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{{Rolfsen Knot Page|
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n = 9 |
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k = 45 |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,9,-5,-3,4,-2,7,-8,-9,5,6,-7,8,-6/goTop.html |
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braid_table = <table cellspacing=0 cellpadding=0 border=0>
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{{Knot Navigation Links|ext=gif}}

{{Rolfsen Knot Page Header|n=9|k=45|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,9,-5,-3,4,-2,7,-8,-9,5,6,-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]]</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]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.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:BraidPart1.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr>
</table>
</table> |
braid_crossings = 9 |

braid_width = 4 |
[[Invariants from Braid Theory|Length]] is 9, width is 4.
braid_index = 4 |

same_alexander = |
[[Invariants from Braid Theory|Braid index]] is 4.
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]]:
{...}

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=16.6667%><table cellpadding=0 cellspacing=0>
<td width=16.6667%><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=8.33333%>-7</td ><td width=8.33333%>-6</td ><td width=8.33333%>-5</td ><td width=8.33333%>-4</td ><td width=8.33333%>-3</td ><td width=8.33333%>-2</td ><td width=8.33333%>-1</td ><td width=8.33333%>0</td ><td width=16.6667%>&chi;</td></tr>
<td width=8.33333%>-7</td ><td width=8.33333%>-6</td ><td width=8.33333%>-5</td ><td width=8.33333%>-4</td ><td width=8.33333%>-3</td ><td width=8.33333%>-2</td ><td width=8.33333%>-1</td ><td width=8.33333%>0</td ><td width=16.6667%>&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 bgcolor=yellow>2</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 bgcolor=yellow>2</td><td>2</td></tr>
<tr align=center><td>-3</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>1</td><td>-1</td></tr>
<tr align=center><td>-3</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>1</td><td>-1</td></tr>
Line 69: Line 37:
<tr align=center><td>-15</td><td bgcolor=yellow>&nbsp;</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>1</td></tr>
<tr align=center><td>-15</td><td bgcolor=yellow>&nbsp;</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>1</td></tr>
<tr align=center><td>-17</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>-1</td></tr>
<tr align=center><td>-17</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>-1</td></tr>
</table>}}
</table> |
coloured_jones_2 = <math> q^{-1} + q^{-2} -5 q^{-3} +4 q^{-4} +6 q^{-5} -13 q^{-6} +6 q^{-7} +12 q^{-8} -19 q^{-9} +5 q^{-10} +15 q^{-11} -18 q^{-12} + q^{-13} +15 q^{-14} -13 q^{-15} -3 q^{-16} +12 q^{-17} -6 q^{-18} -4 q^{-19} +6 q^{-20} - q^{-21} -2 q^{-22} + q^{-23} </math> |

