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coloured_jones_5 = <math> q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} +2 q^{-21} -7 q^{-22} +3 q^{-23} +20 q^{-24} +5 q^{-25} -17 q^{-26} -17 q^{-27} -40 q^{-28} +2 q^{-29} +81 q^{-30} +93 q^{-31} +6 q^{-32} -95 q^{-33} -217 q^{-34} -146 q^{-35} +145 q^{-36} +388 q^{-37} +358 q^{-38} +8 q^{-39} -589 q^{-40} -807 q^{-41} -294 q^{-42} +631 q^{-43} +1298 q^{-44} +1055 q^{-45} -422 q^{-46} -1881 q^{-47} -1953 q^{-48} -368 q^{-49} +2028 q^{-50} +3311 q^{-51} +1689 q^{-52} -1807 q^{-53} -4307 q^{-54} -3616 q^{-55} +510 q^{-56} +5178 q^{-57} +5833 q^{-58} +1377 q^{-59} -4932 q^{-60} -8010 q^{-61} -4336 q^{-62} +3966 q^{-63} +9713 q^{-64} +7475 q^{-65} -1645 q^{-66} -10611 q^{-67} -10992 q^{-68} -1276 q^{-69} +10504 q^{-70} +13934 q^{-71} +5094 q^{-72} -9441 q^{-73} -16569 q^{-74} -8880 q^{-75} +7555 q^{-76} +18254 q^{-77} +12832 q^{-78} -5102 q^{-79} -19456 q^{-80} -16321 q^{-81} +2389 q^{-82} +19912 q^{-83} +19501 q^{-84} +446 q^{-85} -19991 q^{-86} -22235 q^{-87} -3206 q^{-88} +19703 q^{-89} +24507 q^{-90} +5915 q^{-91} -19023 q^{-92} -26444 q^{-93} -8553 q^{-94} +18043 q^{-95} +27810 q^{-96} +11104 q^{-97} -16437 q^{-98} -28639 q^{-99} -13581 q^{-100} +14359 q^{-101} +28564 q^{-102} +15773 q^{-103} -11583 q^{-104} -27561 q^{-105} -17462 q^{-106} +8386 q^{-107} +25406 q^{-108} +18387 q^{-109} -4942 q^{-110} -22316 q^{-111} -18285 q^{-112} +1738 q^{-113} +18385 q^{-114} +17153 q^{-115} +965 q^{-116} -14217 q^{-117} -15075 q^{-118} -2753 q^{-119} +10117 q^{-120} +12370 q^{-121} +3698 q^{-122} -6640 q^{-123} -9456 q^{-124} -3764 q^{-125} +3899 q^{-126} +6721 q^{-127} +3315 q^{-128} -2072 q^{-129} -4411 q^{-130} -2537 q^{-131} +892 q^{-132} +2713 q^{-133} +1791 q^{-134} -368 q^{-135} -1525 q^{-136} -1078 q^{-137} +46 q^{-138} +817 q^{-139} +657 q^{-140} -31 q^{-141} -410 q^{-142} -300 q^{-143} -18 q^{-144} +184 q^{-145} +170 q^{-146} -5 q^{-147} -101 q^{-148} -61 q^{-149} +12 q^{-150} +39 q^{-151} +19 q^{-152} +5 q^{-153} -23 q^{-154} -20 q^{-155} +17 q^{-156} +10 q^{-157} -6 q^{-158} + q^{-159} -8 q^{-161} +6 q^{-162} +3 q^{-163} -4 q^{-164} + q^{-165} </math> |
coloured_jones_5 = <math> q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} +2 q^{-21} -7 q^{-22} +3 q^{-23} +20 q^{-24} +5 q^{-25} -17 q^{-26} -17 q^{-27} -40 q^{-28} +2 q^{-29} +81 q^{-30} +93 q^{-31} +6 q^{-32} -95 q^{-33} -217 q^{-34} -146 q^{-35} +145 q^{-36} +388 q^{-37} +358 q^{-38} +8 q^{-39} -589 q^{-40} -807 q^{-41} -294 q^{-42} +631 q^{-43} +1298 q^{-44} +1055 q^{-45} -422 q^{-46} -1881 q^{-47} -1953 q^{-48} -368 q^{-49} +2028 q^{-50} +3311 q^{-51} +1689 q^{-52} -1807 q^{-53} -4307 q^{-54} -3616 q^{-55} +510 q^{-56} +5178 q^{-57} +5833 q^{-58} +1377 q^{-59} -4932 q^{-60} -8010 q^{-61} -4336 q^{-62} +3966 q^{-63} +9713 q^{-64} +7475 q^{-65} -1645 q^{-66} -10611 q^{-67} -10992 q^{-68} -1276 q^{-69} +10504 q^{-70} +13934 q^{-71} +5094 q^{-72} -9441 q^{-73} -16569 q^{-74} -8880 q^{-75} +7555 q^{-76} +18254 q^{-77} +12832 q^{-78} -5102 q^{-79} -19456 q^{-80} -16321 q^{-81} +2389 q^{-82} +19912 q^{-83} +19501 q^{-84} +446 q^{-85} -19991 q^{-86} -22235 q^{-87} -3206 q^{-88} +19703 q^{-89} +24507 q^{-90} +5915 q^{-91} -19023 q^{-92} -26444 q^{-93} -8553 q^{-94} +18043 q^{-95} +27810 q^{-96} +11104 q^{-97} -16437 q^{-98} -28639 q^{-99} -13581 q^{-100} +14359 q^{-101} +28564 q^{-102} +15773 q^{-103} -11583 q^{-104} -27561 q^{-105} -17462 q^{-106} +8386 q^{-107} +25406 q^{-108} +18387 q^{-109} -4942 q^{-110} -22316 q^{-111} -18285 q^{-112} +1738 q^{-113} +18385 q^{-114} +17153 q^{-115} +965 q^{-116} -14217 q^{-117} -15075 q^{-118} -2753 q^{-119} +10117 q^{-120} +12370 q^{-121} +3698 q^{-122} -6640 q^{-123} -9456 q^{-124} -3764 q^{-125} +3899 q^{-126} +6721 q^{-127} +3315 q^{-128} -2072 q^{-129} -4411 q^{-130} -2537 q^{-131} +892 q^{-132} +2713 q^{-133} +1791 q^{-134} -368 q^{-135} -1525 q^{-136} -1078 q^{-137} +46 q^{-138} +817 q^{-139} +657 q^{-140} -31 q^{-141} -410 q^{-142} -300 q^{-143} -18 q^{-144} +184 q^{-145} +170 q^{-146} -5 q^{-147} -101 q^{-148} -61 q^{-149} +12 q^{-150} +39 q^{-151} +19 q^{-152} +5 q^{-153} -23 q^{-154} -20 q^{-155} +17 q^{-156} +10 q^{-157} -6 q^{-158} + q^{-159} -8 q^{-161} +6 q^{-162} +3 q^{-163} -4 q^{-164} + q^{-165} </math> |
coloured_jones_6 = <math> q^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +8 q^{-25} -6 q^{-26} +22 q^{-28} -4 q^{-29} -15 q^{-30} -33 q^{-31} +12 q^{-32} -11 q^{-33} +12 q^{-34} +109 q^{-35} +45 q^{-36} -30 q^{-37} -169 q^{-38} -92 q^{-39} -145 q^{-40} -8 q^{-41} +399 q^{-42} +417 q^{-43} +267 q^{-44} -312 q^{-45} -484 q^{-46} -945 q^{-47} -718 q^{-48} +517 q^{-49} +1363 q^{-50} +1806 q^{-51} +759 q^{-52} -232 q^{-53} -2577 q^{-54} -3504 q^{-55} -1710 q^{-56} +1087 q^{-57} +4331 q^{-58} +4838 q^{-59} +4099 q^{-60} -1819 q^{-61} -7223 q^{-62} -8412 q^{-63} -5186 q^{-64} +2619 q^{-65} +9264 q^{-66} +14519 q^{-67} +7614 q^{-68} -4181 q^{-69} -14862 q^{-70} -18731 q^{-71} -10973 q^{-72} +3647 q^{-73} +23531 q^{-74} +25770 q^{-75} +14209 q^{-76} -8018 q^{-77} -28920 q^{-78} -34705 q^{-79} -20922 q^{-80} +15622 q^{-81} +38976 q^{-82} +44006 q^{-83} +20800 q^{-84} -18491 q^{-85} -52186 q^{-86} -58073 q^{-87} -17154 q^{-88} +29079 q^{-89} +66699 q^{-90} +63094 q^{-91} +19104 q^{-92} -45268 q^{-93} -87954 q^{-94} -64499 q^{-95} -8764 q^{-96} +64897 q^{-97} +98738 q^{-98} +72032 q^{-99} -10940 q^{-100} -94527 q^{-101} -106650 q^{-102} -61777 q^{-103} +37275 q^{-104} +113524 q^{-105} +121253 q^{-106} +37707 q^{-107} -77814 q^{-108} -131152 q^{-109} -112182 q^{-110} -3359 q^{-111} +108395 q^{-112} +155711 q^{-113} +84670 q^{-114} -49577 q^{-115} -139215 q^{-116} -150494 q^{-117} -43178 q^{-118} +93488 q^{-119} +176291 q^{-120} +122336 q^{-121} -20779 q^{-122} -138757 q^{-123} -177616 q^{-124} -76882 q^{-125} +76464 q^{-126} +188492 q^{-127} +152178 q^{-128} +6344 q^{-129} -133783 q^{-130} -197392 q^{-131} -107215 q^{-132} +56166 q^{-133} +192865 q^{-134} +177073 q^{-135} +36403 q^{-136} -119619 q^{-137} -207712 q^{-138} -136562 q^{-139} +26139 q^{-140} +181743 q^{-141} +192656 q^{-142} +71194 q^{-143} -88400 q^{-144} -198592 