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coloured_jones_5 = <math>-q^{40}+5 q^{39}-4 q^{38}-12 q^{37}+11 q^{36}+11 q^{35}+10 q^{34}+4 q^{33}-40 q^{32}-80 q^{31}+7 q^{30}+140 q^{29}+171 q^{28}+54 q^{27}-254 q^{26}-490 q^{25}-319 q^{24}+449 q^{23}+1152 q^{22}+895 q^{21}-429 q^{20}-2084 q^{19}-2458 q^{18}-222 q^{17}+3485 q^{16}+5070 q^{15}+2143 q^{14}-4223 q^{13}-9204 q^{12}-6649 q^{11}+3800 q^{10}+14187 q^9+14066 q^8+18 q^7-18809 q^6-25002 q^5-8346 q^4+21019 q^3+37988 q^2+22598 q-18483-51260 q^{-1} -42095 q^{-2} +9165 q^{-3} +61828 q^{-4} +65515 q^{-5} +7408 q^{-6} -67235 q^{-7} -89671 q^{-8} -30559 q^{-9} +65579 q^{-10} +111902 q^{-11} +57841 q^{-12} -56827 q^{-13} -129342 q^{-14} -86346 q^{-15} +41963 q^{-16} +140724 q^{-17} +113285 q^{-18} -23266 q^{-19} -145828 q^{-20} -136469 q^{-21} +2968 q^{-22} +145474 q^{-23} +154852 q^{-24} +17071 q^{-25} -140939 q^{-26} -168406 q^{-27} -35656 q^{-28} +133463 q^{-29} +177419 q^{-30} +52498 q^{-31} -123496 q^{-32} -182731 q^{-33} -67920 q^{-34} +111567 q^{-35} +184355 q^{-36} +82072 q^{-37} -96779 q^{-38} -182393 q^{-39} -95395 q^{-40} +79292 q^{-41} +176022 q^{-42} +106998 q^{-43} -58440 q^{-44} -164552 q^{-45} -116191 q^{-46} +35395 q^{-47} +147300 q^{-48} +121050 q^{-49} -11230 q^{-50} -124675 q^{-51} -120244 q^{-52} -11282 q^{-53} +97755 q^{-54} +112623 q^{-55} +30011 q^{-56} -69237 q^{-57} -98701 q^{-58} -42121 q^{-59} +41861 q^{-60} +79782 q^{-61} +47105 q^{-62} -18807 q^{-63} -58903 q^{-64} -44958 q^{-65} +2139 q^{-66} +38711 q^{-67} +37774 q^{-68} +7668 q^{-69} -21982 q^{-70} -28159 q^{-71} -11294 q^{-72} +10043 q^{-73} +18409 q^{-74} +10815 q^{-75} -2801 q^{-76} -10599 q^{-77} -8233 q^{-78} -497 q^{-79} +5144 q^{-80} +5264 q^{-81} +1500 q^{-82} -2082 q^{-83} -2871 q^{-84} -1305 q^{-85} +601 q^{-86} +1359 q^{-87} +827 q^{-88} -107 q^{-89} -522 q^{-90} -404 q^{-91} -67 q^{-92} +202 q^{-93} +183 q^{-94} +12 q^{-95} -49 q^{-96} -40 q^{-97} -38 q^{-98} +18 q^{-99} +32 q^{-100} -10 q^{-101} -5 q^{-102} +7 q^{-103} -7 q^{-104} - q^{-105} +6 q^{-106} -3 q^{-107} -2 q^{-108} +3 q^{-109} - q^{-110} </math> |
coloured_jones_5 = <math>-q^{40}+5 q^{39}-4 q^{38}-12 q^{37}+11 q^{36}+11 q^{35}+10 q^{34}+4 q^{33}-40 q^{32}-80 q^{31}+7 q^{30}+140 q^{29}+171 q^{28}+54 q^{27}-254 q^{26}-490 q^{25}-319 q^{24}+449 q^{23}+1152 q^{22}+895 q^{21}-429 q^{20}-2084 q^{19}-2458 q^{18}-222 q^{17}+3485 q^{16}+5070 q^{15}+2143 q^{14}-4223 q^{13}-9204 q^{12}-6649 q^{11}+3800 q^{10}+14187 q^9+14066 q^8+18 q^7-18809 q^6-25002 q^5-8346 q^4+21019 q^3+37988 q^2+22598 q-18483-51260 q^{-1} -42095 q^{-2} +9165 q^{-3} +61828 q^{-4} +65515 q^{-5} +7408 q^{-6} -67235 q^{-7} -89671 q^{-8} -30559 q^{-9} +65579 q^{-10} +111902 q^{-11} +57841 q^{-12} -56827 q^{-13} -129342 q^{-14} -86346 q^{-15} +41963 q^{-16} +140724 q^{-17} +113285 q^{-18} -23266 q^{-19} -145828 q^{-20} -136469 q^{-21} +2968 q^{-22} +145474 q^{-23} +154852 q^{-24} +17071 q^{-25} -140939 q^{-26} -168406 q^{-27} -35656 q^{-28} +133463 q^{-29} +177419 q^{-30} +52498 q^{-31} -123496 q^{-32} -182731 q^{-33} -67920 q^{-34} +111567 q^{-35} +184355 q^{-36} +82072 q^{-37} -96779 q^{-38} -182393 q^{-39} -95395 q^{-40} +79292 q^{-41} +176022 q^{-42} +106998 q^{-43} -58440 q^{-44} -164552 q^{-45} -116191 q^{-46} +35395 q^{-47} +147300 q^{-48} +121050 q^{-49} -11230 q^{-50} -124675 q^{-51} -120244 q^{-52} -11282 q^{-53} +97755 q^{-54} +112623 q^{-55} +30011 q^{-56} -69237 q^{-57} -98701 q^{-58} -42121 q^{-59} +41861 q^{-60} +79782 q^{-61} +47105 q^{-62} -18807 q^{-63} -58903 q^{-64} -44958 q^{-65} +2139 q^{-66} +38711 q^{-67} +37774 q^{-68} +7668 q^{-69} -21982 q^{-70} -28159 q^{-71} -11294 q^{-72} +10043 q^{-73} +18409 q^{-74} +10815 q^{-75} -2801 q^{-76} -10599 q^{-77} -8233 q^{-78} -497 q^{-79} +5144 q^{-80} +5264 q^{-81} +1500 q^{-82} -2082 q^{-83} -2871 q^{-84} -1305 q^{-85} +601 q^{-86} +1359 q^{-87} +827 q^{-88} -107 q^{-89} -522 q^{-90} -404 q^{-91} -67 q^{-92} +202 q^{-93} +183 q^{-94} +12 q^{-95} -49 q^{-96} -40 q^{-97} -38 q^{-98} +18 q^{-99} +32 q^{-100} -10 q^{-101} -5 q^{-102} +7 q^{-103} -7 q^{-104} - q^{-105} +6 q^{-106} -3 q^{-107} -2 q^{-108} +3 q^{-109} - q^{-110} </math> |
coloured_jones_6 = <math>q^{57}-5 q^{56}+4 q^{55}+12 q^{54}-11 q^{53}-11 q^{52}-15 q^{51}+26 q^{50}+3 q^{49}+16 q^{48}+94 q^{47}-76 q^{46}-130 q^{45}-166 q^{44}+70 q^{43}+159 q^{42}+285 q^{41}+592 q^{40}-162 q^{39}-786 q^{38}-1315 q^{37}-492 q^{36}+358 q^{35}+1823 q^{34}+3589 q^{33}+1467 q^{32}-1862 q^{31}-6117 q^{30}-5872 q^{29}-3275 q^{28}+4221 q^{27}+14196 q^{26}+13763 q^{25}+4612 q^{24}-13711 q^{23}-25068 q^{22}-27520 q^{21}-7712 q^{20}+29398 q^{19}+52006 q^{18}+47384 q^{17}+2707 q^{16}-50689 q^{15}-95331 q^{14}-79513 q^{13}+4575 q^{12}+101533 q^{11}+157541 q^{10}+110029 q^9-14670 q^8-179840 q^7-249195 q^6-151498 q^5+65923 q^4+290494 q^3+350261 q^2+197545 q-153211-449328 q^{-1} -478721 q^{-2} -188657 q^{-3} +287272 q^{-4} +629512 q^{-5} +615513 q^{-6} +130178 q^{-7} -493832 q^{-8} -853697 q^{-9} -674173 q^{-10} +85 q^{-11} +741503 q^{-12} +1085958 q^{-13} +658679 q^{-14} -241986 q^{-15} -1056845 q^{-16} -1211702 q^{-17} -533358 q^{-18} +559209 q^{-19} +1383990 q^{-20} +1234571 q^{-21} +245365 q^{-22} -976920 q^{-23} -1580502 q^{-24} -1106663 q^{-25} +159076 q^{-26} +1414442 q^{-27} +1649785 q^{-28} +766439 q^{-29} -695670 q^{-30} -1702203 q^{-31} -1533185 q^{-32} -274757 q^{-33} +1256130 q^{-34} +1843110 q^{-35} +1164970 q^{-36} -369544 q^{-37} -1649075 q^{-38} -1770803 q^{-39} -622190 q^{-40} +1033995 q^{-41} +1879600 q^{-42} +1420734 q^{-43} -80945 q^{-44} -1516518 q^{-45} -1879548 q^{-46} -885422 q^{-47} +795906 q^{-48} +1828892 q^{-49} +1591422 q^{-50} +192914 q^{-51} -1322990 q^{-52} -1907246 q^{-53} -1120225 q^{-54} +502100 q^{-55} +1683461 q^{-56} +1704725 q^{-57} +504094 q^{-58} -1014325 q^{-59} -1821200 q^{-60} -1332607 q^{-61} +107102 q^{-62} +1374897 q^{-63} +1700374 q^{-64} +832731 q^{-65} -553067 q^{-66} -1536799 q^{-67} -1430264 q^{-68} -339098 q^{-69} +873701 q^{-70} +1474204 q^{-71} +1049771 q^{-72} -20239 q^{-73} -1031393 q^{-74} -1287386 q^{-75} -669820 q^{-76} +290363 q^{-77} +1009106 q^{-78} +1008702 q^{-79} +385642 q^{-80} -441190 q^{-81} -893256 q^{-82} -724847 q^{-83} -154155 q^{-84} +461572 q^{-85} +708363 q^{-86} +503434 q^{-87} +1901 q^{-88} -419792 q^{-89} -518360 q^{-90} -309994 q^{-91} +61894 q^{-92} +330955 q^{-93} +368560 q^{-94} +169279 q^{-95} -85271 q^{-96} -237747 q^{-97} -234071 q^{-98} -89280 q^{-99} +74382 q^{-100} +167164 q^{-101} +135206 q^{-102} +38741 q^{-103} -54925 q^{-104} -101709 q^{-105} -76580 q^{-106} -16054 q^{-107} +41676 q^{-108} +55570 q^{-109} +37906 q^{-110} +5490 q^{-111} -23995 q^{-112} -30296 q^{-113} -18510 q^{-114} +1825 q^{-115} +11931 q^{-116} +13858 q^{-117} +8319 q^{-118} -1409 q^{-119} -6551 q^{-120} -6486 q^{-121} -1947 q^{-122} +633 q^{-123} +2561 q^{-124} +2778 q^{-125} +757 q^{-126} -723 q^{-127} -1264 q^{-128} -474 q^{-129} -299 q^{-130} +163 q^{-131} +538 q^{-132} +222 q^{-133} -39 q^{-134} -183 q^{-135} +6 q^{-136} -80 q^{-137} -39 q^{-138} +84 q^{-139} +29 q^{-140} -2 q^{-141} -35 q^{-142} +23 q^{-143} -5 q^{-144} -18 q^{-145} +13 q^{-146} +2 q^{-147} + q^{-148} -6 q^{-149} +3 q^{-150} +2 q^{-151} -3 q^{-152} + q^{-153} </math> |
coloured_jones_6 = <math>q^{57}-5 q^{56}+4 q^{55}+12 q^{54}-11 q^{53}-11 q^{52}-15 q^{51}+26 q^{50}+3 q^{49}+16 q^{48}+94 q^{47}-76 q^{46}-130 q^{45}-166 q^{44}+70 q^{43}+159 q^{42}+285 q^{41}+592 q^{40}-162 q^{39}-786 q^{38}-1315 q^{37}-492 q^{36}+358 q^{35}+1823 q^{34}+3589 q^{33}+1467 q^{32}-1862 q^{31}-6117 q^{30}-5872 q^{29}-3275 q^{28}+4221 q^{27}+14196 q^{26}+13763 q^{25}+4612 q^{24}-13711 q^{23}-25068 q^{22}-27520 q^{21}-7712 q^{20}+29398 q^{19}+52006 q^{18}+47384 q^{17}+2707 q^{16}-50689 q^{15}-95331 q^{14}-79513 q^{13}+4575 q^{12}+101533 q^{11}+157541 q^{10}+110029 q^9-14670 q^8-179840 q^7-249195 q^6-151498 q^5+65923 q^4+290494 q^3+350261 q^2+197545 q-153211-449328 q^{-1} -478721 q^{-2} -188657 q^{-3} +287272 q^{-4} +629512 q^{-5} +615513 q^{-6} +130178 q^{-7} -493832 q^{-8} -853697 q^{-9} -674173 q^{-10} +85 q^{-11} +741503 q^{-12} +1085958 q^{-13} +658679 q^{-14} -241986 q^{-15} -1056845 q^{-16} -1211702 q^{-17} -533358 q^{-18} +559209 q^{-19} +1383990 q^{-20} +1234571 q^{-21} +245365 q^{-22} -976920 q^{-23} -1580502 q^{-24} -1106663 q^{-25} +159076 q^{-26} +1414442 q^{-27} +1649785 q^{-28} +766439 q^{-29} -695670 q^{-30} -1702203 q^{-31} -1533185 q^{-32} -274757 q^{-33} +1256130 q^{-34} +1843110 q^{-35} +1164970 q^{-36} -369544 q^{-37} -1649075 q^{-38} -1770803 q^{-39} -622190 q^{-40} +1033995 q^{-41} +1879600 q^{-42} +1420734 q^{-43} -80945 q^{-44} -1516518 q^{-45} -1879548 q^{-46} -885422 q^{-47} +795906 q^{-48} +1828892 q^{-49} +1591422 q^{-50} +192914 q^{-51} -1322990 q^{-52} -1907246 q^{-53} -1120225 q^{-54} +502100 q^{-55} +1683461 q^{-56} +1704725 q^{-57} +504094 q^{-58} -1014325 q^{-59} -1821200 q^{-60} -1332607 q^{-61} +107102 q^{-62} +1374897 q^{-63} +1700374 q^{-64} +832731 q^{-65} -553067 q^{-66} -1536799 q^{-67} -1430264 q^{-68} -339098 q^{-69} +873701 q^{-70} +1474204 q^{-71} +1049771 q^{-72} -20239 q^{-73} -1031393 q^{-74} -1287386 q^{-75} -669820 q^{-76} +290363 q^{-77} +1009106 q^{-78} +1008702 q^{-79} +385642 q^{-80} -441190 q^{-81} -893256 q^{-82} -724847 q^{-83} -154155 q^{-84} +461572 q^{-85} +708363 q^{-86} +503434 q^{-87} +1901 q^{-88} -419792 q^{-89} -518360 q^{-90} -309994 q^{-91} +61894 q^{-92} +330955 q^{-93} +368560 q^{-94} +169279 q^{-95} -85271 q^{-96} -237747 q^{-97} -234071 q^{-98} -89280 q^{-99} +74382 q^{-100} +167164 q^{-101} +135206 q^{-102} +38741 q^{-103} -54925 q^{-104} -101709 q^{-105} -76580 q^{-106} -16054 q^{-107} +41676 q^{-108} +55570 q^{-109} +37906 q^{-110} +5490 q^{-111} -23995 q^{-112} -30296 q^{-113} -18510 q^{-114} +1825 q^{-115} +11931 q^{-116} +13858 q^{-117} +8319 q^{-118} -1409 q^{-119} -6551 q^{-120} -6486 q^{-121} -1947 q^{-122} +633 q^{-123} +2561 q^{-124} +2778 q^{-125} +757 q^{-126} -723 q^{-127} -1264 q^{-128} -474 q^{-129} -299 q^{-130} +163 q^{-131} +538 q^{-132} +222 q^{-133} -39 q^{-134} -183 q^{-135} +6 q^{-136} -80 q^{-137} -39 q^{-138} +84 q^{-139} +29 q^{-140} -2 q^{-141} -35 q^{-142} +23 q^{-143} -5 q^{-144} -18 q^{-145} +13 q^{-146} +2 q^{-147} + q^{-148} -6 q^{-149} +3 q^{-150} +2 q^{-151} -3 q^{-152} + q^{-153} </math> |
coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_7 = |
computer_talk =
computer_talk =
<table>
<table>
Line 54: Line 54:
<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, 89]]</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[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10],
<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, 89]]</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[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10],
X[20, 13, 1, 14], X[14, 5, 15, 6], X[6, 19, 7, 20],
X[20, 13, 1, 14], X[14, 5, 15, 6], X[6, 19, 7, 20],
X[18, 16, 19, 15], X[16, 11, 17, 12], X[10, 17, 11, 18]]</nowiki></pre></td></tr>
X[18, 16, 19, 15], X[16, 11, 17, 12], X[10, 17, 11, 18]]</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, 89]]</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, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10,
<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, 89]]</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, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10,
-8, 7, -5]</nowiki></pre></td></tr>
-8, 7, -5]</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, 89]]</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, 8, 14, 12, 2, 16, 20, 18, 10, 6]</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, 89]]</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[5, {-1, 2, -1, 2, 3, -2, -1, -4, -3, 2, -3, -4}]</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, 89]]</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>{5, 12}</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, 89]]</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, 8, 14, 12, 2, 16, 20, 18, 10, 6]</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>5</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, 89]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_89_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, 89]]&) /@ {
<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, 89]]</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[5, {-1, 2, -1, 2, 3, -2, -1, -4, -3, 2, -3, -4}]</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>{5, 12}</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, 89]]</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>5</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, 89]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:10_89_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, 89]]&) /@ {
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, 2, 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, 89]][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 8 24 2 3
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 2, 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, 89]][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 8 24 2 3
-33 + t - -- + -- + 24 t - 8 t + t
-33 + t - -- + -- + 24 t - 8 t + t
2 t
2 t
t</nowiki></pre></td></tr>
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, 89]][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 + z - 2 z + 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, 89]][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, 89]}</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, 89]], KnotSignature[Knot[10, 89]]}</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>{99, -2}</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 + z - 2 z + 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, 89]][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> -8 3 7 12 15 17 16 13 2
<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, 89]}</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, 89]], KnotSignature[Knot[10, 89]]}</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>{99, -2}</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, 89]][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> -8 3 7 12 15 17 16 13 2
-9 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q
-9 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q
7 6 5 4 3 2 q
7 6 5 4 3 2 q
q q q q q q</nowiki></pre></td></tr>
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, 89]}</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, 89]][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> -26 -24 2 -20 -18 4 2 2 2 2 4
<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, 89]}</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, 89]][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> -26 -24 2 -20 -18 4 2 2 2 2 4
-q - q + --- - q - q + --- - --- + --- - -- + -- - -- +
-q - q + --- - q - q + --- - --- + --- - -- + -- - -- +
22 16 14 12 8 6 4
22 16 14 12 8 6 4
Line 107: Line 183:
-- - q + 3 q - q
-- - q + 3 q - q
2
2
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, 89]][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> 4 6 8 2 2 4 2 6 2 4 2 4
<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, 89]][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> 4 6 8 2 2 4 2 6 2 4 2 4
1 - a + 2 a - a + 2 a z - 4 a z + 3 a z - z + 2 a z -
1 - a + 2 a - a + 2 a z - 4 a z + 3 a z - z + 2 a z -
4 4 2 6
4 4 2 6
3 a z + a z</nowiki></pre></td></tr>
3 a z + 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, 89]][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> 4 6 8 3 5 7 9 2 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, 89]][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> 4 6 8 3 5 7 9 2 2
1 - a - 2 a - a - 2 a z - 4 a z - a z + a z + 3 a z +
1 - a - 2 a - a - 2 a z - 4 a z - a z + a z + 3 a z +
Line 133: Line 219:
5 7 7 7 2 8 4 8 6 8 3 9 5 9
5 7 7 7 2 8 4 8 6 8 3 9 5 9
11 a z + 5 a z + 7 a z + 12 a z + 5 a z + 2 a z + 2 a z</nowiki></pre></td></tr>
11 a z + 5 a z + 7 a z + 12 a z + 5 a z + 2 a z + 2 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, 89]], Vassiliev[3][Knot[10, 89]]}</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>{1, -3}</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, 89]][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>6 8 1 2 1 5 2 7 5
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 89]], Vassiliev[3][Knot[10, 89]]}</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>{1, -3}</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, 89]][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>6 8 1 2 1 5 2 7 5
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4
3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4
Line 148: Line 244:
3 2 5 3
3 2 5 3
4 q t + q t</nowiki></pre></td></tr>
4 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, 89], 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> -23 3 2 8 21 9 40 71 7 116
<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, 89], 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> -23 3 2 8 21 9 40 71 7 116
-86 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- -
-86 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- -
22 21 20 19 18 17 16 15 14
22 21 20 19 18 17 16 15 14
Line 162: Line 263:
35 2 3 4 5 6 7
35 2 3 4 5 6 7
-- + 76 q + q - 36 q + 17 q + 4 q - 5 q + q
-- + 76 q + q - 36 q + 17 q + 4 q - 5 q + q
q</nowiki></pre></td></tr>
q</nowiki></code></td></tr>
</table> }}
</table> }}

Latest revision as of 18:01, 1 September 2005

10 88.gif

10_88

10 90.gif

10_90

10 89.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 89's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 89 at Knotilus!

[edit 10 89 Quick Notes]
[edit 10 89 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X12,8,13,7 X8394 X2,9,3,10 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,16,19,15 X16,11,17,12 X10,17,11,18
Gauss code 1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5
Dowker-Thistlethwaite code 4 8 14 12 2 16 20 18 10 6
Conway Notation [.21.210]


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

Length is 12, width is 5,

Braid index is 5

10 89 ML.gif 10 89 AP.gif
[{3, 10}, {6, 2}, {1, 3}, {5, 8}, {7, 9}, {8, 11}, {10, 6}, {12, 7}, {11, 4}, {2, 5}, {4, 12}, {9, 1}]

[edit Notes on presentations of 10 89]


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 89"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X12,8,13,7 X8394 X2,9,3,10 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,16,19,15 X16,11,17,12 X10,17,11,18
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5
In[6]:= DTCode[K]
Out[6]= 4 8 14 12 2 16 20 18 10 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [.21.210]
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}(5,\{-1,2,-1,2,3,-2,-1,-4,-3,2,-3,-4\}) }[/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]= { 5, 12, 5 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.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 89 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 10}, {6, 2}, {1, 3}, {5, 8}, {7, 9}, {8, 11}, {10, 6}, {12, 7}, {11, 4}, {2, 5}, {4, 12}, {9, 1}]
In[14]:= Draw[ap]
10 89 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][-2]
Hyperbolic Volume 15.5661
A-Polynomial See Data:10 89/A-polynomial

[edit Notes for 10 89's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 10 89's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-8 t^2+24 t-33+24 t^{-1} -8 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 99, -2 }
Jones polynomial [math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -15 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-5 z^4 a^8+3 z^2 a^8-a^8+5 z^7 a^7-7 z^5 a^7+4 z^3 a^7-z a^7+5 z^8 a^6-3 z^6 a^6-4 z^4 a^6+6 z^2 a^6-2 a^6+2 z^9 a^5+11 z^7 a^5-27 z^5 a^5+20 z^3 a^5-4 z a^5+12 z^8 a^4-15 z^6 a^4-2 z^4 a^4+6 z^2 a^4-a^4+2 z^9 a^3+15 z^7 a^3-35 z^5 a^3+19 z^3 a^3-2 z a^3+7 z^8 a^2-4 z^6 a^2-9 z^4 a^2+3 z^2 a^2+9 z^7 a-15 z^5 a+5 z^3 a+5 z^6-6 z^4+1+z^5 a^{-1} }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}+2 q^{12}-2 q^8+2 q^6-4 q^4+4 q^2- q^{-2} +3 q^{-4} - q^{-6} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-9 q^{118}+q^{116}+13 q^{114}-32 q^{112}+52 q^{110}-67 q^{108}+62 q^{106}-34 q^{104}-26 q^{102}+111 q^{100}-190 q^{98}+234 q^{96}-208 q^{94}+89 q^{92}+87 q^{90}-276 q^{88}+405 q^{86}-402 q^{84}+261 q^{82}-14 q^{80}-243 q^{78}+405 q^{76}-399 q^{74}+229 q^{72}+28 q^{70}-252 q^{68}+332 q^{66}-238 q^{64}+7 q^{62}+270 q^{60}-447 q^{58}+447 q^{56}-250 q^{54}-76 q^{52}+406 q^{50}-617 q^{48}+623 q^{46}-423 q^{44}+92 q^{42}+264 q^{40}-515 q^{38}+577 q^{36}-433 q^{34}+148 q^{32}+148 q^{30}-350 q^{28}+361 q^{26}-198 q^{24}-53 q^{22}+282 q^{20}-371 q^{18}+283 q^{16}-53 q^{14}-226 q^{12}+424 q^{10}-463 q^8+336 q^6-99 q^4-147 q^2+319-360 q^{-2} +293 q^{-4} -149 q^{-6} - q^{-8} +105 q^{-10} -152 q^{-12} +134 q^{-14} -84 q^{-16} +37 q^{-18} +4 q^{-20} -22 q^{-22} +25 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_89.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+2 q^{15}-4 q^{13}+5 q^{11}-3 q^9+2 q^7+q^5-3 q^3+4 q-4 q^{-1} +4 q^{-3} - q^{-5} }[/math]
2 [math]\displaystyle{ q^{48}-2 q^{46}+7 q^{42}-11 q^{40}-4 q^{38}+28 q^{36}-22 q^{34}-24 q^{32}+52 q^{30}-13 q^{28}-46 q^{26}+45 q^{24}+10 q^{22}-41 q^{20}+10 q^{18}+26 q^{16}-11 q^{14}-28 q^{12}+28 q^{10}+23 q^8-51 q^6+13 q^4+46 q^2-45-9 q^{-2} +41 q^{-4} -18 q^{-6} -15 q^{-8} +16 q^{-10} -4 q^{-14} + q^{-16} }[/math]
3 [math]\displaystyle{ -q^{93}+2 q^{91}-3 q^{87}-q^{85}+10 q^{83}+2 q^{81}-26 q^{79}-7 q^{77}+54 q^{75}+29 q^{73}-93 q^{71}-85 q^{69}+132 q^{67}+173 q^{65}-141 q^{63}-283 q^{61}+98 q^{59}+399 q^{57}-15 q^{55}-463 q^{53}-110 q^{51}+468 q^{49}+232 q^{47}-410 q^{45}-322 q^{43}+300 q^{41}+364 q^{39}-162 q^{37}-366 q^{35}+23 q^{33}+335 q^{31}+108 q^{29}-278 q^{27}-235 q^{25}+212 q^{23}+342 q^{21}-125 q^{19}-431 q^{17}+22 q^{15}+481 q^{13}+99 q^{11}-472 q^9-220 q^7+412 q^5+305 q^3-296 q-339 q^{-1} +161 q^{-3} +316 q^{-5} -44 q^{-7} -245 q^{-9} -27 q^{-11} +153 q^{-13} +55 q^{-15} -76 q^{-17} -52 q^{-19} +33 q^{-21} +26 q^{-23} -5 q^{-25} -12 q^{-27} +4 q^{-31} - q^{-33} }[/math]
4 [math]\displaystyle{ q^{152}-2 q^{150}+3 q^{146}-3 q^{144}+2 q^{142}-8 q^{140}+6 q^{138}+23 q^{136}-16 q^{134}-22 q^{132}-49 q^{130}+41 q^{128}+151 q^{126}+15 q^{124}-146 q^{122}-324 q^{120}-11 q^{118}+554 q^{116}+467 q^{114}-146 q^{112}-1114 q^{110}-782 q^{108}+849 q^{106}+1711 q^{104}+901 q^{102}-1772 q^{100}-2651 q^{98}-221 q^{96}+2762 q^{94}+3300 q^{92}-765 q^{90}-4226 q^{88}-2817 q^{86}+1903 q^{84}+5239 q^{82}+1857 q^{80}-3629 q^{78}-4881 q^{76}-610 q^{74}+4849 q^{72}+3947 q^{70}-1251 q^{68}-4730 q^{66}-2716 q^{64}+2638 q^{62}+4147 q^{60}+1049 q^{58}-3021 q^{56}-3414 q^{54}+250 q^{52}+3176 q^{50}+2520 q^{48}-1089 q^{46}-3367 q^{44}-1789 q^{42}+2012 q^{40}+3655 q^{38}+828 q^{36}-3093 q^{34}-3776 q^{32}+481 q^{30}+4413 q^{28}+3046 q^{26}-1959 q^{24}-5266 q^{22}-1814 q^{20}+3744 q^{18}+4824 q^{16}+447 q^{14}-4886 q^{12}-3842 q^{10}+1286 q^8+4600 q^6+2754 q^4-2443 q^2-3854-1200 q^{-2} +2355 q^{-4} +3049 q^{-6} +33 q^{-8} -2001 q^{-10} -1792 q^{-12} +214 q^{-14} +1649 q^{-16} +787 q^{-18} -325 q^{-20} -942 q^{-22} -424 q^{-24} +407 q^{-26} +404 q^{-28} +159 q^{-30} -217 q^{-32} -218 q^{-34} +20 q^{-36} +70 q^{-38} +80 q^{-40} -12 q^{-42} -41 q^{-44} -6 q^{-46} + q^{-48} +12 q^{-50} -4 q^{-54} + q^{-56} }[/math]
5 [math]\displaystyle{ -q^{225}+2 q^{223}-3 q^{219}+3 q^{217}+2 q^{215}-4 q^{213}-3 q^{209}-10 q^{207}+16 q^{205}+35 q^{203}+4 q^{201}-43 q^{199}-87 q^{197}-65 q^{195}+86 q^{193}+270 q^{191}+241 q^{189}-123 q^{187}-596 q^{185}-715 q^{183}-71 q^{181}+1086 q^{179}+1754 q^{177}+853 q^{175}-1496 q^{173}-3471 q^{171}-2798 q^{169}+1107 q^{167}+5650 q^{165}+6458 q^{163}+1096 q^{161}-7421 q^{159}-11722 q^{157}-6171 q^{155}+7094 q^{153}+17634 q^{151}+14573 q^{149}-2987 q^{147}-22168 q^{145}-25315 q^{143}-5950 q^{141}+22718 q^{139}+36151 q^{137}+19352 q^{135}-17569 q^{133}-44044 q^{131}-34713 q^{129}+6358 q^{127}+46080 q^{125}+48917 q^{123}+9119 q^{121}-41311 q^{119}-58405 q^{117}-25564 q^{115}+30330 q^{113}+61169 q^{111}+39626 q^{109}-15812 q^{107}-57053 q^{105}-48626 q^{103}+919 q^{101}+47596 q^{99}+51649 q^{97}+11772 q^{95}-35438 q^{93}-49490 q^{91}-20768 q^{89}+23129 q^{87}+43925 q^{85}+26084 q^{83}-12255 q^{81}-37181 q^{79}-28848 q^{77}+3427 q^{75}+30902 q^{73}+30564 q^{71}+3847 q^{69}-25727 q^{67}-32663 q^{65}-10767 q^{63}+21497 q^{61}+35822 q^{59}+18379 q^{57}-17048 q^{55}-39615 q^{53}-27604 q^{51}+11020 q^{49}+42957 q^{47}+38068 q^{45}-2269 q^{43}-43836 q^{41}-48586 q^{39}-9591 q^{37}+40532 q^{35}+57018 q^{33}+23457 q^{31}-31987 q^{29}-60809 q^{27}-37193 q^{25}+18594 q^{23}+58265 q^{21}+47857 q^{19}-2576 q^{17}-48963 q^{15}-52714 q^{13}-12990 q^{11}+34586 q^9+50561 q^7+24670 q^5-18247 q^3-42087 q-30233 q^{-1} +3516 q^{-3} +29774 q^{-5} +29448 q^{-7} +6868 q^{-9} -17054 q^{-11} -23886 q^{-13} -11740 q^{-15} +6613 q^{-17} +16218 q^{-19} +11977 q^{-21} +54 q^{-23} -9063 q^{-25} -9378 q^{-27} -2951 q^{-29} +3795 q^{-31} +5934 q^{-33} +3362 q^{-35} -813 q^{-37} -3146 q^{-39} -2475 q^{-41} -336 q^{-43} +1258 q^{-45} +1433 q^{-47} +592 q^{-49} -389 q^{-51} -698 q^{-53} -372 q^{-55} +38 q^{-57} +252 q^{-59} +202 q^{-61} +41 q^{-63} -88 q^{-65} -84 q^{-67} -16 q^{-69} +20 q^{-71} +21 q^{-73} +10 q^{-75} - q^{-77} -12 q^{-79} +4 q^{-83} - q^{-85} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}+2 q^{12}-2 q^8+2 q^6-4 q^4+4 q^2- q^{-2} +3 q^{-4} - q^{-6} }[/math]
2,0 [math]\displaystyle{ q^{66}+q^{64}-q^{62}-3 q^{60}+q^{58}+6 q^{56}-3 q^{54}-12 q^{52}+18 q^{48}+3 q^{46}-22 q^{44}-2 q^{42}+27 q^{40}+2 q^{38}-27 q^{36}+21 q^{32}-4 q^{30}-21 q^{28}+8 q^{26}+9 q^{24}-11 q^{22}+8 q^{20}+9 q^{18}-11 q^{16}-3 q^{14}+23 q^{12}-q^{10}-28 q^8+4 q^6+28 q^4-7 q^2-27+11 q^{-2} +19 q^{-4} -6 q^{-6} -11 q^{-8} +2 q^{-10} +9 q^{-12} -2 q^{-14} -3 q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+5 q^{48}-10 q^{46}+q^{44}+17 q^{42}-25 q^{40}-q^{38}+34 q^{36}-35 q^{34}-3 q^{32}+39 q^{30}-28 q^{28}-7 q^{26}+26 q^{24}-7 q^{22}-11 q^{20}+16 q^{16}-5 q^{14}-25 q^{12}+30 q^{10}+7 q^8-40 q^6+30 q^4+11 q^2-34+22 q^{-2} +7 q^{-4} -17 q^{-6} +10 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+2 q^{29}-q^{27}+2 q^{25}-q^{23}+4 q^{21}-3 q^{19}+3 q^{17}-q^{15}-q^{11}-q^9+q^7-3 q^5+4 q^3-q+3 q^{-1} -2 q^{-3} +3 q^{-5} - q^{-7} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+9 q^{48}-16 q^{46}+23 q^{44}-33 q^{42}+41 q^{40}-45 q^{38}+46 q^{36}-39 q^{34}+27 q^{32}-7 q^{30}-14 q^{28}+39 q^{26}-60 q^{24}+79 q^{22}-89 q^{20}+90 q^{18}-84 q^{16}+67 q^{14}-47 q^{12}+22 q^{10}+q^8-22 q^6+38 q^4-45 q^2+48-44 q^{-2} +39 q^{-4} -27 q^{-6} +18 q^{-8} -10 q^{-10} +4 q^{-12} - q^{-14} }[/math]
1,0 [math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+7 q^{78}-13 q^{74}-10 q^{72}+12 q^{70}+25 q^{68}-q^{66}-36 q^{64}-21 q^{62}+30 q^{60}+42 q^{58}-11 q^{56}-51 q^{54}-13 q^{52}+45 q^{50}+31 q^{48}-28 q^{46}-38 q^{44}+14 q^{42}+38 q^{40}-q^{38}-35 q^{36}-6 q^{34}+31 q^{32}+12 q^{30}-28 q^{28}-20 q^{26}+25 q^{24}+27 q^{22}-20 q^{20}-36 q^{18}+12 q^{16}+46 q^{14}+7 q^{12}-47 q^{10}-29 q^8+36 q^6+45 q^4-13 q^2-46-11 q^{-2} +33 q^{-4} +25 q^{-6} -15 q^{-8} -22 q^{-10} +14 q^{-14} +6 q^{-16} -4 q^{-18} -4 q^{-20} + q^{-24} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-9 q^{118}+q^{116}+13 q^{114}-32 q^{112}+52 q^{110}-67 q^{108}+62 q^{106}-34 q^{104}-26 q^{102}+111 q^{100}-190 q^{98}+234 q^{96}-208 q^{94}+89 q^{92}+87 q^{90}-276 q^{88}+405 q^{86}-402 q^{84}+261 q^{82}-14 q^{80}-243 q^{78}+405 q^{76}-399 q^{74}+229 q^{72}+28 q^{70}-252 q^{68}+332 q^{66}-238 q^{64}+7 q^{62}+270 q^{60}-447 q^{58}+447 q^{56}-250 q^{54}-76 q^{52}+406 q^{50}-617 q^{48}+623 q^{46}-423 q^{44}+92 q^{42}+264 q^{40}-515 q^{38}+577 q^{36}-433 q^{34}+148 q^{32}+148 q^{30}-350 q^{28}+361 q^{26}-198 q^{24}-53 q^{22}+282 q^{20}-371 q^{18}+283 q^{16}-53 q^{14}-226 q^{12}+424 q^{10}-463 q^8+336 q^6-99 q^4-147 q^2+319-360 q^{-2} +293 q^{-4} -149 q^{-6} - q^{-8} +105 q^{-10} -152 q^{-12} +134 q^{-14} -84 q^{-16} +37 q^{-18} +4 q^{-20} -22 q^{-22} +25 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/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 89"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-8 t^2+24 t-33+24 t^{-1} -8 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6-2 z^4+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]= { 99, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -15 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-5 z^4 a^8+3 z^2 a^8-a^8+5 z^7 a^7-7 z^5 a^7+4 z^3 a^7-z a^7+5 z^8 a^6-3 z^6 a^6-4 z^4 a^6+6 z^2 a^6-2 a^6+2 z^9 a^5+11 z^7 a^5-27 z^5 a^5+20 z^3 a^5-4 z a^5+12 z^8 a^4-15 z^6 a^4-2 z^4 a^4+6 z^2 a^4-a^4+2 z^9 a^3+15 z^7 a^3-35 z^5 a^3+19 z^3 a^3-2 z a^3+7 z^8 a^2-4 z^6 a^2-9 z^4 a^2+3 z^2 a^2+9 z^7 a-15 z^5 a+5 z^3 a+5 z^6-6 z^4+1+z^5 a^{-1} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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 89"];
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{ t^3-8 t^2+24 t-33+24 t^{-1} -8 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -15 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/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]= {}
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: (1, -3)
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{ 4 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{302}{3} }[/math] [math]\displaystyle{ \frac{58}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -400 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1208}{3} }[/math] [math]\displaystyle{ \frac{232}{3} }[/math] [math]\displaystyle{ \frac{49471}{30} }[/math] [math]\displaystyle{ -\frac{3062}{15} }[/math] [math]\displaystyle{ \frac{43862}{45} }[/math] [math]\displaystyle{ \frac{641}{18} }[/math] [math]\displaystyle{ \frac{3871}{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]-2 is the signature of 10 89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10123χ
5          1-1
3         4 4
1        51 -4
-1       84  4
-3      96   -3
-5     87    1
-7    79     2
-9   58      -3
-11  27       5
-13 15        -4
-15 2         2
-171          -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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^7-5 q^6+4 q^5+17 q^4-36 q^3+q^2+76 q-86-35 q^{-1} +167 q^{-2} -119 q^{-3} -99 q^{-4} +241 q^{-5} -114 q^{-6} -155 q^{-7} +258 q^{-8} -77 q^{-9} -171 q^{-10} +207 q^{-11} -26 q^{-12} -136 q^{-13} +116 q^{-14} +7 q^{-15} -71 q^{-16} +40 q^{-17} +9 q^{-18} -21 q^{-19} +8 q^{-20} +2 q^{-21} -3 q^{-22} + q^{-23} }[/math]
3 [math]\displaystyle{ -q^{15}+5 q^{14}-4 q^{13}-12 q^{12}+6 q^{11}+36 q^{10}+3 q^9-97 q^8-18 q^7+167 q^6+101 q^5-277 q^4-236 q^3+368 q^2+461 q-432-736 q^{-1} +411 q^{-2} +1062 q^{-3} -325 q^{-4} -1368 q^{-5} +159 q^{-6} +1633 q^{-7} +57 q^{-8} -1827 q^{-9} -294 q^{-10} +1939 q^{-11} +524 q^{-12} -1957 q^{-13} -741 q^{-14} +1896 q^{-15} +910 q^{-16} -1730 q^{-17} -1053 q^{-18} +1507 q^{-19} +1114 q^{-20} -1204 q^{-21} -1117 q^{-22} +885 q^{-23} +1026 q^{-24} -562 q^{-25} -881 q^{-26} +307 q^{-27} +673 q^{-28} -114 q^{-29} -467 q^{-30} +6 q^{-31} +292 q^{-32} +28 q^{-33} -153 q^{-34} -35 q^{-35} +75 q^{-36} +20 q^{-37} -31 q^{-38} -10 q^{-39} +14 q^{-40} + q^{-41} -3 q^{-42} -2 q^{-43} +3 q^{-44} - q^{-45} }[/math]
4 [math]\displaystyle{ q^{26}-5 q^{25}+4 q^{24}+12 q^{23}-11 q^{22}-6 q^{21}-40 q^{20}+33 q^{19}+104 q^{18}-21 q^{17}-56 q^{16}-278 q^{15}+34 q^{14}+480 q^{13}+224 q^{12}-53 q^{11}-1109 q^{10}-484 q^9+1097 q^8+1336 q^7+809 q^6-2544 q^5-2490 q^4+888 q^3+3370 q^2+3825 q-3238-6045 q^{-1} -1766 q^{-2} +4781 q^{-3} +9022 q^{-4} -1392 q^{-5} -9359 q^{-6} -6894 q^{-7} +3737 q^{-8} +14355 q^{-9} +2985 q^{-10} -10439 q^{-11} -12452 q^{-12} +285 q^{-13} +17662 q^{-14} +7990 q^{-15} -9072 q^{-16} -16384 q^{-17} -3972 q^{-18} +18345 q^{-19} +11911 q^{-20} -6245 q^{-21} -18027 q^{-22} -7773 q^{-23} +16767 q^{-24} +14189 q^{-25} -2636 q^{-26} -17371 q^{-27} -10699 q^{-28} +13103 q^{-29} +14582 q^{-30} +1434 q^{-31} -14273 q^{-32} -12208 q^{-33} +7749 q^{-34} +12568 q^{-35} +4913 q^{-36} -9075 q^{-37} -11306 q^{-38} +2290 q^{-39} +8297 q^{-40} +6165 q^{-41} -3589 q^{-42} -7924 q^{-43} -1046 q^{-44} +3577 q^{-45} +4756 q^{-46} -128 q^{-47} -3859 q^{-48} -1584 q^{-49} +594 q^{-50} +2326 q^{-51} +751 q^{-52} -1186 q^{-53} -774 q^{-54} -268 q^{-55} +695 q^{-56} +419 q^{-57} -218 q^{-58} -161 q^{-59} -181 q^{-60} +130 q^{-61} +106 q^{-62} -40 q^{-63} -45 q^{-65} +20 q^{-66} +16 q^{-67} -13 q^{-68} +6 q^{-69} -6 q^{-70} +3 q^{-71} +2 q^{-72} -3 q^{-73} + q^{-74} }[/math]
5 [math]\displaystyle{ -q^{40}+5 q^{39}-4 q^{38}-12 q^{37}+11 q^{36}+11 q^{35}+10 q^{34}+4 q^{33}-40 q^{32}-80 q^{31}+7 q^{30}+140 q^{29}+171 q^{28}+54 q^{27}-254 q^{26}-490 q^{25}-319 q^{24}+449 q^{23}+1152 q^{22}+895 q^{21}-429 q^{20}-2084 q^{19}-2458 q^{18}-222 q^{17}+3485 q^{16}+5070 q^{15}+2143 q^{14}-4223 q^{13}-9204 q^{12}-6649 q^{11}+3800 q^{10}+14187 q^9+14066 q^8+18 q^7-18809 q^6-25002 q^5-8346 q^4+21019 q^3+37988 q^2+22598 q-18483-51260 q^{-1} -42095 q^{-2} +9165 q^{-3} +61828 q^{-4} +65515 q^{-5} +7408 q^{-6} -67235 q^{-7} -89671 q^{-8} -30559 q^{-9} +65579 q^{-10} +111902 q^{-11} +57841 q^{-12} -56827 q^{-13} -129342 q^{-14} -86346 q^{-15} +41963 q^{-16} +140724 q^{-17} +113285 q^{-18} -23266 q^{-19} -145828 q^{-20} -136469 q^{-21} +2968 q^{-22} +145474 q^{-23} +154852 q^{-24} +17071 q^{-25} -140939 q^{-26} -168406 q^{-27} -35656 q^{-28} +133463 q^{-29} +177419 q^{-30} +52498 q^{-31} -123496 q^{-32} -182731 q^{-33} -67920 q^{-34} +111567 q^{-35} +184355 q^{-36} +82072 q^{-37} -96779 q^{-38} -182393 q^{-39} -95395 q^{-40} +79292 q^{-41} +176022 q^{-42} +106998 q^{-43} -58440 q^{-44} -164552 q^{-45} -116191 q^{-46} +35395 q^{-47} +147300 q^{-48} +121050 q^{-49} -11230 q^{-50} -124675 q^{-51} -120244 q^{-52} -11282 q^{-53} +97755 q^{-54} +112623 q^{-55} +30011 q^{-56} -69237 q^{-57} -98701 q^{-58} -42121 q^{-59} +41861 q^{-60} +79782 q^{-61} +47105 q^{-62} -18807 q^{-63} -58903 q^{-64} -44958 q^{-65} +2139 q^{-66} +38711 q^{-67} +37774 q^{-68} +7668 q^{-69} -21982 q^{-70} -28159 q^{-71} -11294 q^{-72} +10043 q^{-73} +18409 q^{-74} +10815 q^{-75} -2801 q^{-76} -10599 q^{-77} -8233 q^{-78} -497 q^{-79} +5144 q^{-80} +5264 q^{-81} +1500 q^{-82} -2082 q^{-83} -2871 q^{-84} -1305 q^{-85} +601 q^{-86} +1359 q^{-87} +827 q^{-88} -107 q^{-89} -522 q^{-90} -404 q^{-91} -67 q^{-92} +202 q^{-93} +183 q^{-94} +12 q^{-95} -49 q^{-96} -40 q^{-97} -38 q^{-98} +18 q^{-99} +32 q^{-100} -10 q^{-101} -5 q^{-102} +7 q^{-103} -7 q^{-104} - q^{-105} +6 q^{-106} -3 q^{-107} -2 q^{-108} +3 q^{-109} - q^{-110} }[/math]
6 [math]\displaystyle{ q^{57}-5 q^{56}+4 q^{55}+12 q^{54}-11 q^{53}-11 q^{52}-15 q^{51}+26 q^{50}+3 q^{49}+16 q^{48}+94 q^{47}-76 q^{46}-130 q^{45}-166 q^{44}+70 q^{43}+159 q^{42}+285 q^{41}+592 q^{40}-162 q^{39}-786 q^{38}-1315 q^{37}-492 q^{36}+358 q^{35}+1823 q^{34}+3589 q^{33}+1467 q^{32}-1862 q^{31}-6117 q^{30}-5872 q^{29}-3275 q^{28}+4221 q^{27}+14196 q^{26}+13763 q^{25}+4612 q^{24}-13711 q^{23}-25068 q^{22}-27520 q^{21}-7712 q^{20}+29398 q^{19}+52006 q^{18}+47384 q^{17}+2707 q^{16}-50689 q^{15}-95331 q^{14}-79513 q^{13}+4575 q^{12}+101533 q^{11}+157541 q^{10}+110029 q^9-14670 q^8-179840 q^7-249195 q^6-151498 q^5+65923 q^4+290494 q^3+350261 q^2+197545 q-153211-449328 q^{-1} -478721 q^{-2} -188657 q^{-3} +287272 q^{-4} +629512 q^{-5} +615513 q^{-6} +130178 q^{-7} -493832 q^{-8} -853697 q^{-9} -674173 q^{-10} +85 q^{-11} +741503 q^{-12} +1085958 q^{-13} +658679 q^{-14} -241986 q^{-15} -1056845 q^{-16} -1211702 q^{-17} -533358 q^{-18} +559209 q^{-19} +1383990 q^{-20} +1234571 q^{-21} +245365 q^{-22} -976920 q^{-23} -1580502 q^{-24} -1106663 q^{-25} +159076 q^{-26} +1414442 q^{-27} +1649785 q^{-28} +766439 q^{-29} -695670 q^{-30} -1702203 q^{-31} -1533185 q^{-32} -274757 q^{-33} +1256130 q^{-34} +1843110 q^{-35} +1164970 q^{-36} -369544 q^{-37} -1649075 q^{-38} -1770803 q^{-39} -622190 q^{-40} +1033995 q^{-41} +1879600 q^{-42} +1420734 q^{-43} -80945 q^{-44} -1516518 q^{-45} -1879548 q^{-46} -885422 q^{-47} +795906 q^{-48} +1828892 q^{-49} +1591422 q^{-50} +192914 q^{-51} -1322990 q^{-52} -1907246 q^{-53} -1120225 q^{-54} +502100 q^{-55} +1683461 q^{-56} +1704725 q^{-57} +504094 q^{-58} -1014325 q^{-59} -1821200 q^{-60} -1332607 q^{-61} +107102 q^{-62} +1374897 q^{-63} +1700374 q^{-64} +832731 q^{-65} -553067 q^{-66} -1536799 q^{-67} -1430264 q^{-68} -339098 q^{-69} +873701 q^{-70} +1474204 q^{-71} +1049771 q^{-72} -20239 q^{-73} -1031393 q^{-74} -1287386 q^{-75} -669820 q^{-76} +290363 q^{-77} +1009106 q^{-78} +1008702 q^{-79} +385642 q^{-80} -441190 q^{-81} -893256 q^{-82} -724847 q^{-83} -154155 q^{-84} +461572 q^{-85} +708363 q^{-86} +503434 q^{-87} +1901 q^{-88} -419792 q^{-89} -518360 q^{-90} -309994 q^{-91} +61894 q^{-92} +330955 q^{-93} +368560 q^{-94} +169279 q^{-95} -85271 q^{-96} -237747 q^{-97} -234071 q^{-98} -89280 q^{-99} +74382 q^{-100} +167164 q^{-101} +135206 q^{-102} +38741 q^{-103} -54925 q^{-104} -101709 q^{-105} -76580 q^{-106} -16054 q^{-107} +41676 q^{-108} +55570 q^{-109} +37906 q^{-110} +5490 q^{-111} -23995 q^{-112} -30296 q^{-113} -18510 q^{-114} +1825 q^{-115} +11931 q^{-116} +13858 q^{-117} +8319 q^{-118} -1409 q^{-119} -6551 q^{-120} -6486 q^{-121} -1947 q^{-122} +633 q^{-123} +2561 q^{-124} +2778 q^{-125} +757 q^{-126} -723 q^{-127} -1264 q^{-128} -474 q^{-129} -299 q^{-130} +163 q^{-131} +538 q^{-132} +222 q^{-133} -39 q^{-134} -183 q^{-135} +6 q^{-136} -80 q^{-137} -39 q^{-138} +84 q^{-139} +29 q^{-140} -2 q^{-141} -35 q^{-142} +23 q^{-143} -5 q^{-144} -18 q^{-145} +13 q^{-146} +2 q^{-147} + q^{-148} -6 q^{-149} +3 q^{-150} +2 q^{-151} -3 q^{-152} + q^{-153} }[/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_88

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10_90

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