coloured_jones_3 = <math>2 q^{-1} -2 q^{-2} - q^{-3} -3 q^{-4} +10 q^{-5} +2 q^{-6} -12 q^{-7} -10 q^{-8} +20 q^{-9} +17 q^{-10} -23 q^{-11} -28 q^{-12} +28 q^{-13} +35 q^{-14} -26 q^{-15} -43 q^{-16} +25 q^{-17} +45 q^{-18} -20 q^{-19} -48 q^{-20} +16 q^{-21} +45 q^{-22} -9 q^{-23} -43 q^{-24} +3 q^{-25} +38 q^{-26} +6 q^{-27} -34 q^{-28} -11 q^{-29} +26 q^{-30} +16 q^{-31} -18 q^{-32} -19 q^{-33} +11 q^{-34} +17 q^{-35} -3 q^{-36} -14 q^{-37} - q^{-38} +9 q^{-39} +3 q^{-40} -5 q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} </math> |
{{Display Coloured Jones|J2=<math> q^{-1} + q^{-2} -5 q^{-3} +4 q^{-4} +6 q^{-5} -13 q^{-6} +6 q^{-7} +12 q^{-8} -19 q^{-9} +5 q^{-10} +15 q^{-11} -18 q^{-12} + q^{-13} +15 q^{-14} -13 q^{-15} -3 q^{-16} +12 q^{-17} -6 q^{-18} -4 q^{-19} +6 q^{-20} - q^{-21} -2 q^{-22} + q^{-23} </math>|J3=<math>2 q^{-1} -2 q^{-2} - q^{-3} -3 q^{-4} +10 q^{-5} +2 q^{-6} -12 q^{-7} -10 q^{-8} +20 q^{-9} +17 q^{-10} -23 q^{-11} -28 q^{-12} +28 q^{-13} +35 q^{-14} -26 q^{-15} -43 q^{-16} +25 q^{-17} +45 q^{-18} -20 q^{-19} -48 q^{-20} +16 q^{-21} +45 q^{-22} -9 q^{-23} -43 q^{-24} +3 q^{-25} +38 q^{-26} +6 q^{-27} -34 q^{-28} -11 q^{-29} +26 q^{-30} +16 q^{-31} -18 q^{-32} -19 q^{-33} +11 q^{-34} +17 q^{-35} -3 q^{-36} -14 q^{-37} - q^{-38} +9 q^{-39} +3 q^{-40} -5 q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} </math>|J4=<math>1+ q^{-1} -3 q^{-2} -4 q^{-3} +4 q^{-4} +5 q^{-5} +10 q^{-6} -8 q^{-7} -27 q^{-8} + q^{-9} +18 q^{-10} +47 q^{-11} -3 q^{-12} -74 q^{-13} -28 q^{-14} +21 q^{-15} +109 q^{-16} +33 q^{-17} -118 q^{-18} -80 q^{-19} - q^{-20} +162 q^{-21} +85 q^{-22} -133 q^{-23} -118 q^{-24} -38 q^{-25} +181 q^{-26} +123 q^{-27} -123 q^{-28} -126 q^{-29} -67 q^{-30} +170 q^{-31} +133 q^{-32} -99 q^{-33} -110 q^{-34} -88 q^{-35} +141 q^{-36} +129 q^{-37} -65 q^{-38} -83 q^{-39} -104 q^{-40} +97 q^{-41} +115 q^{-42} -21 q^{-43} -45 q^{-44} -113 q^{-45} +41 q^{-46} +87 q^{-47} +18 q^{-48} + q^{-49} -98 q^{-50} -8 q^{-51} +41 q^{-52} +31 q^{-53} +38 q^{-54} -57 q^{-55} -26 q^{-56} - q^{-57} +14 q^{-58} +44 q^{-59} -15 q^{-60} -14 q^{-61} -15 q^{-62} -5 q^{-63} +24 q^{-64} + q^{-65} -7 q^{-67} -7 q^{-68} +6 q^{-69} + q^{-70} +2 q^{-71} - q^{-72} -2 q^{-73} + q^{-74} </math>|J5=<math>2 q-2-3 q^{-2} -3 q^{-3} +6 q^{-4} +15 q^{-5} -2 q^{-6} -9 q^{-7} -20 q^{-8} -28 q^{-9} +19 q^{-10} +61 q^{-11} +41 q^{-12} -9 q^{-13} -83 q^{-14} -114 q^{-15} -15 q^{-16} +138 q^{-17} +177 q^{-18} +67 q^{-19} -152 q^{-20} -282 q^{-21} -147 q^{-22} +167 q^{-23} +369 q^{-24} +245 q^{-25} -143 q^{-26} -450 q^{-27} -352 q^{-28} +109 q^{-29} +497 q^{-30} +447 q^{-31} -49 q^{-32} -531 q^{-33} -517 q^{-34} -2 q^{-35} +524 q^{-36} +566 q^{-37} +58 q^{-38} -517 q^{-39} -588 q^{-40} -90 q^{-41} +484 q^{-42} +590 q^{-43} +125 q^{-44} -458 q^{-45} -579 q^{-46} -140 q^{-47} +415 q^{-48} +559 q^{-49} +164 q^{-50} -375 q^{-51} -531 q^{-52} -182 q^{-53} +316 q^{-54} +500 q^{-55} +213 q^{-56} -259 q^{-57} -457 q^{-58} -237 q^{-59} +179 q^{-60} +408 q^{-61} +264 q^{-62} -101 q^{-63} -345 q^{-64} -272 q^{-65} +15 q^{-66} +264 q^{-67} +274 q^{-68} +55 q^{-69} -177 q^{-70} -244 q^{-71} -111 q^{-72} +87 q^{-73} +194 q^{-74} +142 q^{-75} -10 q^{-76} -132 q^{-77} -137 q^{-78} -45 q^{-79} +60 q^{-80} +113 q^{-81} +74 q^{-82} -9 q^{-83} -68 q^{-84} -74 q^{-85} -28 q^{-86} +28 q^{-87} +54 q^{-88} +41 q^{-89} +4 q^{-90} -32 q^{-91} -36 q^{-92} -14 q^{-93} +7 q^{-94} +23 q^{-95} +20 q^{-96} + q^{-97} -13 q^{-98} -10 q^{-99} -5 q^{-100} +9 q^{-102} +5 q^{-103} -2 q^{-104} -2 q^{-105} - q^{-106} -2 q^{-107} + q^{-108} +2 q^{-109} - q^{-110} </math>|J6=<math>q^3+q^2-3 q-2+ q^{-2} +2 q^{-3} +9 q^{-4} +12 q^{-5} -9 q^{-6} -29 q^{-7} -21 q^{-8} -3 q^{-9} +12 q^{-10} +64 q^{-11} +80 q^{-12} +12 q^{-13} -96 q^{-14} -144 q^{-15} -105 q^{-16} -30 q^{-17} +189 q^{-18} +329 q^{-19} +209 q^{-20} -109 q^{-21} -385 q^{-22} -456 q^{-23} -320 q^{-24} +241 q^{-25} +743 q^{-26} +736 q^{-27} +164 q^{-28} -552 q^{-29} -989 q^{-30} -969 q^{-31} -14 q^{-32} +1061 q^{-33} +1437 q^{-34} +780 q^{-35} -406 q^{-36} -1393 q^{-37} -1736 q^{-38} -569 q^{-39} +1049 q^{-40} +1957 q^{-41} +1449 q^{-42} +7 q^{-43} -1459 q^{-44} -2262 q^{-45} -1126 q^{-46} +778 q^{-47} +2124 q^{-48} +1861 q^{-49} +417 q^{-50} -1283 q^{-51} -2432 q^{-52} -1447 q^{-53} +486 q^{-54} +2046 q^{-55} +1969 q^{-56} +654 q^{-57} -1066 q^{-58} -2371 q^{-59} -1536 q^{-60} +291 q^{-61} +1886 q^{-62} +1902 q^{-63} +755 q^{-64} -874 q^{-65} -2210 q^{-66} -1524 q^{-67} +129 q^{-68} +1679 q^{-69} +1773 q^{-70} +846 q^{-71} -640 q^{-72} -1974 q^{-73} -1505 q^{-74} -108 q^{-75} +1368 q^{-76} +1599 q^{-77} +993 q^{-78} -282 q^{-79} -1617 q^{-80} -1467 q^{-81} -445 q^{-82} +899 q^{-83} +1317 q^{-84} +1134 q^{-85} +183 q^{-86} -1086 q^{-87} -1302 q^{-88} -766 q^{-89} +312 q^{-90} +848 q^{-91} +1107 q^{-92} +600 q^{-93} -430 q^{-94} -901 q^{-95} -866 q^{-96} -208 q^{-97} +247 q^{-98} +788 q^{-99} +741 q^{-100} +132 q^{-101} -340 q^{-102} -623 q^{-103} -415 q^{-104} -240 q^{-105} +291 q^{-106} +515 q^{-107} +348 q^{-108} +98 q^{-109} -203 q^{-110} -256 q^{-111} -373 q^{-112} -74 q^{-113} +144 q^{-114} +215 q^{-115} +199 q^{-116} +72 q^{-117} +7 q^{-118} -209 q^{-119} -135 q^{-120} -63 q^{-121} +17 q^{-122} +73 q^{-123} +90 q^{-124} +110 q^{-125} -36 q^{-126} -38 q^{-127} -60 q^{-128} -42 q^{-129} -24 q^{-130} +14 q^{-131} +68 q^{-132} +11 q^{-133} +15 q^{-134} -9 q^{-135} -15 q^{-136} -27 q^{-137} -13 q^{-138} +18 q^{-139} +2 q^{-140} +11 q^{-141} +4 q^{-142} +3 q^{-143} -9 q^{-144} -7 q^{-145} +4 q^{-146} -2 q^{-147} +2 q^{-148} + q^{-149} +2 q^{-150} - q^{-151} -2 q^{-152} + q^{-153} </math>|J7=<math>2 q^5-2 q^4-2 q^2-3 q+1+6 q^{-1} +7 q^{-2} +6 q^{-3} -2 q^{-4} -8 q^{-5} -18 q^{-6} -34 q^{-7} -16 q^{-8} +29 q^{-9} +65 q^{-10} +73 q^{-11} +46 q^{-12} -11 q^{-13} -101 q^{-14} -202 q^{-15} -199 q^{-16} -19 q^{-17} +190 q^{-18} +369 q^{-19} +405 q^{-20} +224 q^{-21} -147 q^{-22} -632 q^{-23} -869 q^{-24} -589 q^{-25} +54 q^{-26} +846 q^{-27} +1385 q^{-28} +1241 q^{-29} +400 q^{-30} -963 q^{-31} -2082 q^{-32} -2135 q^{-33} -1084 q^{-34} +838 q^{-35} +2644 q^{-36} +3207 q^{-37} +2149 q^{-38} -366 q^{-39} -3085 q^{-40} -4318 q^{-41} -3411 q^{-42} -412 q^{-43} +3201 q^{-44} +5271 q^{-45} +4772 q^{-46} +1462 q^{-47} -2997 q^{-48} -5978 q^{-49} -6027 q^{-50} -2620 q^{-51} +2505 q^{-52} +6357 q^{-53} +7045 q^{-54} +3739 q^{-55} -1857 q^{-56} -6403 q^{-57} -7743 q^{-58} -4699 q^{-59} +1135 q^{-60} +6235 q^{-61} +8154 q^{-62} +5394 q^{-63} -510 q^{-64} -5911 q^{-65} -8262 q^{-66} -5857 q^{-67} -27 q^{-68} +5562 q^{-69} +8234 q^{-70} +6087 q^{-71} +373 q^{-72} -5224 q^{-73} -8059 q^{-74} -6166 q^{-75} -628 q^{-76} +4927 q^{-77} +7873 q^{-78} +6152 q^{-79} +773 q^{-80} -4673 q^{-81} -7635 q^{-82} -6090 q^{-83} -914 q^{-84} +4409 q^{-85} +7408 q^{-86} +6027 q^{-87} +1058 q^{-88} -4113 q^{-89} -7122 q^{-90} -5982 q^{-91} -1282 q^{-92} +3734 q^{-93} +6811 q^{-94} +5939 q^{-95} +1572 q^{-96} -3231 q^{-97} -6397 q^{-98} -5908 q^{-99} -1973 q^{-100} +2617 q^{-101} +5884 q^{-102} +5825 q^{-103} +2418 q^{-104} -1845 q^{-105} -5209 q^{-106} -5683 q^{-107} -2900 q^{-108} +974 q^{-109} +4399 q^{-110} +5383 q^{-111} +3312 q^{-112} -29 q^{-113} -3407 q^{-114} -4897 q^{-115} -3620 q^{-116} -892 q^{-117} +2314 q^{-118} +4207 q^{-119} +3675 q^{-120} +1685 q^{-121} -1153 q^{-122} -3295 q^{-123} -3471 q^{-124} -2273 q^{-125} +84 q^{-126} +2259 q^{-127} +2971 q^{-128} +2515 q^{-129} +799 q^{-130} -1181 q^{-131} -2230 q^{-132} -2421 q^{-133} -1390 q^{-134} +237 q^{-135} +1378 q^{-136} +2015 q^{-137} +1592 q^{-138} +460 q^{-139} -531 q^{-140} -1398 q^{-141} -1480 q^{-142} -847 q^{-143} -114 q^{-144} +745 q^{-145} +1108 q^{-146} +892 q^{-147} +516 q^{-148} -180 q^{-149} -650 q^{-150} -709 q^{-151} -637 q^{-152} -181 q^{-153} +233 q^{-154} +421 q^{-155} +540 q^{-156} +318 q^{-157} +48 q^{-158} -124 q^{-159} -348 q^{-160} -310 q^{-161} -168 q^{-162} -49 q^{-163} +158 q^{-164} +181 q^{-165} +159 q^{-166} +145 q^{-167} -10 q^{-168} -86 q^{-169} -111 q^{-170} -125 q^{-171} -33 q^{-172} -2 q^{-173} +26 q^{-174} +92 q^{-175} +56 q^{-176} +30 q^{-177} -3 q^{-178} -47 q^{-179} -24 q^{-180} -26 q^{-181} -26 q^{-182} +11 q^{-183} +18 q^{-184} +23 q^{-185} +17 q^{-186} -9 q^{-187} -4 q^{-189} -13 q^{-190} -4 q^{-191} -2 q^{-192} +6 q^{-193} +7 q^{-194} -2 q^{-195} +2 q^{-197} -2 q^{-198} - q^{-199} -2 q^{-200} + q^{-201} +2 q^{-202} - q^{-203} </math>}}
coloured_jones_4 = <math>1+ q^{-1} -3 q^{-2} -4 q^{-3} +4 q^{-4} +5 q^{-5} +10 q^{-6} -8 q^{-7} -27 q^{-8} + q^{-9} +18 q^{-10} +47 q^{-11} -3 q^{-12} -74 q^{-13} -28 q^{-14} +21 q^{-15} +109 q^{-16} +33 q^{-17} -118 q^{-18} -80 q^{-19} - q^{-20} +162 q^{-21} +85 q^{-22} -133 q^{-23} -118 q^{-24} -38 q^{-25} +181 q^{-26} +123 q^{-27} -123 q^{-28} -126 q^{-29} -67 q^{-30} +170 q^{-31} +133 q^{-32} -99 q^{-33} -110 q^{-34} -88 q^{-35} +141 q^{-36} +129 q^{-37} -65 q^{-38} -83 q^{-39} -104 q^{-40} +97 q^{-41} +115 q^{-42} -21 q^{-43} -45 q^{-44} -113 q^{-45} +41 q^{-46} +87 q^{-47} +18 q^{-48} + q^{-49} -98 q^{-50} -8 q^{-51} +41 q^{-52} +31 q^{-53} +38 q^{-54} -57 q^{-55} -26 q^{-56} - q^{-57} +14 q^{-58} +44 q^{-59} -15 q^{-60} -14 q^{-61} -15 q^{-62} -5 q^{-63} +24 q^{-64} + q^{-65} -7 q^{-67} -7 q^{-68} +6 q^{-69} + q^{-70} +2 q^{-71} - q^{-72} -2 q^{-73} + q^{-74} </math> |

coloured_jones_5 = <math>2 q-2-3 q^{-2} -3 q^{-3} +6 q^{-4} +15 q^{-5} -2 q^{-6} -9 q^{-7} -20 q^{-8} -28 q^{-9} +19 q^{-10} +61 q^{-11} +41 q^{-12} -9 q^{-13} -83 q^{-14} -114 q^{-15} -15 q^{-16} +138 q^{-17} +177 q^{-18} +67 q^{-19} -152 q^{-20} -282 q^{-21} -147 q^{-22} +167 q^{-23} +369 q^{-24} +245 q^{-25} -143 q^{-26} -450 q^{-27} -352 q^{-28} +109 q^{-29} +497 q^{-30} +447 q^{-31} -49 q^{-32} -531 q^{-33} -517 q^{-34} -2 q^{-35} +524 q^{-36} +566 q^{-37} +58 q^{-38} -517 q^{-39} -588 q^{-40} -90 q^{-41} +484 q^{-42} +590 q^{-43} +125 q^{-44} -458 q^{-45} -579 q^{-46} -140 q^{-47} +415 q^{-48} +559 q^{-49} +164 q^{-50} -375 q^{-51} -531 q^{-52} -182 q^{-53} +316 q^{-54} +500 q^{-55} +213 q^{-56} -259 q^{-57} -457 q^{-58} -237 q^{-59} +179 q^{-60} +408 q^{-61} +264 q^{-62} -101 q^{-63} -345 q^{-64} -272 q^{-65} +15 q^{-66} +264 q^{-67} +274 q^{-68} +55 q^{-69} -177 q^{-70} -244 q^{-71} -111 q^{-72} +87 q^{-73} +194 q^{-74} +142 q^{-75} -10 q^{-76} -132 q^{-77} -137 q^{-78} -45 q^{-79} +60 q^{-80} +113 q^{-81} +74 q^{-82} -9 q^{-83} -68 q^{-84} -74 q^{-85} -28 q^{-86} +28 q^{-87} +54 q^{-88} +41 q^{-89} +4 q^{-90} -32 q^{-91} -36 q^{-92} -14 q^{-93} +7 q^{-94} +23 q^{-95} +20 q^{-96} + q^{-97} -13 q^{-98} -10 q^{-99} -5 q^{-100} +9 q^{-102} +5 q^{-103} -2 q^{-104} -2 q^{-105} - q^{-106} -2 q^{-107} + q^{-108} +2 q^{-109} - q^{-110} </math> |
{{Computer Talk Header}}
coloured_jones_6 = <math>q^3+q^2-3 q-2+ q^{-2} +2 q^{-3} +9 q^{-4} +12 q^{-5} -9 q^{-6} -29 q^{-7} -21 q^{-8} -3 q^{-9} +12 q^{-10} +64 q^{-11} +80 q^{-12} +12 q^{-13} -96 q^{-14} -144 q^{-15} -105 q^{-16} -30 q^{-17} +189 q^{-18} +329 q^{-19} +209 q^{-20} -109 q^{-21} -385 q^{-22} -456 q^{-23} -320 q^{-24} +241 q^{-25} +743 q^{-26} +736 q^{-27} +164 q^{-28} -552 q^{-29} -989 q^{-30} -969 q^{-31} -14 q^{-32} +1061 q^{-33} +1437 q^{-34} +780 q^{-35} -406 q^{-36} -1393 q^{-37} -1736 q^{-38} -569 q^{-39} +1049 q^{-40} +1957 q^{-41} +1449 q^{-42} +7 q^{-43} -1459 q^{-44} -2262 q^{-45} -1126 q^{-46} +778 q^{-47} +2124 q^{-48} +1861 q^{-49} +417 q^{-50} -1283 q^{-51} -2432 q^{-52} -1447 q^{-53} +486 q^{-54} +2046 q^{-55} +1969 q^{-56} +654 q^{-57} -1066 q^{-58} -2371 q^{-59} -1536 q^{-60} +291 q^{-61} +1886 q^{-62} +1902 q^{-63} +755 q^{-64} -874 q^{-65} -2210 q^{-66} -1524 q^{-67} +129 q^{-68} +1679 q^{-69} +1773 q^{-70} +846 q^{-71} -640 q^{-72} -1974 q^{-73} -1505 q^{-74} -108 q^{-75} +1368 q^{-76} +1599 q^{-77} +993 q^{-78} -282 q^{-79} -1617 q^{-80} -1467 q^{-81} -445 q^{-82} +899 q^{-83} +1317 q^{-84} +1134 q^{-85} +183 q^{-86} -1086 q^{-87} -1302 q^{-88} -766 q^{-89} +312 q^{-90} +848 q^{-91} +1107 q^{-92} +600 q^{-93} -430 q^{-94} -901 q^{-95} -866 q^{-96} -208 q^{-97} +247 q^{-98} +788 q^{-99} +741 q^{-100} +132 q^{-101} -340 q^{-102} -623 q^{-103} -415 q^{-104} -240 q^{-105} +291 q^{-106} +515 q^{-107} +348 q^{-108} +98 q^{-109} -203 q^{-110} -256 q^{-111} -373 q^{-112} -74 q^{-113} +144 q^{-114} +215 q^{-115} +199 q^{-116} +72 q^{-117} +7 q^{-118} -209 q^{-119} -135 q^{-120} -63 q^{-121} +17 q^{-122} +73 q^{-123} +90 q^{-124} +110 q^{-125} -36 q^{-126} -38 q^{-127} -60 q^{-128} -42 q^{-129} -24 q^{-130} +14 q^{-131} +68 q^{-132} +11 q^{-133} +15 q^{-134} -9 q^{-135} -15 q^{-136} -27 q^{-137} -13 q^{-138} +18 q^{-139} +2 q^{-140} +11 q^{-141} +4 q^{-142} +3 q^{-143} -9 q^{-144} -7 q^{-145} +4 q^{-146} -2 q^{-147} +2 q^{-148} + q^{-149} +2 q^{-150} - q^{-151} -2 q^{-152} + q^{-153} </math> |

coloured_jones_7 = <math>2 q^5-2 q^4-2 q^2-3 q+1+6 q^{-1} +7 q^{-2} +6 q^{-3} -2 q^{-4} -8 q^{-5} -18 q^{-6} -34 q^{-7} -16 q^{-8} +29 q^{-9} +65 q^{-10} +73 q^{-11} +46 q^{-12} -11 q^{-13} -101 q^{-14} -202 q^{-15} -199 q^{-16} -19 q^{-17} +190 q^{-18} +369 q^{-19} +405 q^{-20} +224 q^{-21} -147 q^{-22} -632 q^{-23} -869 q^{-24} -589 q^{-25} +54 q^{-26} +846 q^{-27} +1385 q^{-28} +1241 q^{-29} +400 q^{-30} -963 q^{-31} -2082 q^{-32} -2135 q^{-33} -1084 q^{-34} +838 q^{-35} +2644 q^{-36} +3207 q^{-37} +2149 q^{-38} -366 q^{-39} -3085 q^{-40} -4318 q^{-41} -3411 q^{-42} -412 q^{-43} +3201 q^{-44} +5271 q^{-45} +4772 q^{-46} +1462 q^{-47} -2997 q^{-48} -5978 q^{-49} -6027 q^{-50} -2620 q^{-51} +2505 q^{-52} +6357 q^{-53} +7045 q^{-54} +3739 q^{-55} -1857 q^{-56} -6403 q^{-57} -7743 q^{-58} -4699 q^{-59} +1135 q^{-60} +6235 q^{-61} +8154 q^{-62} +5394 q^{-63} -510 q^{-64} -5911 q^{-65} -8262 q^{-66} -5857 q^{-67} -27 q^{-68} +5562 q^{-69} +8234 q^{-70} +6087 q^{-71} +373 q^{-72} -5224 q^{-73} -8059 q^{-74} -6166 q^{-75} -628 q^{-76} +4927 q^{-77} +7873 q^{-78} +6152 q^{-79} +773 q^{-80} -4673 q^{-81} -7635 q^{-82} -6090 q^{-83} -914 q^{-84} +4409 q^{-85} +7408 q^{-86} +6027 q^{-87} +1058 q^{-88} -4113 q^{-89} -7122 q^{-90} -5982 q^{-91} -1282 q^{-92} +3734 q^{-93} +6811 q^{-94} +5939 q^{-95} +1572 q^{-96} -3231 q^{-97} -6397 q^{-98} -5908 q^{-99} -1973 q^{-100} +2617 q^{-101} +5884 q^{-102} +5825 q^{-103} +2418 q^{-104} -1845 q^{-105} -5209 q^{-106} -5683 q^{-107} -2900 q^{-108} +974 q^{-109} +4399 q^{-110} +5383 q^{-111} +3312 q^{-112} -29 q^{-113} -3407 q^{-114} -4897 q^{-115} -3620 q^{-116} -892 q^{-117} +2314 q^{-118} +4207 q^{-119} +3675 q^{-120} +1685 q^{-121} -1153 q^{-122} -3295 q^{-123} -3471 q^{-124} -2273 q^{-125} +84 q^{-126} +2259 q^{-127} +2971 q^{-128} +2515 q^{-129} +799 q^{-130} -1181 q^{-131} -2230 q^{-132} -2421 q^{-133} -1390 q^{-134} +237 q^{-135} +1378 q^{-136} +2015 q^{-137} +1592 q^{-138} +460 q^{-139} -531 q^{-140} -1398 q^{-141} -1480 q^{-142} -847 q^{-143} -114 q^{-144} +745 q^{-145} +1108 q^{-146} +892 q^{-147} +516 q^{-148} -180 q^{-149} -650 q^{-150} -709 q^{-151} -637 q^{-152} -181 q^{-153} +233 q^{-154} +421 q^{-155} +540 q^{-156} +318 q^{-157} +48 q^{-158} -124 q^{-159} -348 q^{-160} -310 q^{-161} -168 q^{-162} -49 q^{-163} +158 q^{-164} +181 q^{-165} +159 q^{-166} +145 q^{-167} -10 q^{-168} -86 q^{-169} -111 q^{-170} -125 q^{-171} -33 q^{-172} -2 q^{-173} +26 q^{-174} +92 q^{-175} +56 q^{-176} +30 q^{-177} -3 q^{-178} -47 q^{-179} -24 q^{-180} -26 q^{-181} -26 q^{-182} +11 q^{-183} +18 q^{-184} +23 q^{-185} +17 q^{-186} -9 q^{-187} -4 q^{-189} -13 q^{-190} -4 q^{-191} -2 q^{-192} +6 q^{-193} +7 q^{-194} -2 q^{-195} +2 q^{-197} -2 q^{-198} - q^{-199} -2 q^{-200} + q^{-201} +2 q^{-202} - q^{-203} </math> |
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computer_talk =
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
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<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><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></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[9, 45]]</nowiki></pre></td></tr>
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<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[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],
<table><tr align=left>
<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[9, 45]]</nowiki></code></td></tr>
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<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[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],
X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12],
X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12],
X[12, 17, 13, 18], X[13, 6, 14, 7]]</nowiki></pre></td></tr>
X[12, 17, 13, 18], X[13, 6, 14, 7]]</nowiki></code></td></tr>
</table>

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<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[9, 45]]</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, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]</nowiki></pre></td></tr>
<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[9, 45]]</nowiki></code></td></tr>

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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[9, 45]]</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, 10, -14, 2, 16, -6, 18, 12]</nowiki></pre></td></tr>
<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, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]</nowiki></code></td></tr>

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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[9, 45]]</nowiki></pre></td></tr>
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]</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>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</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[9, 45]]</nowiki></code></td></tr>
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 9}</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[9, 45]]</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, 10, -14, 2, 16, -6, 18, 12]</nowiki></code></td></tr>
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr>
<table><tr align=left>

<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<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[9, 45]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_45_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>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[9, 45]]</nowiki></code></td></tr>

<tr align=left>
<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[9, 45]]&) /@ {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, 1, 2, 3, {4, 5}, 1}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]</nowiki></code></td></tr>

</table>
<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[9, 45]][t]</nowiki></pre></td></tr>
<table><tr align=left>
<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> -2 6 2
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{4, 9}</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[9, 45]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[9, 45]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:9_45_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[9, 45]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 2, 3, {4, 5}, 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[9, 45]][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> -2 6 2
-9 - t + - + 6 t - t
-9 - t + - + 6 t - t
t</nowiki></pre></td></tr>
t</nowiki></code></td></tr>
</table>

<table><tr align=left>
<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[9, 45]][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
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[9, 45]][z]</nowiki></code></td></tr>
1 + 2 z - z</nowiki></pre></td></tr>
<tr align=left>

<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 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[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[9, 45]}</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
1 + 2 z - z</nowiki></code></td></tr>

</table>
<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[9, 45]], KnotSignature[Knot[9, 45]]}</nowiki></pre></td></tr>
<table><tr align=left>
<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>{23, -2}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>

<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[9, 45]][q]</nowiki></pre></td></tr>
<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>
<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 2 3 4 4 4 3 2
<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[9, 45]}</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[9, 45]], KnotSignature[Knot[9, 45]]}</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>{23, -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[9, 45]][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 2 3 4 4 4 3 2
-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>

<table><tr align=left>
<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[9, 45]}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<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>
<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[9, 45]][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> -26 -24 -22 -18 -16 -14 -10 -8 -6 2
<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[9, 45]}</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[9, 45]][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 -22 -18 -16 -14 -10 -8 -6 2
-q - q + q + q + q - q - q + q + q + --
-q - q + q + q + q - q - q + q + q + --
2
2
q</nowiki></pre></td></tr>
q</nowiki></code></td></tr>
</table>

<table><tr align=left>
<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[9, 45]][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 2 2 4 2 6 2 4 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[9, 45]][a, z]</nowiki></code></td></tr>
2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z</nowiki></pre></td></tr>
<tr align=left>

<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[9, 45]][a, z]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<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 7 9 2 2 4 2
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 8 2 2 4 2 6 2 4 4
2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<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[9, 45]][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> 2 4 6 8 7 9 2 2 4 2
-2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z +
-2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z +
Line 154: Line 196:
5 7 7 7
5 7 7 7
a z + a z</nowiki></pre></td></tr>
a z + a z</nowiki></code></td></tr>
</table>

<table><tr align=left>
<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[9, 45]], Vassiliev[3][Knot[9, 45]]}</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>{2, -4}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}</nowiki></code></td></tr>

<tr align=left>
<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[9, 45]][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 2 1 1 1 2 1 2 2
<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>{2, -4}</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[9, 45]][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> -3 2 1 1 1 2 1 2 2
q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
q 17 7 15 6 13 6 13 5 11 5 11 4 9 4
q 17 7 15 6 13 6 13 5 11 5 11 4 9 4
Line 168: Line 218:
----- + ----- + ----- + ----- + ---- + ----
----- + ----- + ----- + ----- + ---- + ----
9 3 7 3 7 2 5 2 5 3
9 3 7 3 7 2 5 2 5 3
q t q t q t q t q t q t</nowiki></pre></td></tr>
q t q t q t q t q t q t</nowiki></code></td></tr>
</table>

<table><tr align=left>
<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[9, 45], 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> -23 2 -21 6 4 6 12 3 13 15 -13
<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[9, 45], 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 2 -21 6 4 6 12 3 13 15 -13
q - --- - q + --- - --- - --- + --- - --- - --- + --- + q -
q - --- - q + --- - --- - --- + --- - --- - --- + --- + q -
22 20 19 18 17 16 15 14
22 20 19 18 17 16 15 14
Line 179: Line 233:
--- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + -
--- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + -
12 11 10 9 8 7 6 5 4 3 q
12 11 10 9 8 7 6 5 4 3 q
q q q q q q q q q q</nowiki></pre></td></tr>
q q q q q q q q q q</nowiki></code></td></tr>
</table> }}

</table>

See/edit the [[Rolfsen_Splice_Template]].

[[Category:Knot Page]]

Latest revision as of 17:04, 1 September 2005

9 44.gif

9_44

9 46.gif

9_46

9 45.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 45's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 45 at Knotilus!


Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X7,14,8,15 X18,15,1,16 X16,11,17,12 X12,17,13,18 X13,6,14,7
Gauss code 1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6
Dowker-Thistlethwaite code 4 8 10 -14 2 16 -6 18 12
Conway Notation [211,21,2-]


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

Length is 9, width is 4,

Braid index is 4

9 45 ML.gif 9 45 AP.gif
[{2, 11}, {1, 7}, {10, 4}, {11, 9}, {6, 8}, {7, 10}, {3, 5}, {4, 6}, {5, 2}, {8, 3}, {9, 1}]

[edit Notes on presentations of 9 45]

Knot 9_45.
A graph, knot 9_45.
A part of a knot and a part of a graph.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][1]
Hyperbolic Volume 8.60203
A-Polynomial See Data:9 45/A-polynomial

[edit Notes for 9 45's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 2

[edit Notes for 9 45's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 23, -2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (2, -4)
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

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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where -2 is the signature of 9 45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-10χ
-1       22
-3      21-1
-5     21 1
-7    22  0
-9   22   0
-11  12    1
-13 12     -1
-15 1      1
-171       -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials