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{{Rolfsen Knot Page|
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n = 10 |
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k = 78 |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,-3,9,-10,2,-7,8,-4,5,-9,3,-5,4,-6,7,-8,6/goTop.html |
<span id="top"></span>
braid_table = <table cellspacing=0 cellpadding=0 border=0>
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{{Knot Navigation Links|ext=gif}}

{{Rolfsen Knot Page Header|n=10|k=78|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,-3,9,-10,2,-7,8,-4,5,-9,3,-5,4,-6,7,-8,6/goTop.html}}

<br style="clear:both" />

{{:{{PAGENAME}} Further Notes and Views}}

{{Knot Presentations}}

<center><table border=1 cellpadding=10><tr align=center valign=top>
<td>
[[Braid Representatives|Minimum Braid Representative]]:
<table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]]</td></tr>
</table>
</table> |
braid_crossings = 12 |

braid_width = 5 |
[[Invariants from Braid Theory|Length]] is 12, width is 5.
braid_index = 5 |

same_alexander = [[K11n98]], [[K11n105]], |
[[Invariants from Braid Theory|Braid index]] is 5.
same_jones = |
</td>
khovanov_table = <table border=1>
<td>
[[Lightly Documented Features|A Morse Link Presentation]]:

[[Image:{{PAGENAME}}_ML.gif]]
</td>
</tr></table></center>

{{3D Invariants}}
{{4D Invariants}}
{{Polynomial Invariants}}

=== "Similar" Knots (within the Atlas) ===

Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]:
{[[K11n98]], [[K11n105]], ...}

Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>):
{...}

{{Vassiliev Invariants}}

{{Khovanov Homology|table=<table border=1>
<tr align=center>
<tr align=center>
<td width=13.3333%><table cellpadding=0 cellspacing=0>
<td width=13.3333%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
</table></td>
<td width=6.66667%>-8</td ><td width=6.66667%>-7</td ><td width=6.66667%>-6</td ><td width=6.66667%>-5</td ><td width=6.66667%>-4</td ><td width=6.66667%>-3</td ><td width=6.66667%>-2</td ><td width=6.66667%>-1</td ><td width=6.66667%>0</td ><td width=6.66667%>1</td ><td width=6.66667%>2</td ><td width=13.3333%>&chi;</td></tr>
<td width=6.66667%>-8</td ><td width=6.66667%>-7</td ><td width=6.66667%>-6</td ><td width=6.66667%>-5</td ><td width=6.66667%>-4</td ><td width=6.66667%>-3</td ><td width=6.66667%>-2</td ><td width=6.66667%>-1</td ><td width=6.66667%>0</td ><td width=6.66667%>1</td ><td width=6.66667%>2</td ><td width=13.3333%>&chi;</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>&nbsp;</td><td>-2</td></tr>
Line 74: Line 38:
<tr align=center><td>-19</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-19</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table>}}
</table> |
coloured_jones_2 = <math>q^2-3 q+11 q^{-1} -13 q^{-2} -10 q^{-3} +37 q^{-4} -21 q^{-5} -37 q^{-6} +69 q^{-7} -16 q^{-8} -73 q^{-9} +91 q^{-10} - q^{-11} -100 q^{-12} +93 q^{-13} +16 q^{-14} -104 q^{-15} +75 q^{-16} +24 q^{-17} -79 q^{-18} +45 q^{-19} +18 q^{-20} -41 q^{-21} +20 q^{-22} +7 q^{-23} -14 q^{-24} +7 q^{-25} + q^{-26} -3 q^{-27} + q^{-28} </math> |

