q^{70}-6 q^{68}-5 q^{66}+14 q^{64}+q^{62}-3 q^{60}+5 q^{58}-7 q^{54}-14 q^{52}-2 q^{50}-4 q^{48}-16 q^{46}+6 q^{44}+9 q^{42}-6 q^{40}+9 q^{38}+12 q^{36}-5 q^{34}+6 q^{32}+4 q^{30}-3 q^{28}+3 q^{26}+q^{24}-q^{22}+q^{20}} |
| 1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{53}+q^{51}+2 q^{49}-q^{47}-5 q^{43}-2 q^{41}-5 q^{39}-q^{37}-q^{35}+2 q^{33}+3 q^{31}+2 q^{29}+4 q^{27}+3 q^{23}-q^{21}+2 q^{19}-q^{17}+q^{15}} |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}+q^{112}-q^{110}-2 q^{108}+q^{106}+q^{104}-6 q^{102}-6 q^{100}+4 q^{98}+q^{96}-8 q^{94}+3 q^{92}+13 q^{90}+5 q^{88}+5 q^{86}+18 q^{84}+14 q^{82}+q^{78}+q^{76}-21 q^{74}-26 q^{72}-12 q^{70}-19 q^{68}-26 q^{66}-4 q^{64}+8 q^{62}+4 q^{58}+16 q^{56}+13 q^{54}+4 q^{52}+7 q^{50}+9 q^{48}+2 q^{46}+q^{44}+5 q^{42}+2 q^{36}+q^{34}-q^{32}+q^{30}} |
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}+q^{64}+2 q^{62}+2 q^{60}-q^{58}-5 q^{54}-4 q^{52}-5 q^{50}-5 q^{48}-q^{46}-q^{44}+3 q^{42}+2 q^{40}+5 q^{38}+2 q^{36}+4 q^{34}+q^{32}+q^{30}+2 q^{28}-q^{26}+2 q^{24}-q^{22}+q^{20}} |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-2 q^{86}+4 q^{84}-6 q^{82}+8 q^{80}-11 q^{78}+13 q^{76}-14 q^{74}+14 q^{72}-12 q^{70}+8 q^{68}-3 q^{66}-4 q^{64}+11 q^{62}-19 q^{60}+23 q^{58}-28 q^{56}+27 q^{54}-28 q^{52}+22 q^{50}-18 q^{48}+10 q^{46}-2 q^{44}-3 q^{42}+10 q^{40}-11 q^{38}+16 q^{36}-13 q^{34}+14 q^{32}-10 q^{30}+9 q^{28}-5 q^{26}+3 q^{24}-q^{22}+q^{20}} |
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{142}-2 q^{138}-2 q^{136}+2 q^{134}+5 q^{132}-7 q^{128}-6 q^{126}+5 q^{124}+11 q^{122}+q^{120}-12 q^{118}-9 q^{116}+8 q^{114}+15 q^{112}-13 q^{108}-5 q^{106}+12 q^{104}+11 q^{102}-5 q^{100}-10 q^{98}+4 q^{96}+11 q^{94}-12 q^{90}-4 q^{88}+6 q^{86}+q^{84}-12 q^{82}-8 q^{80}+5 q^{78}+5 q^{76}-9 q^{74}-14 q^{72}+2 q^{70}+15 q^{68}+7 q^{66}-12 q^{64}-10 q^{62}+9 q^{60}+18 q^{58}+3 q^{56}-10 q^{54}-6 q^{52}+9 q^{50}+9 q^{48}-q^{46}-6 q^{44}+4 q^{40}+2 q^{38}-q^{36}-q^{34}+q^{30}} |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{122}-2 q^{120}+2 q^{118}-3 q^{116}+5 q^{114}-7 q^{112}+6 q^{110}-8 q^{108}+11 q^{106}-11 q^{104}+9 q^{102}-10 q^{100}+12 q^{98}-7 q^{96}+4 q^{94}-2 q^{92}+2 q^{90}+10 q^{88}-6 q^{86}+16 q^{84}-14 q^{82}+22 q^{80}-22 q^{78}+16 q^{76}-31 q^{74}+10 q^{72}-27 q^{70}+5 q^{68}-19 q^{66}+3 q^{64}-2 q^{62}+3 q^{60}+9 q^{58}-q^{56}+17 q^{54}-5 q^{52}+15 q^{50}-8 q^{48}+14 q^{46}-8 q^{44}+9 q^{42}-6 q^{40}+6 q^{38}-2 q^{36}+2 q^{34}-q^{32}+q^{30}} |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{210}-2 q^{208}+4 q^{206}-6 q^{204}+4 q^{202}-2 q^{200}-4 q^{198}+12 q^{196}-18 q^{194}+22 q^{192}-20 q^{190}+9 q^{188}+5 q^{186}-22 q^{184}+38 q^{182}-43 q^{180}+41 q^{178}-26 q^{176}+2 q^{174}+26 q^{172}-46 q^{170}+60 q^{168}-54 q^{166}+33 q^{164}-30 q^{160}+49 q^{158}-44 q^{156}+24 q^{154}+8 q^{152}-36 q^{150}+40 q^{148}-26 q^{146}-13 q^{144}+53 q^{142}-80 q^{140}+72 q^{138}-36 q^{136}-23 q^{134}+71 q^{132}-105 q^{130}+97 q^{128}-66 q^{126}+8 q^{124}+41 q^{122}-78 q^{120}+84 q^{118}-61 q^{116}+18 q^{114}+20 q^{112}-48 q^{110}+48 q^{108}-26 q^{106}-6 q^{104}+43 q^{102}-56 q^{100}+48 q^{98}-10 q^{96}-32 q^{94}+72 q^{92}-79 q^{90}+62 q^{88}-20 q^{86}-22 q^{84}+58 q^{82}-66 q^{80}+58 q^{78}-29 q^{76}+q^{74}+20 q^{72}-30 q^{70}+27 q^{68}-17 q^{66}+9 q^{64}+q^{62}-4 q^{60}+5 q^{58}-4 q^{56}+3 q^{54}-q^{52}+q^{50}} |
.
Computer Talk
The above data is available with the Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 49"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +3 t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^6+10 z^4+7 z^2+1} |
In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 59, -6 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -10 q^{-8} +9 q^{-9} -8 q^{-10} +5 q^{-11} -3 q^{-12} + q^{-13} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^{12}+2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-7 a^{10}+2 z^6 a^8+9 z^4 a^8+12 z^2 a^8+5 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6+a^6} |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-4 z^3 a^{15}+z a^{15}+4 z^6 a^{14}-4 z^4 a^{14}+4 z^7 a^{13}-4 z^5 a^{13}+z^3 a^{13}+3 z^8 a^{12}-3 z^6 a^{12}+2 z^4 a^{12}-2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+5 z^7 a^{11}-19 z^5 a^{11}+24 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+26 z^4 a^{10}-20 z^2 a^{10}+7 a^{10}+z^9 a^9+3 z^7 a^9-18 z^5 a^9+22 z^3 a^9-9 z a^9+3 z^8 a^8-11 z^6 a^8+15 z^4 a^8-13 z^2 a^8+5 a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {...}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}
Vassiliev invariants
| V2 and V3: | (7, -16) |
| V2,1 through V6,9: |
|
V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -6 is the signature of 10 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
|
The Coloured Jones Polynomials
The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1}
-dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)}
)
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} -2 q^{-7} + q^{-8} +7 q^{-9} -10 q^{-10} -3 q^{-11} +24 q^{-12} -19 q^{-13} -18 q^{-14} +46 q^{-15} -20 q^{-16} -41 q^{-17} +65 q^{-18} -14 q^{-19} -62 q^{-20} +72 q^{-21} -3 q^{-22} -69 q^{-23} +63 q^{-24} +6 q^{-25} -56 q^{-26} +41 q^{-27} +7 q^{-28} -32 q^{-29} +20 q^{-30} +4 q^{-31} -13 q^{-32} +7 q^{-33} + q^{-34} -3 q^{-35} + q^{-36} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} +2 q^{-13} -10 q^{-14} -3 q^{-15} +17 q^{-16} +16 q^{-17} -30 q^{-18} -28 q^{-19} +29 q^{-20} +62 q^{-21} -35 q^{-22} -83 q^{-23} +11 q^{-24} +121 q^{-25} +6 q^{-26} -133 q^{-27} -55 q^{-28} +158 q^{-29} +83 q^{-30} -146 q^{-31} -138 q^{-32} +150 q^{-33} +165 q^{-34} -125 q^{-35} -209 q^{-36} +111 q^{-37} +232 q^{-38} -85 q^{-39} -250 q^{-40} +57 q^{-41} +259 q^{-42} -33 q^{-43} -246 q^{-44} +3 q^{-45} +228 q^{-46} +10 q^{-47} -183 q^{-48} -30 q^{-49} +150 q^{-50} +23 q^{-51} -100 q^{-52} -25 q^{-53} +72 q^{-54} +11 q^{-55} -43 q^{-56} -7 q^{-57} +30 q^{-58} - q^{-59} -18 q^{-60} + q^{-61} +13 q^{-62} -2 q^{-63} -8 q^{-64} +2 q^{-65} +3 q^{-66} + q^{-67} -3 q^{-68} + q^{-69} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} -2 q^{-13} + q^{-14} +3 q^{-15} -2 q^{-16} +2 q^{-17} -11 q^{-18} +3 q^{-19} +19 q^{-20} +6 q^{-22} -50 q^{-23} -15 q^{-24} +55 q^{-25} +36 q^{-26} +52 q^{-27} -121 q^{-28} -100 q^{-29} +50 q^{-30} +92 q^{-31} +212 q^{-32} -137 q^{-33} -229 q^{-34} -75 q^{-35} +50 q^{-36} +439 q^{-37} -3 q^{-38} -246 q^{-39} -266 q^{-40} -195 q^{-41} +558 q^{-42} +219 q^{-43} -41 q^{-44} -346 q^{-45} -565 q^{-46} +449 q^{-47} +352 q^{-48} +322 q^{-49} -215 q^{-50} -892 q^{-51} +178 q^{-52} +312 q^{-53} +683 q^{-54} +57 q^{-55} -1086 q^{-56} -123 q^{-57} +165 q^{-58} +957 q^{-59} +347 q^{-60} -1163 q^{-61} -385 q^{-62} -12 q^{-63} +1125 q^{-64} +604 q^{-65} -1120 q^{-66} -589 q^{-67} -216 q^{-68} +1146 q^{-69} +802 q^{-70} -909 q^{-71} -658 q^{-72} -437 q^{-73} +937 q^{-74} +864 q^{-75} -548 q^{-76} -516 q^{-77} -569 q^{-78} +553 q^{-79} +707 q^{-80} -210 q^{-81} -223 q^{-82} -503 q^{-83} +195 q^{-84} +407 q^{-85} -44 q^{-86} +22 q^{-87} -304 q^{-88} +20 q^{-89} +151 q^{-90} -27 q^{-91} +105 q^{-92} -125 q^{-93} -8 q^{-94} +31 q^{-95} -42 q^{-96} +76 q^{-97} -35 q^{-98} +5 q^{-99} +3 q^{-100} -34 q^{-101} +31 q^{-102} -8 q^{-103} +7 q^{-104} +2 q^{-105} -14 q^{-106} +7 q^{-107} -2 q^{-108} +3 q^{-109} + q^{-110} -3 q^{-111} + q^{-112} } |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} + q^{-21} -5 q^{-22} +4 q^{-23} +16 q^{-24} +3 q^{-25} -16 q^{-26} -15 q^{-27} -26 q^{-28} +9 q^{-29} +59 q^{-30} +60 q^{-31} -9 q^{-32} -75 q^{-33} -128 q^{-34} -62 q^{-35} +111 q^{-36} +220 q^{-37} +154 q^{-38} -51 q^{-39} -318 q^{-40} -338 q^{-41} -45 q^{-42} +336 q^{-43} +505 q^{-44} +313 q^{-45} -272 q^{-46} -682 q^{-47} -541 q^{-48} +13 q^{-49} +648 q^{-50} +884 q^{-51} +332 q^{-52} -515 q^{-53} -962 q^{-54} -759 q^{-55} +44 q^{-56} +967 q^{-57} +1110 q^{-58} +444 q^{-59} -541 q^{-60} -1290 q^{-61} -1129 q^{-62} +5 q^{-63} +1184 q^{-64} +1603 q^{-65} +881 q^{-66} -792 q^{-67} -2058 q^{-68} -1669 q^{-69} +111 q^{-70} +2105 q^{-71} +2624 q^{-72} +735 q^{-73} -2083 q^{-74} -3268 q^{-75} -1679 q^{-76} +1673 q^{-77} +3936 q^{-78} +2621 q^{-79} -1305 q^{-80} -4281 q^{-81} -3476 q^{-82} +726 q^{-83} +4614 q^{-84} +4245 q^{-85} -263 q^{-86} -4766 q^{-87} -4888 q^{-88} -246 q^{-89} +4898 q^{-90} +5453 q^{-91} +696 q^{-92} -4953 q^{-93} -5942 q^{-94} -1145 q^{-95} +4912 q^{-96} +6344 q^{-97} +1666 q^{-98} -4761 q^{-99} -6647 q^{-100} -2184 q^{-101} +4356 q^{-102} +6783 q^{-103} +2788 q^{-104} -3794 q^{-105} -6656 q^{-106} -3288 q^{-107} +2924 q^{-108} +6252 q^{-109} +3736 q^{-110} -2038 q^{-111} -5513 q^{-112} -3845 q^{-113} +993 q^{-114} +4554 q^{-115} +3790 q^{-116} -202 q^{-117} -3461 q^{-118} -3336 q^{-119} -493 q^{-120} +2381 q^{-121} +2807 q^{-122} +813 q^{-123} -1446 q^{-124} -2102 q^{-125} -967 q^{-126} +736 q^{-127} +1485 q^{-128} +874 q^{-129} -256 q^{-130} -919 q^{-131} -739 q^{-132} -3 q^{-133} +535 q^{-134} +512 q^{-135} +127 q^{-136} -245 q^{-137} -353 q^{-138} -151 q^{-139} +105 q^{-140} +196 q^{-141} +124 q^{-142} -12 q^{-143} -100 q^{-144} -95 q^{-145} -17 q^{-146} +51 q^{-147} +50 q^{-148} +18 q^{-149} -6 q^{-150} -30 q^{-151} -24 q^{-152} +9 q^{-153} +13 q^{-154} +2 q^{-155} +9 q^{-156} -4 q^{-157} -10 q^{-158} + q^{-159} +3 q^{-160} -2 q^{-161} +3 q^{-162} + q^{-163} -3 q^{-164} + q^{-165} } |
| 6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +7 q^{-25} -4 q^{-26} + q^{-27} +18 q^{-28} -6 q^{-29} -14 q^{-30} -26 q^{-31} +11 q^{-32} -3 q^{-33} +16 q^{-34} +80 q^{-35} +18 q^{-36} -35 q^{-37} -124 q^{-38} -53 q^{-39} -70 q^{-40} +31 q^{-41} +273 q^{-42} +215 q^{-43} +92 q^{-44} -255 q^{-45} -282 q^{-46} -451 q^{-47} -242 q^{-48} +426 q^{-49} +679 q^{-50} +722 q^{-51} +56 q^{-52} -289 q^{-53} -1139 q^{-54} -1213 q^{-55} -192 q^{-56} +726 q^{-57} +1596 q^{-58} +1172 q^{-59} +883 q^{-60} -1033 q^{-61} -2196 q^{-62} -1764 q^{-63} -709 q^{-64} +1146 q^{-65} +1823 q^{-66} +2971 q^{-67} +931 q^{-68} -1169 q^{-69} -2337 q^{-70} -2724 q^{-71} -1479 q^{-72} -319 q^{-73} +3247 q^{-74} +3080 q^{-75} +2287 q^{-76} +569 q^{-77} -1923 q^{-78} -3765 q^{-79} -5031 q^{-80} -741 q^{-81} +1534 q^{-82} +4841 q^{-83} +6038 q^{-84} +3853 q^{-85} -1567 q^{-86} -8292 q^{-87} -7378 q^{-88} -5412 q^{-89} +2308 q^{-90} +9640 q^{-91} +12321 q^{-92} +6273 q^{-93} -6117 q^{-94} -12041 q^{-95} -14969 q^{-96} -5916 q^{-97} +7818 q^{-98} +18862 q^{-99} +16640 q^{-100} +1555 q^{-101} -11654 q^{-102} -22733 q^{-103} -16559 q^{-104} +992 q^{-105} +20875 q^{-106} +25452 q^{-107} +11417 q^{-108} -6958 q^{-109} -26535 q^{-110} -25864 q^{-111} -7635 q^{-112} +19197 q^{-113} +30886 q^{-114} +20112 q^{-115} -894 q^{-116} -27214 q^{-117} -32251 q^{-118} -15125 q^{-119} +16299 q^{-120} +33658 q^{-121} +26376 q^{-122} +4206 q^{-123} -26760 q^{-124} -36359 q^{-125} -20636 q^{-126} +13767 q^{-127} +35282 q^{-128} +30977 q^{-129} +8172 q^{-130} -26057 q^{-131} -39398 q^{-132} -25252 q^{-133} +10985 q^{-134} +35995 q^{-135} +35014 q^{-136} +12603 q^{-137} -23783 q^{-138} -41087 q^{-139} -30043 q^{-140} +5916 q^{-141} +33877 q^{-142} +37709 q^{-143} +18451 q^{-144} -17672 q^{-145} -38986 q^{-146} -33663 q^{-147} -1981 q^{-148} +26646 q^{-149} +36104 q^{-150} +23614 q^{-151} -7840 q^{-152} -30942 q^{-153} -32690 q^{-154} -9664 q^{-155} +15236 q^{-156} +28304 q^{-157} +24224 q^{-158} +1736 q^{-159} -18758 q^{-160} -25605 q^{-161} -12863 q^{-162} +4383 q^{-163} +16862 q^{-164} +19110 q^{-165} +6667 q^{-166} -7621 q^{-167} -15495 q^{-168} -10696 q^{-169} -1703 q^{-170} +6957 q^{-171} +11498 q^{-172} +6459 q^{-173} -1231 q^{-174} -7100 q^{-175} -6233 q^{-176} -2976 q^{-177} +1467 q^{-178} +5371 q^{-179} +4009 q^{-180} +812 q^{-181} -2451 q^{-182} -2655 q^{-183} -2075 q^{-184} -390 q^{-185} +2030 q^{-186} +1869 q^{-187} +854 q^{-188} -626 q^{-189} -813 q^{-190} -1033 q^{-191} -604 q^{-192} +648 q^{-193} +695 q^{-194} +490 q^{-195} -95 q^{-196} -125 q^{-197} -408 q^{-198} -396 q^{-199} +171 q^{-200} +195 q^{-201} +211 q^{-202} +6 q^{-203} +49 q^{-204} -122 q^{-205} -184 q^{-206} +37 q^{-207} +30 q^{-208} +68 q^{-209} +2 q^{-210} +48 q^{-211} -24 q^{-212} -62 q^{-213} +10 q^{-214} -3 q^{-215} +17 q^{-216} -5 q^{-217} +19 q^{-218} -2 q^{-219} -16 q^{-220} +5 q^{-221} -3 q^{-222} +3 q^{-223} -2 q^{-224} +3 q^{-225} + q^{-226} -3 q^{-227} + q^{-228} } |
| 7 | Not Available |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 29, 2005, 15:27:48)... | |
In[2]:= | PD[Knot[10, 49]] |
Out[2]= | PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 20, 16, 1],X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11],X[19, 12, 20, 13], X[13, 6, 14, 7], X[7, 2, 8, 3]] |
In[3]:= | GaussCode[Knot[10, 49]] |
Out[3]= | GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4] |
In[4]:= | DTCode[Knot[10, 49]] |
Out[4]= | DTCode[4, 8, 14, 2, 16, 18, 6, 20, 10, 12] |
In[5]:= | br = BR[Knot[10, 49]] |
Out[5]= | BR[4, {-1, -1, -1, -1, 2, -1, -3, -2, -2, -2, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 11} |
In[7]:= | BraidIndex[Knot[10, 49]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[10, 49]]] |
 | |
| Out[8]= | -Graphics- |
In[9]:= | (#[Knot[10, 49]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 3, 3, 3, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 49]][t] |
Out[10]= | 3 8 12 2 3 |
In[11]:= | Conway[Knot[10, 49]][z] |
Out[11]= | 2 4 6 1 + 7 z + 10 z + 3 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 49]} |
In[13]:= | {KnotDet[Knot[10, 49]], KnotSignature[Knot[10, 49]]} |
Out[13]= | {59, -6} |
In[14]:= | Jones[Knot[10, 49]][q] |
Out[14]= | -13 3 5 8 9 10 9 6 5 2 -3 |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 49]} |
In[16]:= | A2Invariant[Knot[10, 49]][q] |
Out[16]= | -40 -38 -36 3 2 -28 2 3 3 2 2 |
In[17]:= | HOMFLYPT[Knot[10, 49]][a, z] |
Out[17]= | 6 8 10 12 6 2 8 2 10 2 12 2 |
In[18]:= | Kauffman[Knot[10, 49]][a, z] |
Out[18]= | 6 8 10 12 9 11 15 6 2 |
In[19]:= | {Vassiliev[2][Knot[10, 49]], Vassiliev[3][Knot[10, 49]]} |
Out[19]= | {7, -16} |
In[20]:= | Kh[Knot[10, 49]][q, t] |
Out[20]= | -7 -5 1 2 1 3 2 5 |
In[21]:= | ColouredJones[Knot[10, 49], 2][q] |
Out[21]= | -36 3 -34 7 13 4 20 32 7 41 56 |
See/edit the Rolfsen_Splice_Template.