q^{-145} -158247 q^{-146} -14279 q^{-147} +146792 q^{-148} +186918 q^{-149} +101207 q^{-150} -41454 q^{-151} -161908 q^{-152} -158190 q^{-153} -52386 q^{-154} +92208 q^{-155} +152282 q^{-156} +110215 q^{-157} +5020 q^{-158} -104830 q^{-159} -129424 q^{-160} -70153 q^{-161} +37579 q^{-162} +98405 q^{-163} +91783 q^{-164} +31675 q^{-165} -49065 q^{-166} -83258 q^{-167} -61888 q^{-168} +2693 q^{-169} +47459 q^{-170} +57844 q^{-171} +33653 q^{-172} -13203 q^{-173} -40753 q^{-174} -39411 q^{-175} -8703 q^{-176} +15411 q^{-177} +27114 q^{-178} +22137 q^{-179} +838 q^{-180} -14669 q^{-181} -18678 q^{-182} -6914 q^{-183} +2273 q^{-184} +9133 q^{-185} +10343 q^{-186} +2700 q^{-187} -3705 q^{-188} -6736 q^{-189} -2762 q^{-190} -656 q^{-191} +2006 q^{-192} +3625 q^{-193} +1353 q^{-194} -613 q^{-195} -1941 q^{-196} -506 q^{-197} -527 q^{-198} +158 q^{-199} +1014 q^{-200} +376 q^{-201} -74 q^{-202} -504 q^{-203} +89 q^{-204} -169 q^{-205} -79 q^{-206} +248 q^{-207} +64 q^{-208} -20 q^{-209} -140 q^{-210} +100 q^{-211} -35 q^{-212} -41 q^{-213} +57 q^{-214} +2 q^{-215} -4 q^{-216} -42 q^{-217} +39 q^{-218} -2 q^{-219} -16 q^{-220} +14 q^{-221} -3 q^{-222} -8 q^{-224} +6 q^{-225} +3 q^{-226} -4 q^{-227} + q^{-228} </math> |
coloured_jones_6 = <math> q^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +8 q^{-25} -6 q^{-26} +22 q^{-28} -4 q^{-29} -15 q^{-30} -33 q^{-31} +12 q^{-32} -11 q^{-33} +12 q^{-34} +109 q^{-35} +45 q^{-36} -30 q^{-37} -169 q^{-38} -92 q^{-39} -145 q^{-40} -8 q^{-41} +399 q^{-42} +417 q^{-43} +267 q^{-44} -312 q^{-45} -484 q^{-46} -945 q^{-47} -718 q^{-48} +517 q^{-49} +1363 q^{-50} +1806 q^{-51} +759 q^{-52} -232 q^{-53} -2577 q^{-54} -3504 q^{-55} -1710 q^{-56} +1087 q^{-57} +4331 q^{-58} +4838 q^{-59} +4099 q^{-60} -1819 q^{-61} -7223 q^{-62} -8412 q^{-63} -5186 q^{-64} +2619 q^{-65} +9264 q^{-66} +14519 q^{-67} +7614 q^{-68} -4181 q^{-69} -14862 q^{-70} -18731 q^{-71} -10973 q^{-72} +3647 q^{-73} +23531 q^{-74} +25770 q^{-75} +14209 q^{-76} -8018 q^{-77} -28920 q^{-78} -34705 q^{-79} -20922 q^{-80} +15622 q^{-81} +38976 q^{-82} +44006 q^{-83} +20800 q^{-84} -18491 q^{-85} -52186 q^{-86} -58073 q^{-87} -17154 q^{-88} +29079 q^{-89} +66699 q^{-90} +63094 q^{-91} +19104 q^{-92} -45268 q^{-93} -87954 q^{-94} -64499 q^{-95} -8764 q^{-96} +64897 q^{-97} +98738 q^{-98} +72032 q^{-99} -10940 q^{-100} -94527 q^{-101} -106650 q^{-102} -61777 q^{-103} +37275 q^{-104} +113524 q^{-105} +121253 q^{-106} +37707 q^{-107} -77814 q^{-108} -131152 q^{-109} -112182 q^{-110} -3359 q^{-111} +108395 q^{-112} +155711 q^{-113} +84670 q^{-114} -49577 q^{-115} -139215 q^{-116} -150494 q^{-117} -43178 q^{-118} +93488 q^{-119} +176291 q^{-120} +122336 q^{-121} -20779 q^{-122} -138757 q^{-123} -177616 q^{-124} -76882 q^{-125} +76464 q^{-126} +188492 q^{-127} +152178 q^{-128} +6344 q^{-129} -133783 q^{-130} -197392 q^{-131} -107215 q^{-132} +56166 q^{-133} +192865 q^{-134} +177073 q^{-135} +36403 q^{-136} -119619 q^{-137} -207712 q^{-138} -136562 q^{-139} +26139 q^{-140} +181743 q^{-141} +192656 q^{-142} +71194 q^{-143} -88400 q^{-144} -198592 q^{-145} -158247 q^{-146} -14279 q^{-147} +146792 q^{-148} +186918 q^{-149} +101207 q^{-150} -41454 q^{-151} -161908 q^{-152} -158190 q^{-153} -52386 q^{-154} +92208 q^{-155} +152282 q^{-156} +110215 q^{-157} +5020 q^{-158} -104830 q^{-159} -129424 q^{-160} -70153 q^{-161} +37579 q^{-162} +98405 q^{-163} +91783 q^{-164} +31675 q^{-165} -49065 q^{-166} -83258 q^{-167} -61888 q^{-168} +2693 q^{-169} +47459 q^{-170} +57844 q^{-171} +33653 q^{-172} -13203 q^{-173} -40753 q^{-174} -39411 q^{-175} -8703 q^{-176} +15411 q^{-177} +27114 q^{-178} +22137 q^{-179} +838 q^{-180} -14669 q^{-181} -18678 q^{-182} -6914 q^{-183} +2273 q^{-184} +9133 q^{-185} +10343 q^{-186} +2700 q^{-187} -3705 q^{-188} -6736 q^{-189} -2762 q^{-190} -656 q^{-191} +2006 q^{-192} +3625 q^{-193} +1353 q^{-194} -613 q^{-195} -1941 q^{-196} -506 q^{-197} -527 q^{-198} +158 q^{-199} +1014 q^{-200} +376 q^{-201} -74 q^{-202} -504 q^{-203} +89 q^{-204} -169 q^{-205} -79 q^{-206} +248 q^{-207} +64 q^{-208} -20 q^{-209} -140 q^{-210} +100 q^{-211} -35 q^{-212} -41 q^{-213} +57 q^{-214} +2 q^{-215} -4 q^{-216} -42 q^{-217} +39 q^{-218} -2 q^{-219} -16 q^{-220} +14 q^{-221} -3 q^{-222} -8 q^{-224} +6 q^{-225} +3 q^{-226} -4 q^{-227} + q^{-228} </math> |
coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_7 = |
computer_talk =
computer_talk =
<table>
<table>
Line 53: Line 53:
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</td></tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<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, 66]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17],
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 66]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17],
X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19],
X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19],
X[19, 12, 20, 13], X[13, 8, 14, 9], X[9, 2, 10, 3]]</nowiki></pre></td></tr>
X[19, 12, 20, 13], X[13, 8, 14, 9], X[9, 2, 10, 3]]</nowiki></code></td></tr>
</table>
<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, 66]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6,
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 66]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6,
7, -8, 6]</nowiki></pre></td></tr>
7, -8, 6]</nowiki></code></td></tr>
</table>
<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, 66]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 10, 14, 16, 2, 18, 8, 6, 20, 12]</nowiki></pre></td></tr>
<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, 66]]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 66]]</nowiki></code></td></tr>
<tr align=left>
<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>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<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, 66]]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 10, 14, 16, 2, 18, 8, 6, 20, 12]</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr>
<table><tr align=left>
<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, 66]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_66_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>
<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, 66]]&) /@ {
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 66]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{4, 11}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 66]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 66]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:10_66_ML.gif]]</td></tr><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 66]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></pre></td></tr>
}</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 3, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr>
<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, 66]][t]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 9 16 2 3
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 3, 3, 3, NotAvailable, 1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 66]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 9 16 2 3
-19 + -- - -- + -- + 16 t - 9 t + 3 t
-19 + -- - -- + -- + 16 t - 9 t + 3 t
3 2 t
3 2 t
t t</nowiki></pre></td></tr>
t t</nowiki></code></td></tr>
</table>
<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, 66]][z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
1 + 7 z + 9 z + 3 z</nowiki></pre></td></tr>
<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>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 66]][z]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 66], Knot[11, Alternating, 245]}</nowiki></pre></td></tr>
<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, 66]], KnotSignature[Knot[10, 66]]}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{75, -6}</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6
1 + 7 z + 9 z + 3 z</nowiki></code></td></tr>
<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, 66]][q]</nowiki></pre></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -13 4 7 10 12 13 11 8 6 2 -3
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 66], Knot[11, Alternating, 245]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{75, -6}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 66]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -13 4 7 10 12 13 11 8 6 2 -3
q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q
q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q
12 11 10 9 8 7 6 5 4
12 11 10 9 8 7 6 5 4
q q q q q q q q q</nowiki></pre></td></tr>
q q q q q q q q q</nowiki></code></td></tr>
</table>
<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>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 66]}</nowiki></pre></td></tr>
<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, 66]][q]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -40 2 -34 2 3 2 2 3 2 3 -12
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 66]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 66]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -40 2 -34 2 3 2 2 3 2 3 -12
q - --- + q - --- - --- + --- - --- + --- + --- + --- - q +
q - --- + q - --- - --- + --- - --- + --- + --- + --- - q +
36 32 26 24 22 20 18 14
36 32 26 24 22 20 18 14
Line 104: Line 180:
-10
-10
q</nowiki></pre></td></tr>
q</nowiki></code></td></tr>
</table>
<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, 66]][a, z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]=&nbsp;&nbsp;</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
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 66]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 6 8 10 12 6 2 8 2 10 2 12 2
2 a + 2 a - 4 a + a + 5 a z + 9 a z - 8 a z + a z +
2 a + 2 a - 4 a + a + 5 a z + 9 a z - 8 a z + a z +
6 4 8 4 10 4 6 6 8 6
6 4 8 4 10 4 6 6 8 6
4 a z + 8 a z - 3 a z + a z + 2 a z</nowiki></pre></td></tr>
4 a z + 8 a z - 3 a z + a z + 2 a z</nowiki></code></td></tr>
</table>
<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, 66]][a, z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6 8 10 12 7 9 11 6 2
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 66]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 6 8 10 12 7 9 11 6 2
-2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z -
-2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z -
Line 131: Line 217:
10 8 12 8 9 9 11 9
10 8 12 8 9 9 11 9
7 a z + 4 a z + a z + a z</nowiki></pre></td></tr>
7 a z + 4 a z + a z + a z</nowiki></code></td></tr>
</table>
<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, 66]], Vassiliev[3][Knot[10, 66]]}</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{7, -17}</nowiki></pre></td></tr>
<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, 66]][q, t]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -7 -5 1 3 1 4 3 6
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{7, -17}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 66]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -7 -5 1 3 1 4 3 6
q + q + ------- + ------ + ------ + ------ + ------ + ------ +
q + q + ------- + ------ + ------ + ------ + ------ + ------ +
27 10 25 9 23 9 23 8 21 8 21 7
27 10 25 9 23 9 23 8 21 8 21 7
Line 148: Line 244:
------ + ------ + ------ + ----- + ----
------ + ------ + ------ + ----- + ----
13 3 11 3 11 2 9 2 7
13 3 11 3 11 2 9 2 7
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></code></td></tr>
</table>
<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, 66], 2][q]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -36 4 3 10 24 10 36 62 15 77 104
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 66], 2][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -36 4 3 10 24 10 36 62 15 77 104
q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- +
q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- +
35 34 33 32 31 30 29 28 27 26
35 34 33 32 31 30 29 28 27 26
Line 163: Line 264:
--- - --- + --- - --- - --- + -- + q - -- + q
--- - --- + --- - --- - --- + -- + q - -- + q
14 13 12 11 10 9 7
14 13 12 11 10 9 7
q q q q q q q</nowiki></pre></td></tr>
q q q q q q q</nowiki></code></td></tr>
</table> }}
</table> }}

Latest revision as of 17:59, 1 September 2005

10 65.gif

10_65

10 67.gif

10_67

10 66.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 66's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 66 at Knotilus!

[edit 10 66 Quick Notes]
[edit 10 66 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X9,2,10,3
Gauss code -1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code 4 10 14 16 2 18 8 6 20 12
Conway Notation [31,21,21]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 66 ML.gif 10 66 AP.gif
[{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 5}, {4, 6}, {5, 9}, {6, 1}, {7, 10}, {8, 2}, {12, 7}, {1, 8}]

[edit Notes on presentations of 10 66]


Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 66"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6
In[6]:= DTCode[K]
Out[6]= 4 10 14 16 2 18 8 6 20 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [31,21,21]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(4,\{-1,-1,-1,2,-1,-3,-2,-2,-2,-3,-3\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
10 66 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 5}, {4, 6}, {5, 9}, {6, 1}, {7, 10}, {8, 2}, {12, 7}, {1, 8}]
In[14]:= Draw[ap]
10 66 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-17][5]
Hyperbolic Volume 13.0293
A-Polynomial See Data:10 66/A-polynomial

[edit Notes for 10 66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math]
Topological 4 genus [math]\displaystyle{ 3 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -6

[edit Notes for 10 66's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^3-9 t^2+16 t-19+16 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^6+9 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 75, -6 }
Jones polynomial [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -13 q^{-8} +12 q^{-9} -10 q^{-10} +7 q^{-11} -4 q^{-12} + q^{-13} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^{12}+a^{12}-3 z^4 a^{10}-8 z^2 a^{10}-4 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+2 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 a^6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{16}+4 z^5 a^{15}-3 z^3 a^{15}+7 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}+7 z^7 a^{13}-7 z^5 a^{13}+z^3 a^{13}+4 z^8 a^{12}+3 z^6 a^{12}-13 z^4 a^{12}+5 z^2 a^{12}+a^{12}+z^9 a^{11}+11 z^7 a^{11}-28 z^5 a^{11}+22 z^3 a^{11}-6 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+8 z^4 a^{10}-8 z^2 a^{10}+4 a^{10}+z^9 a^9+6 z^7 a^9-22 z^5 a^9+20 z^3 a^9-5 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-6 z^2 a^8+2 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]
The A2 invariant [math]\displaystyle{ q^{40}-2 q^{36}+q^{34}-2 q^{32}-3 q^{26}+2 q^{24}-2 q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
The G2 invariant [math]\displaystyle{ q^{210}-3 q^{208}+6 q^{206}-10 q^{204}+9 q^{202}-6 q^{200}-2 q^{198}+19 q^{196}-34 q^{194}+50 q^{192}-53 q^{190}+33 q^{188}-2 q^{186}-44 q^{184}+90 q^{182}-117 q^{180}+119 q^{178}-79 q^{176}+9 q^{174}+73 q^{172}-134 q^{170}+158 q^{168}-136 q^{166}+64 q^{164}+21 q^{162}-95 q^{160}+126 q^{158}-92 q^{156}+21 q^{154}+64 q^{152}-114 q^{150}+100 q^{148}-35 q^{146}-74 q^{144}+168 q^{142}-211 q^{140}+179 q^{138}-68 q^{136}-76 q^{134}+198 q^{132}-256 q^{130}+228 q^{128}-135 q^{126}-5 q^{124}+115 q^{122}-177 q^{120}+178 q^{118}-106 q^{116}+q^{114}+82 q^{112}-117 q^{110}+87 q^{108}-17 q^{106}-76 q^{104}+136 q^{102}-139 q^{100}+90 q^{98}+6 q^{96}-107 q^{94}+178 q^{92}-175 q^{90}+118 q^{88}-30 q^{86}-60 q^{84}+118 q^{82}-125 q^{80}+102 q^{78}-47 q^{76}-2 q^{74}+38 q^{72}-49 q^{70}+42 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_66.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{27}-3 q^{25}+3 q^{23}-3 q^{21}+2 q^{19}-q^{17}-2 q^{15}+3 q^{13}-2 q^{11}+4 q^9-q^7+q^5 }[/math]
2 [math]\displaystyle{ q^{74}-3 q^{72}+9 q^{68}-11 q^{66}-4 q^{64}+22 q^{62}-16 q^{60}-11 q^{58}+30 q^{56}-12 q^{54}-19 q^{52}+20 q^{50}+2 q^{48}-16 q^{46}-q^{44}+16 q^{42}-2 q^{40}-20 q^{38}+18 q^{36}+10 q^{34}-30 q^{32}+10 q^{30}+18 q^{28}-22 q^{26}+q^{24}+17 q^{22}-8 q^{20}-3 q^{18}+7 q^{16}-q^{12}+q^{10} }[/math]
3 [math]\displaystyle{ q^{141}-3 q^{139}+6 q^{135}+q^{133}-11 q^{131}-7 q^{129}+26 q^{127}+10 q^{125}-39 q^{123}-22 q^{121}+57 q^{119}+39 q^{117}-79 q^{115}-64 q^{113}+95 q^{111}+96 q^{109}-99 q^{107}-127 q^{105}+86 q^{103}+151 q^{101}-55 q^{99}-158 q^{97}+13 q^{95}+145 q^{93}+33 q^{91}-112 q^{89}-72 q^{87}+69 q^{85}+105 q^{83}-23 q^{81}-123 q^{79}-21 q^{77}+125 q^{75}+64 q^{73}-130 q^{71}-97 q^{69}+110 q^{67}+131 q^{65}-89 q^{63}-147 q^{61}+53 q^{59}+157 q^{57}-14 q^{55}-149 q^{53}-24 q^{51}+122 q^{49}+50 q^{47}-88 q^{45}-63 q^{43}+47 q^{41}+62 q^{39}-20 q^{37}-42 q^{35}-q^{33}+28 q^{31}+8 q^{29}-10 q^{27}-5 q^{25}+5 q^{23}+3 q^{21}-q^{17}+q^{15} }[/math]
4 [math]\displaystyle{ q^{228}-3 q^{226}+6 q^{222}-2 q^{220}+q^{218}-14 q^{216}+3 q^{214}+26 q^{212}-8 q^{210}-3 q^{208}-50 q^{206}+10 q^{204}+91 q^{202}-2 q^{200}-41 q^{198}-150 q^{196}+25 q^{194}+248 q^{192}+71 q^{190}-129 q^{188}-395 q^{186}-34 q^{184}+511 q^{182}+344 q^{180}-157 q^{178}-786 q^{176}-321 q^{174}+682 q^{172}+793 q^{170}+94 q^{168}-1036 q^{166}-799 q^{164}+458 q^{162}+1063 q^{160}+586 q^{158}-789 q^{156}-1070 q^{154}-114 q^{152}+821 q^{150}+919 q^{148}-153 q^{146}-859 q^{144}-609 q^{142}+227 q^{140}+840 q^{138}+449 q^{136}-360 q^{134}-811 q^{132}-319 q^{130}+547 q^{128}+814 q^{126}+94 q^{124}-820 q^{122}-673 q^{120}+235 q^{118}+1007 q^{116}+467 q^{114}-726 q^{112}-929 q^{110}-129 q^{108}+1031 q^{106}+823 q^{104}-424 q^{102}-1031 q^{100}-588 q^{98}+754 q^{96}+1044 q^{94}+104 q^{92}-793 q^{90}-931 q^{88}+188 q^{86}+882 q^{84}+557 q^{82}-245 q^{80}-856 q^{78}-307 q^{76}+376 q^{74}+579 q^{72}+223 q^{70}-416 q^{68}-390 q^{66}-62 q^{64}+267 q^{62}+299 q^{60}-42 q^{58}-175 q^{56}-151 q^{54}+15 q^{52}+134 q^{50}+52 q^{48}-10 q^{46}-60 q^{44}-30 q^{42}+24 q^{40}+17 q^{38}+13 q^{36}-7 q^{34}-8 q^{32}+3 q^{30}+q^{28}+3 q^{26}-q^{22}+q^{20} }[/math]
5 [math]\displaystyle{ q^{335}-3 q^{333}+6 q^{329}-2 q^{327}-2 q^{325}-2 q^{323}-4 q^{321}+3 q^{319}+14 q^{317}+2 q^{315}-21 q^{313}-17 q^{311}+8 q^{309}+37 q^{307}+32 q^{305}-9 q^{303}-87 q^{301}-97 q^{299}+54 q^{297}+199 q^{295}+169 q^{293}-70 q^{291}-379 q^{289}-405 q^{287}+82 q^{285}+715 q^{283}+779 q^{281}+q^{279}-1114 q^{277}-1451 q^{275}-317 q^{273}+1579 q^{271}+2425 q^{269}+966 q^{267}-1920 q^{265}-3624 q^{263}-2100 q^{261}+1908 q^{259}+4915 q^{257}+3688 q^{255}-1357 q^{253}-5925 q^{251}-5542 q^{249}+107 q^{247}+6325 q^{245}+7336 q^{243}+1717 q^{241}-5860 q^{239}-8593 q^{237}-3810 q^{235}+4458 q^{233}+8949 q^{231}+5739 q^{229}-2360 q^{227}-8267 q^{225}-7033 q^{223}-12 q^{221}+6636 q^{219}+7459 q^{217}+2214 q^{215}-4427 q^{213}-7041 q^{211}-3883 q^{209}+2090 q^{207}+5971 q^{205}+4893 q^{203}+39 q^{201}-4630 q^{199}-5384 q^{197}-1700 q^{195}+3328 q^{193}+5523 q^{191}+2937 q^{189}-2252 q^{187}-5555 q^{185}-3895 q^{183}+1452 q^{181}+5661 q^{179}+4693 q^{177}-776 q^{175}-5782 q^{173}-5573 q^{171}+22 q^{169}+5936 q^{167}+6471 q^{165}+923 q^{163}-5746 q^{161}-7404 q^{159}-2238 q^{157}+5203 q^{155}+8090 q^{153}+3751 q^{151}-3987 q^{149}-8307 q^{147}-5358 q^{145}+2246 q^{143}+7823 q^{141}+6653 q^{139}-86 q^{137}-6545 q^{135}-7336 q^{133}-2094 q^{131}+4562 q^{129}+7163 q^{127}+3876 q^{125}-2222 q^{123}-6102 q^{121}-4890 q^{119}-44 q^{117}+4350 q^{115}+4975 q^{113}+1791 q^{111}-2353 q^{109}-4220 q^{107}-2702 q^{105}+546 q^{103}+2900 q^{101}+2826 q^{99}+715 q^{97}-1541 q^{95}-2271 q^{93}-1272 q^{91}+387 q^{89}+1461 q^{87}+1294 q^{85}+247 q^{83}-693 q^{81}-936 q^{79}-497 q^{77}+164 q^{75}+536 q^{73}+433 q^{71}+81 q^{69}-214 q^{67}-278 q^{65}-130 q^{63}+47 q^{61}+125 q^{59}+102 q^{57}+14 q^{55}-47 q^{53}-46 q^{51}-13 q^{49}+6 q^{47}+21 q^{45}+14 q^{43}-3 q^{41}-5 q^{39}-q^{35}+q^{33}+3 q^{31}-q^{27}+q^{25} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{40}-2 q^{36}+q^{34}-2 q^{32}-3 q^{26}+2 q^{24}-2 q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
1,1 [math]\displaystyle{ q^{108}-6 q^{106}+18 q^{104}-38 q^{102}+71 q^{100}-128 q^{98}+206 q^{96}-296 q^{94}+403 q^{92}-522 q^{90}+628 q^{88}-696 q^{86}+717 q^{84}-676 q^{82}+550 q^{80}-328 q^{78}+36 q^{76}+310 q^{74}-676 q^{72}+1016 q^{70}-1301 q^{68}+1474 q^{66}-1528 q^{64}+1452 q^{62}-1242 q^{60}+964 q^{58}-608 q^{56}+228 q^{54}+119 q^{52}-424 q^{50}+620 q^{48}-762 q^{46}+786 q^{44}-738 q^{42}+632 q^{40}-496 q^{38}+368 q^{36}-236 q^{34}+152 q^{32}-76 q^{30}+43 q^{28}-16 q^{26}+8 q^{24}-2 q^{22}+q^{20} }[/math]
2,0 [math]\displaystyle{ q^{100}-2 q^{96}-q^{94}+q^{92}+q^{90}-5 q^{88}+q^{86}+7 q^{84}-3 q^{80}+6 q^{78}+10 q^{76}-7 q^{74}-10 q^{72}+6 q^{70}+2 q^{68}-12 q^{66}-q^{64}+10 q^{62}-4 q^{60}-6 q^{58}+6 q^{56}+q^{54}-13 q^{52}-2 q^{50}+8 q^{48}-9 q^{46}-8 q^{44}+11 q^{42}+7 q^{40}-7 q^{38}+2 q^{36}+11 q^{34}+3 q^{32}-5 q^{30}+3 q^{28}+5 q^{26}-q^{24}-q^{22}+q^{20} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{88}-3 q^{86}+8 q^{82}-9 q^{80}-2 q^{78}+17 q^{76}-15 q^{74}-8 q^{72}+23 q^{70}-13 q^{68}-10 q^{66}+22 q^{64}-3 q^{62}-8 q^{60}+7 q^{58}+3 q^{56}-6 q^{54}-15 q^{52}+8 q^{50}+2 q^{48}-24 q^{46}+10 q^{44}+11 q^{42}-20 q^{40}+10 q^{38}+14 q^{36}-12 q^{34}+9 q^{32}+7 q^{30}-4 q^{28}+4 q^{26}+2 q^{24}-q^{22}+q^{20} }[/math]
1,0,0 [math]\displaystyle{ q^{53}+q^{49}-2 q^{47}+q^{45}-4 q^{43}+q^{41}-3 q^{39}-2 q^{35}+q^{31}+4 q^{27}+4 q^{23}-q^{21}+3 q^{19}-q^{17}+q^{15} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{114}-3 q^{110}-2 q^{108}+4 q^{106}+4 q^{104}-6 q^{102}-2 q^{100}+11 q^{98}-q^{96}-16 q^{94}+3 q^{92}+12 q^{90}-8 q^{88}-3 q^{86}+20 q^{84}+11 q^{82}-9 q^{80}+7 q^{78}+12 q^{76}-19 q^{74}-18 q^{72}+7 q^{70}-13 q^{68}-30 q^{66}+3 q^{64}+9 q^{62}-13 q^{60}-6 q^{58}+17 q^{56}+7 q^{54}-6 q^{52}+7 q^{50}+13 q^{48}+10 q^{42}+2 q^{40}+3 q^{36}+2 q^{34}-q^{32}+q^{30} }[/math]
1,0,0,0 [math]\displaystyle{ q^{66}+q^{62}+q^{60}-2 q^{58}+q^{56}-4 q^{54}-q^{52}-2 q^{50}-3 q^{48}-2 q^{44}+q^{42}-q^{40}+3 q^{38}+4 q^{34}+q^{32}+2 q^{30}+3 q^{28}-q^{26}+3 q^{24}-q^{22}+q^{20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{88}-3 q^{86}+6 q^{84}-10 q^{82}+15 q^{80}-20 q^{78}+23 q^{76}-25 q^{74}+24 q^{72}-21 q^{70}+13 q^{68}-2 q^{66}-10 q^{64}+23 q^{62}-34 q^{60}+43 q^{58}-49 q^{56}+48 q^{54}-45 q^{52}+34 q^{50}-26 q^{48}+12 q^{46}-11 q^{42}+20 q^{40}-22 q^{38}+26 q^{36}-22 q^{34}+21 q^{32}-15 q^{30}+12 q^{28}-6 q^{26}+4 q^{24}-q^{22}+q^{20} }[/math]
1,0 [math]\displaystyle{ q^{142}-3 q^{138}-3 q^{136}+3 q^{134}+9 q^{132}+2 q^{130}-12 q^{128}-11 q^{126}+9 q^{124}+20 q^{122}+2 q^{120}-23 q^{118}-16 q^{116}+15 q^{114}+25 q^{112}-3 q^{110}-25 q^{108}-7 q^{106}+21 q^{104}+16 q^{102}-13 q^{100}-17 q^{98}+8 q^{96}+18 q^{94}-3 q^{92}-19 q^{90}-2 q^{88}+15 q^{86}+3 q^{84}-18 q^{82}-8 q^{80}+15 q^{78}+12 q^{76}-16 q^{74}-23 q^{72}+5 q^{70}+26 q^{68}+6 q^{66}-25 q^{64}-18 q^{62}+15 q^{60}+26 q^{58}-18 q^{54}-8 q^{52}+14 q^{50}+13 q^{48}-2 q^{46}-8 q^{44}+5 q^{40}+3 q^{38}-q^{36}-q^{34}+q^{30} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{122}-3 q^{120}+3 q^{118}-4 q^{116}+9 q^{114}-12 q^{112}+12 q^{110}-14 q^{108}+20 q^{106}-20 q^{104}+16 q^{102}-19 q^{100}+19 q^{98}-12 q^{96}+5 q^{94}-3 q^{92}-2 q^{90}+17 q^{88}-17 q^{86}+26 q^{84}-30 q^{82}+38 q^{80}-37 q^{78}+33 q^{76}-43 q^{74}+27 q^{72}-31 q^{70}+16 q^{68}-21 q^{66}+5 q^{64}-4 q^{60}+11 q^{58}-14 q^{56}+22 q^{54}-16 q^{52}+21 q^{50}-16 q^{48}+20 q^{46}-11 q^{44}+14 q^{42}-7 q^{40}+8 q^{38}-2 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{210}-3 q^{208}+6 q^{206}-10 q^{204}+9 q^{202}-6 q^{200}-2 q^{198}+19 q^{196}-34 q^{194}+50 q^{192}-53 q^{190}+33 q^{188}-2 q^{186}-44 q^{184}+90 q^{182}-117 q^{180}+119 q^{178}-79 q^{176}+9 q^{174}+73 q^{172}-134 q^{170}+158 q^{168}-136 q^{166}+64 q^{164}+21 q^{162}-95 q^{160}+126 q^{158}-92 q^{156}+21 q^{154}+64 q^{152}-114 q^{150}+100 q^{148}-35 q^{146}-74 q^{144}+168 q^{142}-211 q^{140}+179 q^{138}-68 q^{136}-76 q^{134}+198 q^{132}-256 q^{130}+228 q^{128}-135 q^{126}-5 q^{124}+115 q^{122}-177 q^{120}+178 q^{118}-106 q^{116}+q^{114}+82 q^{112}-117 q^{110}+87 q^{108}-17 q^{106}-76 q^{104}+136 q^{102}-139 q^{100}+90 q^{98}+6 q^{96}-107 q^{94}+178 q^{92}-175 q^{90}+118 q^{88}-30 q^{86}-60 q^{84}+118 q^{82}-125 q^{80}+102 q^{78}-47 q^{76}-2 q^{74}+38 q^{72}-49 q^{70}+42 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50} }[/math]

.

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]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 66"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^3-9 t^2+16 t-19+16 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^6+9 z^4+7 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
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.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 75, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -13 q^{-8} +12 q^{-9} -10 q^{-10} +7 q^{-11} -4 q^{-12} + q^{-13} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^{12}+a^{12}-3 z^4 a^{10}-8 z^2 a^{10}-4 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+2 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 a^6 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{16}+4 z^5 a^{15}-3 z^3 a^{15}+7 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}+7 z^7 a^{13}-7 z^5 a^{13}+z^3 a^{13}+4 z^8 a^{12}+3 z^6 a^{12}-13 z^4 a^{12}+5 z^2 a^{12}+a^{12}+z^9 a^{11}+11 z^7 a^{11}-28 z^5 a^{11}+22 z^3 a^{11}-6 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+8 z^4 a^{10}-8 z^2 a^{10}+4 a^{10}+z^9 a^9+6 z^7 a^9-22 z^5 a^9+20 z^3 a^9-5 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-6 z^2 a^8+2 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a245,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 66"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ 3 t^3-9 t^2+16 t-19+16 t^{-1} -9 t^{-2} +3 t^{-3} }[/math], [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -13 q^{-8} +12 q^{-9} -10 q^{-10} +7 q^{-11} -4 q^{-12} + q^{-13} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {K11a245,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (7, -17)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 28 }[/math] [math]\displaystyle{ -136 }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ \frac{2594}{3} }[/math] [math]\displaystyle{ \frac{358}{3} }[/math] [math]\displaystyle{ -3808 }[/math] [math]\displaystyle{ -\frac{18448}{3} }[/math] [math]\displaystyle{ -\frac{3232}{3} }[/math] [math]\displaystyle{ -680 }[/math] [math]\displaystyle{ \frac{10976}{3} }[/math] [math]\displaystyle{ 9248 }[/math] [math]\displaystyle{ \frac{72632}{3} }[/math] [math]\displaystyle{ \frac{10024}{3} }[/math] [math]\displaystyle{ \frac{1349977}{30} }[/math] [math]\displaystyle{ \frac{41206}{15} }[/math] [math]\displaystyle{ \frac{677354}{45} }[/math] [math]\displaystyle{ \frac{3431}{18} }[/math] [math]\displaystyle{ \frac{55417}{30} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-6 is the signature of 10 66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -10-9-8-7-6-5-4-3-2-10χ
-5          11
-7         21-1
-9        4  4
-11       42  -2
-13      74   3
-15     64    -2
-17    67     -1
-19   46      2
-21  36       -3
-23 14        3
-25 3         -3
-271          1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math]
[math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{-6} -2 q^{-7} + q^{-8} +8 q^{-9} -12 q^{-10} -4 q^{-11} +33 q^{-12} -28 q^{-13} -27 q^{-14} +73 q^{-15} -36 q^{-16} -67 q^{-17} +113 q^{-18} -28 q^{-19} -105 q^{-20} +131 q^{-21} -10 q^{-22} -122 q^{-23} +116 q^{-24} +8 q^{-25} -104 q^{-26} +77 q^{-27} +15 q^{-28} -62 q^{-29} +36 q^{-30} +10 q^{-31} -24 q^{-32} +10 q^{-33} +3 q^{-34} -4 q^{-35} + q^{-36} }[/math]
3 [math]\displaystyle{ q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} +3 q^{-13} -12 q^{-14} -4 q^{-15} +21 q^{-16} +23 q^{-17} -41 q^{-18} -45 q^{-19} +43 q^{-20} +105 q^{-21} -56 q^{-22} -155 q^{-23} +18 q^{-24} +243 q^{-25} +16 q^{-26} -301 q^{-27} -107 q^{-28} +378 q^{-29} +187 q^{-30} -405 q^{-31} -307 q^{-32} +436 q^{-33} +407 q^{-34} -426 q^{-35} -514 q^{-36} +403 q^{-37} +601 q^{-38} -365 q^{-39} -660 q^{-40} +301 q^{-41} +701 q^{-42} -237 q^{-43} -696 q^{-44} +160 q^{-45} +661 q^{-46} -92 q^{-47} -584 q^{-48} +28 q^{-49} +490 q^{-50} +11 q^{-51} -378 q^{-52} -37 q^{-53} +277 q^{-54} +39 q^{-55} -183 q^{-56} -38 q^{-57} +118 q^{-58} +24 q^{-59} -65 q^{-60} -20 q^{-61} +39 q^{-62} +7 q^{-63} -16 q^{-64} -4 q^{-65} +6 q^{-66} +3 q^{-67} -4 q^{-68} + q^{-69} }[/math]
4 [math]\displaystyle{ q^{-12} -2 q^{-13} + q^{-14} +3 q^{-15} -2 q^{-16} +3 q^{-17} -13 q^{-18} +2 q^{-19} +23 q^{-20} +2 q^{-21} +10 q^{-22} -67 q^{-23} -28 q^{-24} +73 q^{-25} +64 q^{-26} +92 q^{-27} -186 q^{-28} -194 q^{-29} +49 q^{-30} +197 q^{-31} +433 q^{-32} -218 q^{-33} -523 q^{-34} -279 q^{-35} +171 q^{-36} +1072 q^{-37} +138 q^{-38} -726 q^{-39} -962 q^{-40} -378 q^{-41} +1683 q^{-42} +940 q^{-43} -401 q^{-44} -1656 q^{-45} -1497 q^{-46} +1821 q^{-47} +1837 q^{-48} +539 q^{-49} -1946 q^{-50} -2839 q^{-51} +1378 q^{-52} +2444 q^{-53} +1786 q^{-54} -1738 q^{-55} -3999 q^{-56} +578 q^{-57} +2647 q^{-58} +2979 q^{-59} -1198 q^{-60} -4771 q^{-61} -330 q^{-62} +2500 q^{-63} +3893 q^{-64} -478 q^{-65} -5038 q^{-66} -1196 q^{-67} +2008 q^{-68} +4344 q^{-69} +331 q^{-70} -4647 q^{-71} -1809 q^{-72} +1172 q^{-73} +4094 q^{-74} +1037 q^{-75} -3575 q^{-76} -1907 q^{-77} +237 q^{-78} +3138 q^{-79} +1318 q^{-80} -2200 q^{-81} -1430 q^{-82} -368 q^{-83} +1881 q^{-84} +1081 q^{-85} -1070 q^{-86} -731 q^{-87} -479 q^{-88} +878 q^{-89} +616 q^{-90} -441 q^{-91} -230 q^{-92} -312 q^{-93} +333 q^{-94} +255 q^{-95} -175 q^{-96} -30 q^{-97} -135 q^{-98} +110 q^{-99} +80 q^{-100} -66 q^{-101} +9 q^{-102} -42 q^{-103} +29 q^{-104} +20 q^{-105} -19 q^{-106} +4 q^{-107} -8 q^{-108} +6 q^{-109} +3 q^{-110} -4 q^{-111} + q^{-112} }[/math]
5 [math]\displaystyle{ q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} +2 q^{-21} -7 q^{-22} +3 q^{-23} +20 q^{-24} +5 q^{-25} -17 q^{-26} -17 q^{-27} -40 q^{-28} +2 q^{-29} +81 q^{-30} +93 q^{-31} +6 q^{-32} -95 q^{-33} -217 q^{-34} -146 q^{-35} +145 q^{-36} +388 q^{-37} +358 q^{-38} +8 q^{-39} -589 q^{-40} -807 q^{-41} -294 q^{-42} +631 q^{-43} +1298 q^{-44} +1055 q^{-45} -422 q^{-46} -1881 q^{-47} -1953 q^{-48} -368 q^{-49} +2028 q^{-50} +3311 q^{-51} +1689 q^{-52} -1807 q^{-53} -4307 q^{-54} -3616 q^{-55} +510 q^{-56} +5178 q^{-57} +5833 q^{-58} +1377 q^{-59} -4932 q^{-60} -8010 q^{-61} -4336 q^{-62} +3966 q^{-63} +9713 q^{-64} +7475 q^{-65} -1645 q^{-66} -10611 q^{-67} -10992 q^{-68} -1276 q^{-69} +10504 q^{-70} +13934 q^{-71} +5094 q^{-72} -9441 q^{-73} -16569 q^{-74} -8880 q^{-75} +7555 q^{-76} +18254 q^{-77} +12832 q^{-78} -5102 q^{-79} -19456 q^{-80} -16321 q^{-81} +2389 q^{-82} +19912 q^{-83} +19501 q^{-84} +446 q^{-85} -19991 q^{-86} -22235 q^{-87} -3206 q^{-88} +19703 q^{-89} +24507 q^{-90} +5915 q^{-91} -19023 q^{-92} -26444 q^{-93} -8553 q^{-94} +18043 q^{-95} +27810 q^{-96} +11104 q^{-97} -16437 q^{-98} -28639 q^{-99} -13581 q^{-100} +14359 q^{-101} +28564 q^{-102} +15773 q^{-103} -11583 q^{-104} -27561 q^{-105} -17462 q^{-106} +8386 q^{-107} +25406 q^{-108} +18387 q^{-109} -4942 q^{-110} -22316 q^{-111} -18285 q^{-112} +1738 q^{-113} +18385 q^{-114} +17153 q^{-115} +965 q^{-116} -14217 q^{-117} -15075 q^{-118} -2753 q^{-119} +10117 q^{-120} +12370 q^{-121} +3698 q^{-122} -6640 q^{-123} -9456 q^{-124} -3764 q^{-125} +3899 q^{-126} +6721 q^{-127} +3315 q^{-128} -2072 q^{-129} -4411 q^{-130} -2537 q^{-131} +892 q^{-132} +2713 q^{-133} +1791 q^{-134} -368 q^{-135} -1525 q^{-136} -1078 q^{-137} +46 q^{-138} +817 q^{-139} +657 q^{-140} -31 q^{-141} -410 q^{-142} -300 q^{-143} -18 q^{-144} +184 q^{-145} +170 q^{-146} -5 q^{-147} -101 q^{-148} -61 q^{-149} +12 q^{-150} +39 q^{-151} +19 q^{-152} +5 q^{-153} -23 q^{-154} -20 q^{-155} +17 q^{-156} +10 q^{-157} -6 q^{-158} + q^{-159} -8 q^{-161} +6 q^{-162} +3 q^{-163} -4 q^{-164} + q^{-165} }[/math]
6 [math]\displaystyle{ q^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +8 q^{-25} -6 q^{-26} +22 q^{-28} -4 q^{-29} -15 q^{-30} -33 q^{-31} +12 q^{-32} -11 q^{-33} +12 q^{-34} +109 q^{-35} +45 q^{-36} -30 q^{-37} -169 q^{-38} -92 q^{-39} -145 q^{-40} -8 q^{-41} +399 q^{-42} +417 q^{-43} +267 q^{-44} -312 q^{-45} -484 q^{-46} -945 q^{-47} -718 q^{-48} +517 q^{-49} +1363 q^{-50} +1806 q^{-51} +759 q^{-52} -232 q^{-53} -2577 q^{-54} -3504 q^{-55} -1710 q^{-56} +1087 q^{-57} +4331 q^{-58} +4838 q^{-59} +4099 q^{-60} -1819 q^{-61} -7223 q^{-62} -8412 q^{-63} -5186 q^{-64} +2619 q^{-65} +9264 q^{-66} +14519 q^{-67} +7614 q^{-68} -4181 q^{-69} -14862 q^{-70} -18731 q^{-71} -10973 q^{-72} +3647 q^{-73} +23531 q^{-74} +25770 q^{-75} +14209 q^{-76} -8018 q^{-77} -28920 q^{-78} -34705 q^{-79} -20922 q^{-80} +15622 q^{-81} +38976 q^{-82} +44006 q^{-83} +20800 q^{-84} -18491 q^{-85} -52186 q^{-86} -58073 q^{-87} -17154 q^{-88} +29079 q^{-89} +66699 q^{-90} +63094 q^{-91} +19104 q^{-92} -45268 q^{-93} -87954 q^{-94} -64499 q^{-95} -8764 q^{-96} +64897 q^{-97} +98738 q^{-98} +72032 q^{-99} -10940 q^{-100} -94527 q^{-101} -106650 q^{-102} -61777 q^{-103} +37275 q^{-104} +113524 q^{-105} +121253 q^{-106} +37707 q^{-107} -77814 q^{-108} -131152 q^{-109} -112182 q^{-110} -3359 q^{-111} +108395 q^{-112} +155711 q^{-113} +84670 q^{-114} -49577 q^{-115} -139215 q^{-116} -150494 q^{-117} -43178 q^{-118} +93488 q^{-119} +176291 q^{-120} +122336 q^{-121} -20779 q^{-122} -138757 q^{-123} -177616 q^{-124} -76882 q^{-125} +76464 q^{-126} +188492 q^{-127} +152178 q^{-128} +6344 q^{-129} -133783 q^{-130} -197392 q^{-131} -107215 q^{-132} +56166 q^{-133} +192865 q^{-134} +177073 q^{-135} +36403 q^{-136} -119619 q^{-137} -207712 q^{-138} -136562 q^{-139} +26139 q^{-140} +181743 q^{-141} +192656 q^{-142} +71194 q^{-143} -88400 q^{-144} -198592 q^{-145} -158247 q^{-146} -14279 q^{-147} +146792 q^{-148} +186918 q^{-149} +101207 q^{-150} -41454 q^{-151} -161908 q^{-152} -158190 q^{-153} -52386 q^{-154} +92208 q^{-155} +152282 q^{-156} +110215 q^{-157} +5020 q^{-158} -104830 q^{-159} -129424 q^{-160} -70153 q^{-161} +37579 q^{-162} +98405 q^{-163} +91783 q^{-164} +31675 q^{-165} -49065 q^{-166} -83258 q^{-167} -61888 q^{-168} +2693 q^{-169} +47459 q^{-170} +57844 q^{-171} +33653 q^{-172} -13203 q^{-173} -40753 q^{-174} -39411 q^{-175} -8703 q^{-176} +15411 q^{-177} +27114 q^{-178} +22137 q^{-179} +838 q^{-180} -14669 q^{-181} -18678 q^{-182} -6914 q^{-183} +2273 q^{-184} +9133 q^{-185} +10343 q^{-186} +2700 q^{-187} -3705 q^{-188} -6736 q^{-189} -2762 q^{-190} -656 q^{-191} +2006 q^{-192} +3625 q^{-193} +1353 q^{-194} -613 q^{-195} -1941 q^{-196} -506 q^{-197} -527 q^{-198} +158 q^{-199} +1014 q^{-200} +376 q^{-201} -74 q^{-202} -504 q^{-203} +89 q^{-204} -169 q^{-205} -79 q^{-206} +248 q^{-207} +64 q^{-208} -20 q^{-209} -140 q^{-210} +100 q^{-211} -35 q^{-212} -41 q^{-213} +57 q^{-214} +2 q^{-215} -4 q^{-216} -42 q^{-217} +39 q^{-218} -2 q^{-219} -16 q^{-220} +14 q^{-221} -3 q^{-222} -8 q^{-224} +6 q^{-225} +3 q^{-226} -4 q^{-227} + q^{-228} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

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10_65

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10_67

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