coloured_jones_3 = <math>q^6-3 q^5+5 q^3+6 q^2-13 q-17+20 q^{-1} +39 q^{-2} -21 q^{-3} -71 q^{-4} +6 q^{-5} +115 q^{-6} +21 q^{-7} -149 q^{-8} -75 q^{-9} +182 q^{-10} +139 q^{-11} -190 q^{-12} -221 q^{-13} +191 q^{-14} +288 q^{-15} -153 q^{-16} -371 q^{-17} +126 q^{-18} +416 q^{-19} -62 q^{-20} -474 q^{-21} +19 q^{-22} +486 q^{-23} +50 q^{-24} -504 q^{-25} -92 q^{-26} +478 q^{-27} +141 q^{-28} -441 q^{-29} -166 q^{-30} +378 q^{-31} +174 q^{-32} -299 q^{-33} -170 q^{-34} +232 q^{-35} +131 q^{-36} -150 q^{-37} -107 q^{-38} +105 q^{-39} +64 q^{-40} -60 q^{-41} -40 q^{-42} +40 q^{-43} +15 q^{-44} -21 q^{-45} -7 q^{-46} +14 q^{-47} + q^{-48} -9 q^{-49} +2 q^{-50} +3 q^{-51} + q^{-52} -3 q^{-53} + q^{-54} </math> |
{{Display Coloured Jones|J2=<math>q^2-3 q+11 q^{-1} -13 q^{-2} -10 q^{-3} +37 q^{-4} -21 q^{-5} -37 q^{-6} +69 q^{-7} -16 q^{-8} -73 q^{-9} +91 q^{-10} - q^{-11} -100 q^{-12} +93 q^{-13} +16 q^{-14} -104 q^{-15} +75 q^{-16} +24 q^{-17} -79 q^{-18} +45 q^{-19} +18 q^{-20} -41 q^{-21} +20 q^{-22} +7 q^{-23} -14 q^{-24} +7 q^{-25} + q^{-26} -3 q^{-27} + q^{-28} </math>|J3=<math>q^6-3 q^5+5 q^3+6 q^2-13 q-17+20 q^{-1} +39 q^{-2} -21 q^{-3} -71 q^{-4} +6 q^{-5} +115 q^{-6} +21 q^{-7} -149 q^{-8} -75 q^{-9} +182 q^{-10} +139 q^{-11} -190 q^{-12} -221 q^{-13} +191 q^{-14} +288 q^{-15} -153 q^{-16} -371 q^{-17} +126 q^{-18} +416 q^{-19} -62 q^{-20} -474 q^{-21} +19 q^{-22} +486 q^{-23} +50 q^{-24} -504 q^{-25} -92 q^{-26} +478 q^{-27} +141 q^{-28} -441 q^{-29} -166 q^{-30} +378 q^{-31} +174 q^{-32} -299 q^{-33} -170 q^{-34} +232 q^{-35} +131 q^{-36} -150 q^{-37} -107 q^{-38} +105 q^{-39} +64 q^{-40} -60 q^{-41} -40 q^{-42} +40 q^{-43} +15 q^{-44} -21 q^{-45} -7 q^{-46} +14 q^{-47} + q^{-48} -9 q^{-49} +2 q^{-50} +3 q^{-51} + q^{-52} -3 q^{-53} + q^{-54} </math>|J4=<math>q^{12}-3 q^{11}+5 q^9+6 q^7-20 q^6-10 q^5+20 q^4+15 q^3+48 q^2-63 q-73+5 q^{-1} +42 q^{-2} +207 q^{-3} -55 q^{-4} -186 q^{-5} -151 q^{-6} -56 q^{-7} +484 q^{-8} +152 q^{-9} -167 q^{-10} -426 q^{-11} -467 q^{-12} +642 q^{-13} +527 q^{-14} +215 q^{-15} -559 q^{-16} -1150 q^{-17} +425 q^{-18} +786 q^{-19} +925 q^{-20} -296 q^{-21} -1796 q^{-22} -157 q^{-23} +679 q^{-24} +1685 q^{-25} +337 q^{-26} -2147 q^{-27} -862 q^{-28} +227 q^{-29} +2250 q^{-30} +1114 q^{-31} -2165 q^{-32} -1487 q^{-33} -384 q^{-34} +2556 q^{-35} +1832 q^{-36} -1935 q^{-37} -1935 q^{-38} -1003 q^{-39} +2574 q^{-40} +2368 q^{-41} -1481 q^{-42} -2109 q^{-43} -1539 q^{-44} +2234 q^{-45} +2579 q^{-46} -846 q^{-47} -1871 q^{-48} -1828 q^{-49} +1542 q^{-50} +2311 q^{-51} -225 q^{-52} -1249 q^{-53} -1692 q^{-54} +768 q^{-55} +1619 q^{-56} +114 q^{-57} -543 q^{-58} -1186 q^{-59} +237 q^{-60} +855 q^{-61} +137 q^{-62} -88 q^{-63} -622 q^{-64} +34 q^{-65} +335 q^{-66} +33 q^{-67} +68 q^{-68} -243 q^{-69} +3 q^{-70} +98 q^{-71} -30 q^{-72} +65 q^{-73} -71 q^{-74} +9 q^{-75} +23 q^{-76} -33 q^{-77} +29 q^{-78} -15 q^{-79} +8 q^{-80} +5 q^{-81} -15 q^{-82} +7 q^{-83} -2 q^{-84} +3 q^{-85} + q^{-86} -3 q^{-87} + q^{-88} </math>|J5=<math>q^{20}-3 q^{19}+5 q^{17}-q^{14}-13 q^{13}-10 q^{12}+20 q^{11}+24 q^{10}+15 q^9-3 q^8-56 q^7-73 q^6-6 q^5+95 q^4+136 q^3+88 q^2-84 q-266-247 q^{-1} +10 q^{-2} +354 q^{-3} +498 q^{-4} +234 q^{-5} -339 q^{-6} -792 q^{-7} -670 q^{-8} +96 q^{-9} +1021 q^{-10} +1246 q^{-11} +453 q^{-12} -947 q^{-13} -1890 q^{-14} -1363 q^{-15} +526 q^{-16} +2330 q^{-17} +2460 q^{-18} +481 q^{-19} -2372 q^{-20} -3676 q^{-21} -1889 q^{-22} +1841 q^{-23} +4588 q^{-24} +3713 q^{-25} -654 q^{-26} -5126 q^{-27} -5569 q^{-28} -1114 q^{-29} +4963 q^{-30} +7379 q^{-31} +3298 q^{-32} -4271 q^{-33} -8692 q^{-34} -5714 q^{-35} +2866 q^{-36} +9720 q^{-37} +8094 q^{-38} -1232 q^{-39} -10049 q^{-40} -10306 q^{-41} -869 q^{-42} +10197 q^{-43} +12248 q^{-44} +2792 q^{-45} -9783 q^{-46} -13878 q^{-47} -4919 q^{-48} +9370 q^{-49} +15252 q^{-50} +6696 q^{-51} -8586 q^{-52} -16326 q^{-53} -8590 q^{-54} +7858 q^{-55} +17149 q^{-56} +10152 q^{-57} -6755 q^{-58} -17634 q^{-59} -11764 q^{-60} +5584 q^{-61} +17702 q^{-62} +13016 q^{-63} -4006 q^{-64} -17210 q^{-65} -14083 q^{-66} +2285 q^{-67} +16123 q^{-68} +14575 q^{-69} -404 q^{-70} -14357 q^{-71} -14485 q^{-72} -1426 q^{-73} +12114 q^{-74} +13721 q^{-75} +2846 q^{-76} -9509 q^{-77} -12209 q^{-78} -3915 q^{-79} +6902 q^{-80} +10360 q^{-81} +4233 q^{-82} -4577 q^{-83} -8067 q^{-84} -4187 q^{-85} +2674 q^{-86} +6027 q^{-87} +3574 q^{-88} -1346 q^{-89} -4079 q^{-90} -2876 q^{-91} +489 q^{-92} +2662 q^{-93} +2044 q^{-94} -46 q^{-95} -1543 q^{-96} -1416 q^{-97} -129 q^{-98} +879 q^{-99} +848 q^{-100} +166 q^{-101} -413 q^{-102} -513 q^{-103} -150 q^{-104} +210 q^{-105} +260 q^{-106} +97 q^{-107} -72 q^{-108} -122 q^{-109} -70 q^{-110} +15 q^{-111} +64 q^{-112} +34 q^{-113} -8 q^{-114} -7 q^{-115} -17 q^{-116} -19 q^{-117} +11 q^{-118} +12 q^{-119} -5 q^{-120} +10 q^{-121} - q^{-122} -11 q^{-123} + q^{-124} +3 q^{-125} -2 q^{-126} +3 q^{-127} + q^{-128} -3 q^{-129} + q^{-130} </math>|J6=<math>q^{30}-3 q^{29}+5 q^{27}-7 q^{24}+6 q^{23}-13 q^{22}-10 q^{21}+29 q^{20}+15 q^{19}+15 q^{18}-27 q^{17}+4 q^{16}-67 q^{15}-73 q^{14}+59 q^{13}+86 q^{12}+135 q^{11}+15 q^{10}+72 q^9-230 q^8-365 q^7-121 q^6+67 q^5+422 q^4+376 q^3+666 q^2-130 q-844-955 q^{-1} -799 q^{-2} +100 q^{-3} +744 q^{-4} +2323 q^{-5} +1444 q^{-6} -63 q^{-7} -1674 q^{-8} -2939 q^{-9} -2571 q^{-10} -1234 q^{-11} +3406 q^{-12} +4632 q^{-13} +4209 q^{-14} +1127 q^{-15} -3451 q^{-16} -7082 q^{-17} -8074 q^{-18} -771 q^{-19} +5024 q^{-20} +10428 q^{-21} +9987 q^{-22} +3545 q^{-23} -7307 q^{-24} -16786 q^{-25} -12513 q^{-26} -4203 q^{-27} +10689 q^{-28} +20320 q^{-29} +19700 q^{-30} +4209 q^{-31} -17836 q^{-32} -25635 q^{-33} -23755 q^{-34} -2739 q^{-35} +21464 q^{-36} +37278 q^{-37} +27093 q^{-38} -3643 q^{-39} -28813 q^{-40} -44708 q^{-41} -28428 q^{-42} +6501 q^{-43} +44747 q^{-44} +51694 q^{-45} +23351 q^{-46} -16022 q^{-47} -55896 q^{-48} -56423 q^{-49} -21155 q^{-50} +37030 q^{-51} +67591 q^{-52} +53235 q^{-53} +8682 q^{-54} -53313 q^{-55} -77196 q^{-56} -51845 q^{-57} +18441 q^{-58} +71658 q^{-59} +77422 q^{-60} +36285 q^{-61} -41293 q^{-62} -88186 q^{-63} -77893 q^{-64} -3050 q^{-65} +67801 q^{-66} +93735 q^{-67} +60290 q^{-68} -26534 q^{-69} -92520 q^{-70} -97465 q^{-71} -22458 q^{-72} +61055 q^{-73} +104478 q^{-74} +79605 q^{-75} -12244 q^{-76} -93390 q^{-77} -112292 q^{-78} -39902 q^{-79} +52519 q^{-80} +111176 q^{-81} +95965 q^{-82} +3219 q^{-83} -89748 q^{-84} -122656 q^{-85} -57530 q^{-86} +38902 q^{-87} +110931 q^{-88} +108825 q^{-89} +22424 q^{-90} -76852 q^{-91} -124197 q^{-92} -73977 q^{-93} +17489 q^{-94} +98016 q^{-95} +112533 q^{-96} +42840 q^{-97} -52366 q^{-98} -110636 q^{-99} -82112 q^{-100} -7603 q^{-101} +71011 q^{-102} +100433 q^{-103} +55677 q^{-104} -22378 q^{-105} -81793 q^{-106} -74838 q^{-107} -25953 q^{-108} +38032 q^{-109} +73539 q^{-110} +53566 q^{-111} +1206 q^{-112} -47522 q^{-113} -53940 q^{-114} -30082 q^{-115} +11806 q^{-116} +42539 q^{-117} +38927 q^{-118} +11098 q^{-119} -20559 q^{-120} -29958 q^{-121} -22676 q^{-122} -1132 q^{-123} +19006 q^{-124} +21589 q^{-125} +10109 q^{-126} -6202 q^{-127} -12561 q^{-128} -12368 q^{-129} -3839 q^{-130} +6575 q^{-131} +9300 q^{-132} +5608 q^{-133} -1184 q^{-134} -3864 q^{-135} -5131 q^{-136} -2622 q^{-137} +1863 q^{-138} +3229 q^{-139} +2249 q^{-140} -160 q^{-141} -773 q^{-142} -1700 q^{-143} -1236 q^{-144} +513 q^{-145} +947 q^{-146} +724 q^{-147} -75 q^{-148} -10 q^{-149} -466 q^{-150} -507 q^{-151} +164 q^{-152} +240 q^{-153} +206 q^{-154} -56 q^{-155} +86 q^{-156} -105 q^{-157} -192 q^{-158} +53 q^{-159} +45 q^{-160} +57 q^{-161} -31 q^{-162} +56 q^{-163} -16 q^{-164} -64 q^{-165} +16 q^{-166} +16 q^{-168} -12 q^{-169} +20 q^{-170} + q^{-171} -17 q^{-172} +5 q^{-173} -3 q^{-174} +3 q^{-175} -2 q^{-176} +3 q^{-177} + q^{-178} -3 q^{-179} + q^{-180} </math>|J7=Not Available}}
coloured_jones_4 = <math>q^{12}-3 q^{11}+5 q^9+6 q^7-20 q^6-10 q^5+20 q^4+15 q^3+48 q^2-63 q-73+5 q^{-1} +42 q^{-2} +207 q^{-3} -55 q^{-4} -186 q^{-5} -151 q^{-6} -56 q^{-7} +484 q^{-8} +152 q^{-9} -167 q^{-10} -426 q^{-11} -467 q^{-12} +642 q^{-13} +527 q^{-14} +215 q^{-15} -559 q^{-16} -1150 q^{-17} +425 q^{-18} +786 q^{-19} +925 q^{-20} -296 q^{-21} -1796 q^{-22} -157 q^{-23} +679 q^{-24} +1685 q^{-25} +337 q^{-26} -2147 q^{-27} -862 q^{-28} +227 q^{-29} +2250 q^{-30} +1114 q^{-31} -2165 q^{-32} -1487 q^{-33} -384 q^{-34} +2556 q^{-35} +1832 q^{-36} -1935 q^{-37} -1935 q^{-38} -1003 q^{-39} +2574 q^{-40} +2368 q^{-41} -1481 q^{-42} -2109 q^{-43} -1539 q^{-44} +2234 q^{-45} +2579 q^{-46} -846 q^{-47} -1871 q^{-48} -1828 q^{-49} +1542 q^{-50} +2311 q^{-51} -225 q^{-52} -1249 q^{-53} -1692 q^{-54} +768 q^{-55} +1619 q^{-56} +114 q^{-57} -543 q^{-58} -1186 q^{-59} +237 q^{-60} +855 q^{-61} +137 q^{-62} -88 q^{-63} -622 q^{-64} +34 q^{-65} +335 q^{-66} +33 q^{-67} +68 q^{-68} -243 q^{-69} +3 q^{-70} +98 q^{-71} -30 q^{-72} +65 q^{-73} -71 q^{-74} +9 q^{-75} +23 q^{-76} -33 q^{-77} +29 q^{-78} -15 q^{-79} +8 q^{-80} +5 q^{-81} -15 q^{-82} +7 q^{-83} -2 q^{-84} +3 q^{-85} + q^{-86} -3 q^{-87} + q^{-88} </math> |

coloured_jones_5 = <math>q^{20}-3 q^{19}+5 q^{17}-q^{14}-13 q^{13}-10 q^{12}+20 q^{11}+24 q^{10}+15 q^9-3 q^8-56 q^7-73 q^6-6 q^5+95 q^4+136 q^3+88 q^2-84 q-266-247 q^{-1} +10 q^{-2} +354 q^{-3} +498 q^{-4} +234 q^{-5} -339 q^{-6} -792 q^{-7} -670 q^{-8} +96 q^{-9} +1021 q^{-10} +1246 q^{-11} +453 q^{-12} -947 q^{-13} -1890 q^{-14} -1363 q^{-15} +526 q^{-16} +2330 q^{-17} +2460 q^{-18} +481 q^{-19} -2372 q^{-20} -3676 q^{-21} -1889 q^{-22} +1841 q^{-23} +4588 q^{-24} +3713 q^{-25} -654 q^{-26} -5126 q^{-27} -5569 q^{-28} -1114 q^{-29} +4963 q^{-30} +7379 q^{-31} +3298 q^{-32} -4271 q^{-33} -8692 q^{-34} -5714 q^{-35} +2866 q^{-36} +9720 q^{-37} +8094 q^{-38} -1232 q^{-39} -10049 q^{-40} -10306 q^{-41} -869 q^{-42} +10197 q^{-43} +12248 q^{-44} +2792 q^{-45} -9783 q^{-46} -13878 q^{-47} -4919 q^{-48} +9370 q^{-49} +15252 q^{-50} +6696 q^{-51} -8586 q^{-52} -16326 q^{-53} -8590 q^{-54} +7858 q^{-55} +17149 q^{-56} +10152 q^{-57} -6755 q^{-58} -17634 q^{-59} -11764 q^{-60} +5584 q^{-61} +17702 q^{-62} +13016 q^{-63} -4006 q^{-64} -17210 q^{-65} -14083 q^{-66} +2285 q^{-67} +16123 q^{-68} +14575 q^{-69} -404 q^{-70} -14357 q^{-71} -14485 q^{-72} -1426 q^{-73} +12114 q^{-74} +13721 q^{-75} +2846 q^{-76} -9509 q^{-77} -12209 q^{-78} -3915 q^{-79} +6902 q^{-80} +10360 q^{-81} +4233 q^{-82} -4577 q^{-83} -8067 q^{-84} -4187 q^{-85} +2674 q^{-86} +6027 q^{-87} +3574 q^{-88} -1346 q^{-89} -4079 q^{-90} -2876 q^{-91} +489 q^{-92} +2662 q^{-93} +2044 q^{-94} -46 q^{-95} -1543 q^{-96} -1416 q^{-97} -129 q^{-98} +879 q^{-99} +848 q^{-100} +166 q^{-101} -413 q^{-102} -513 q^{-103} -150 q^{-104} +210 q^{-105} +260 q^{-106} +97 q^{-107} -72 q^{-108} -122 q^{-109} -70 q^{-110} +15 q^{-111} +64 q^{-112} +34 q^{-113} -8 q^{-114} -7 q^{-115} -17 q^{-116} -19 q^{-117} +11 q^{-118} +12 q^{-119} -5 q^{-120} +10 q^{-121} - q^{-122} -11 q^{-123} + q^{-124} +3 q^{-125} -2 q^{-126} +3 q^{-127} + q^{-128} -3 q^{-129} + q^{-130} </math> |
{{Computer Talk Header}}
coloured_jones_6 = <math>q^{30}-3 q^{29}+5 q^{27}-7 q^{24}+6 q^{23}-13 q^{22}-10 q^{21}+29 q^{20}+15 q^{19}+15 q^{18}-27 q^{17}+4 q^{16}-67 q^{15}-73 q^{14}+59 q^{13}+86 q^{12}+135 q^{11}+15 q^{10}+72 q^9-230 q^8-365 q^7-121 q^6+67 q^5+422 q^4+376 q^3+666 q^2-130 q-844-955 q^{-1} -799 q^{-2} +100 q^{-3} +744 q^{-4} +2323 q^{-5} +1444 q^{-6} -63 q^{-7} -1674 q^{-8} -2939 q^{-9} -2571 q^{-10} -1234 q^{-11} +3406 q^{-12} +4632 q^{-13} +4209 q^{-14} +1127 q^{-15} -3451 q^{-16} -7082 q^{-17} -8074 q^{-18} -771 q^{-19} +5024 q^{-20} +10428 q^{-21} +9987 q^{-22} +3545 q^{-23} -7307 q^{-24} -16786 q^{-25} -12513 q^{-26} -4203 q^{-27} +10689 q^{-28} +20320 q^{-29} +19700 q^{-30} +4209 q^{-31} -17836 q^{-32} -25635 q^{-33} -23755 q^{-34} -2739 q^{-35} +21464 q^{-36} +37278 q^{-37} +27093 q^{-38} -3643 q^{-39} -28813 q^{-40} -44708 q^{-41} -28428 q^{-42} +6501 q^{-43} +44747 q^{-44} +51694 q^{-45} +23351 q^{-46} -16022 q^{-47} -55896 q^{-48} -56423 q^{-49} -21155 q^{-50} +37030 q^{-51} +67591 q^{-52} +53235 q^{-53} +8682 q^{-54} -53313 q^{-55} -77196 q^{-56} -51845 q^{-57} +18441 q^{-58} +71658 q^{-59} +77422 q^{-60} +36285 q^{-61} -41293 q^{-62} -88186 q^{-63} -77893 q^{-64} -3050 q^{-65} +67801 q^{-66} +93735 q^{-67} +60290 q^{-68} -26534 q^{-69} -92520 q^{-70} -97465 q^{-71} -22458 q^{-72} +61055 q^{-73} +104478 q^{-74} +79605 q^{-75} -12244 q^{-76} -93390 q^{-77} -112292 q^{-78} -39902 q^{-79} +52519 q^{-80} +111176 q^{-81} +95965 q^{-82} +3219 q^{-83} -89748 q^{-84} -122656 q^{-85} -57530 q^{-86} +38902 q^{-87} +110931 q^{-88} +108825 q^{-89} +22424 q^{-90} -76852 q^{-91} -124197 q^{-92} -73977 q^{-93} +17489 q^{-94} +98016 q^{-95} +112533 q^{-96} +42840 q^{-97} -52366 q^{-98} -110636 q^{-99} -82112 q^{-100} -7603 q^{-101} +71011 q^{-102} +100433 q^{-103} +55677 q^{-104} -22378 q^{-105} -81793 q^{-106} -74838 q^{-107} -25953 q^{-108} +38032 q^{-109} +73539 q^{-110} +53566 q^{-111} +1206 q^{-112} -47522 q^{-113} -53940 q^{-114} -30082 q^{-115} +11806 q^{-116} +42539 q^{-117} +38927 q^{-118} +11098 q^{-119} -20559 q^{-120} -29958 q^{-121} -22676 q^{-122} -1132 q^{-123} +19006 q^{-124} +21589 q^{-125} +10109 q^{-126} -6202 q^{-127} -12561 q^{-128} -12368 q^{-129} -3839 q^{-130} +6575 q^{-131} +9300 q^{-132} +5608 q^{-133} -1184 q^{-134} -3864 q^{-135} -5131 q^{-136} -2622 q^{-137} +1863 q^{-138} +3229 q^{-139} +2249 q^{-140} -160 q^{-141} -773 q^{-142} -1700 q^{-143} -1236 q^{-144} +513 q^{-145} +947 q^{-146} +724 q^{-147} -75 q^{-148} -10 q^{-149} -466 q^{-150} -507 q^{-151} +164 q^{-152} +240 q^{-153} +206 q^{-154} -56 q^{-155} +86 q^{-156} -105 q^{-157} -192 q^{-158} +53 q^{-159} +45 q^{-160} +57 q^{-161} -31 q^{-162} +56 q^{-163} -16 q^{-164} -64 q^{-165} +16 q^{-166} +16 q^{-168} -12 q^{-169} +20 q^{-170} + q^{-171} -17 q^{-172} +5 q^{-173} -3 q^{-174} +3 q^{-175} -2 q^{-176} +3 q^{-177} + q^{-178} -3 q^{-179} + q^{-180} </math> |

coloured_jones_7 = |
<table>
computer_talk =
<tr valign=top>
<table>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<tr valign=top>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</pre></td></tr>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>

<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr>
<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, 78]]</nowiki></pre></td></tr>
<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, 8, 4, 9], X[5, 14, 6, 15], X[11, 17, 12, 16],
<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, 78]]</nowiki></pre></td></tr>
<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, 8, 4, 9], X[5, 14, 6, 15], X[11, 17, 12, 16],
X[15, 13, 16, 12], X[17, 20, 18, 1], X[9, 18, 10, 19],
X[15, 13, 16, 12], X[17, 20, 18, 1], X[9, 18, 10, 19],
X[19, 10, 20, 11], X[13, 6, 14, 7], X[7, 2, 8, 3]]</nowiki></pre></td></tr>
X[19, 10, 20, 11], X[13, 6, 14, 7], X[7, 2, 8, 3]]</nowiki></pre></td></tr>
<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, 78]]</nowiki></pre></td></tr>

<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, 78]]</nowiki></pre></td></tr>
<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, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6,
<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, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6,
7, -8, 6]</nowiki></pre></td></tr>
7, -8, 6]</nowiki></pre></td></tr>
<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, 78]]</nowiki></pre></td></tr>

<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, 78]]</nowiki></pre></td></tr>
<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, 2, 18, 16, 6, 12, 20, 10]</nowiki></pre></td></tr>
<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, 2, 18, 16, 6, 12, 20, 10]</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, 78]]</nowiki></pre></td></tr>
<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, -1, -2, 1, -2, -1, 3, -2, -4, 3, -4, -4}]</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, 78]]</nowiki></pre></td></tr>
<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[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[5, {-1, -1, -2, 1, -2, -1, 3, -2, -4, 3, -4, -4}]</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>
<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, 78]]</nowiki></pre></td></tr>

<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[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</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>
<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, 78]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_78_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, 78]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>

<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, 78]]</nowiki></pre></td></tr>
<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>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</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, 78]][t]</nowiki></pre></td></tr>
<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 7 16 2 3

<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, 78]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_78_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, 78]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>
<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, 78]][t]</nowiki></pre></td></tr>
<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 7 16 2 3
21 - t + -- - -- - 16 t + 7 t - t
21 - t + -- - -- - 16 t + 7 t - t
2 t
2 t
t</nowiki></pre></td></tr>
t</nowiki></pre></td></tr>
<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, 78]][z]</nowiki></pre></td></tr>

<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, 78]][z]</nowiki></pre></td></tr>
<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
<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
1 + 3 z + z - z</nowiki></pre></td></tr>
1 + 3 z + 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>

<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>
<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, 78], Knot[11, NonAlternating, 98],
<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, 78], Knot[11, NonAlternating, 98],
Knot[11, NonAlternating, 105]}</nowiki></pre></td></tr>
Knot[11, NonAlternating, 105]}</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, 78]], KnotSignature[Knot[10, 78]]}</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, 78]], KnotSignature[Knot[10, 78]]}</nowiki></pre></td></tr>
<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>{69, -4}</nowiki></pre></td></tr>
<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>{69, -4}</nowiki></pre></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, 78]][q]</nowiki></pre></td></tr>
<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> -10 3 5 9 11 11 11 8 6 3

<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, 78]][q]</nowiki></pre></td></tr>
<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> -10 3 5 9 11 11 11 8 6 3
1 + q - -- + -- - -- + -- - -- + -- - -- + -- - -
1 + q - -- + -- - -- + -- - -- + -- - -- + -- - -
9 8 7 6 5 4 3 2 q
9 8 7 6 5 4 3 2 q
q q q q q q q q</nowiki></pre></td></tr>
q q q q q q q q</nowiki></pre></td></tr>
<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>

<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>
<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, 78]}</nowiki></pre></td></tr>
<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, 78]}</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, 78]][q]</nowiki></pre></td></tr>
<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> -32 -30 2 -26 -24 3 2 2 2 -12

<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, 78]][q]</nowiki></pre></td></tr>
<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> -32 -30 2 -26 -24 3 2 2 2 -12
1 + q + q - --- - q - q - --- + --- + --- + --- - q +
1 + q + q - --- - q - q - --- + --- + --- + --- - q +
28 22 20 16 14
28 22 20 16 14
Line 153: Line 104:
10 8
10 8
q q</nowiki></pre></td></tr>
q q</nowiki></pre></td></tr>
<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, 78]][a, z]</nowiki></pre></td></tr>

<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, 78]][a, z]</nowiki></pre></td></tr>
<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> 2 4 6 8 10 2 2 4 2 6 2 8 2
<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> 2 4 6 8 10 2 2 4 2 6 2 8 2
a - a + 4 a - 4 a + a + 2 a z - 3 a z + 7 a z - 3 a z +
a - a + 4 a - 4 a + a + 2 a z - 3 a z + 7 a z - 3 a z +
2 4 4 4 6 4 4 6
2 4 4 4 6 4 4 6
a z - 3 a z + 3 a z - a z</nowiki></pre></td></tr>
a z - 3 a z + 3 a z - a z</nowiki></pre></td></tr>
<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, 78]][a, z]</nowiki></pre></td></tr>

<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, 78]][a, z]</nowiki></pre></td></tr>
<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> 2 4 6 8 10 3 5 7 9
<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> 2 4 6 8 10 3 5 7 9
-a - a - 4 a - 4 a - a - a z - 3 a z + 2 a z + 6 a z +
-a - a - 4 a - 4 a - a - a z - 3 a z + 2 a z + 6 a z +
Line 182: Line 131:
4 8 6 8 8 8 5 9 7 9
4 8 6 8 8 8 5 9 7 9
3 a z + 6 a z + 3 a z + a z + a z</nowiki></pre></td></tr>
3 a z + 6 a z + 3 a z + a z + a z</nowiki></pre></td></tr>
<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, 78]], Vassiliev[3][Knot[10, 78]]}</nowiki></pre></td></tr>

<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, 78]], Vassiliev[3][Knot[10, 78]]}</nowiki></pre></td></tr>
<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>{3, -5}</nowiki></pre></td></tr>
<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>{3, -5}</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, 78]][q, t]</nowiki></pre></td></tr>
<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>3 4 1 2 1 3 2 6

<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, 78]][q, t]</nowiki></pre></td></tr>
<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>3 4 1 2 1 3 2 6
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ +
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ +
5 3 21 8 19 7 17 7 17 6 15 6 15 5
5 3 21 8 19 7 17 7 17 6 15 6 15 5
Line 201: Line 148:
5 3 q
5 3 q
q t q</nowiki></pre></td></tr>
q t q</nowiki></pre></td></tr>
<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, 78], 2][q]</nowiki></pre></td></tr>

<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, 78], 2][q]</nowiki></pre></td></tr>
<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> -28 3 -26 7 14 7 20 41 18 45 79
<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> -28 3 -26 7 14 7 20 41 18 45 79
q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- +
q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- +
27 25 24 23 22 21 20 19 18
27 25 24 23 22 21 20 19 18
Line 217: Line 163:
5 4 3 2 q
5 4 3 2 q
q q q q</nowiki></pre></td></tr>
q q q q</nowiki></pre></td></tr>
</table> }}

</table>

{| width=100%
|align=left|See/edit the [[Rolfsen_Splice_Template]].

Back to the [[#top|top]].
|align=right|{{Knot Navigation Links|ext=gif}}
|}

[[Category:Knot Page]]

Revision as of 10:35, 30 August 2005

10 77.gif

10_77

10 79.gif

10_79

10 78.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 78's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 78 at Knotilus!


Knot presentations

Planar diagram presentation X1425 X3849 X5,14,6,15 X11,17,12,16 X15,13,16,12 X17,20,18,1 X9,18,10,19 X19,10,20,11 X13,6,14,7 X7283
Gauss code -1, 10, -2, 1, -3, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code 4 8 14 2 18 16 6 12 20 10
Conway Notation [21,21,2++]


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

Length is 12, width is 5,

Braid index is 5

10 78 ML.gif 10 78 AP.gif
[{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 9}, {5, 10}, {4, 6}, {7, 5}, {6, 1}, {8, 2}, {12, 7}, {1, 8}]

[edit Notes on presentations of 10 78]


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 [-13][1]
Hyperbolic Volume 12.5021
A-Polynomial See Data:10 78/A-polynomial

[edit Notes for 10 78's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -4

[edit Notes for 10 78's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 69, -4 }
Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +11 q^{-6} -9 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{10}-3 z^2 a^8-4 a^8+3 z^4 a^6+7 z^2 a^6+4 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+2 z a^{11}+4 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}-a^{10}+4 z^7 a^9-7 z^3 a^9+6 z a^9+3 z^8 a^8+2 z^6 a^8-10 z^4 a^8+10 z^2 a^8-4 a^8+z^9 a^7+7 z^7 a^7-15 z^5 a^7+5 z^3 a^7+2 z a^7+6 z^8 a^6-8 z^6 a^6-7 z^4 a^6+11 z^2 a^6-4 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+15 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2}
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}+q^{30}-2 q^{28}-q^{26}-q^{24}-3 q^{22}+2 q^{20}+2 q^{16}+2 q^{14}-q^{12}+3 q^{10}-2 q^8+q^6+q^4-q^2+1}
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-3 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+25 q^{144}-24 q^{142}+17 q^{140}-19 q^{136}+43 q^{134}-58 q^{132}+62 q^{130}-53 q^{128}+24 q^{126}+19 q^{124}-62 q^{122}+98 q^{120}-102 q^{118}+79 q^{116}-33 q^{114}-31 q^{112}+75 q^{110}-98 q^{108}+78 q^{106}-31 q^{104}-31 q^{102}+66 q^{100}-68 q^{98}+24 q^{96}+39 q^{94}-100 q^{92}+120 q^{90}-93 q^{88}+18 q^{86}+76 q^{84}-150 q^{82}+184 q^{80}-149 q^{78}+68 q^{76}+34 q^{74}-116 q^{72}+160 q^{70}-143 q^{68}+84 q^{66}-3 q^{64}-64 q^{62}+99 q^{60}-81 q^{58}+29 q^{56}+38 q^{54}-84 q^{52}+89 q^{50}-52 q^{48}-18 q^{46}+89 q^{44}-129 q^{42}+125 q^{40}-73 q^{38}-4 q^{36}+76 q^{34}-115 q^{32}+116 q^{30}-79 q^{28}+25 q^{26}+22 q^{24}-54 q^{22}+58 q^{20}-43 q^{18}+24 q^{16}-2 q^{14}-9 q^{12}+12 q^{10}-10 q^8+6 q^6-2 q^4+q^2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n98, K11n105,}

Same Jones Polynomial (up to mirroring, ): {}

Vassiliev invariants

V2 and V3: (3, -5)
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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -40} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 158} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 26} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2128}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{352}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -104} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 800} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1896} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 312} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32751}{10}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6274}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{59}{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1711}{10}}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 10 78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-1012χ
1          11
-1         2 -2
-3        41 3
-5       53  -2
-7      63   3
-9     55    0
-11    66     0
-13   35      2
-15  26       -4
-17 13        2
-19 2         -2
-211          1
Integral Khovanov Homology

(db, data source)

  
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials