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{{Knot Presentations}}
{{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:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.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:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]]</td></tr>
</table>

[[Invariants from Braid Theory|Length]] is 11, width is 4.

[[Invariants from Braid Theory|Braid index]] is 4.
</td>
<td>
[[Lightly Documented Features|A Morse Link Presentation]]:

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

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

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

Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]:
{[[10_87]], [[K11a58]], [[K11a165]], [[K11n72]], ...}

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

{{Vassiliev Invariants}}
{{Vassiliev Invariants}}


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<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>}}

{{Display Coloured Jones|J2=<math>q^2-3 q+1+11 q^{-1} -17 q^{-2} -7 q^{-3} +45 q^{-4} -34 q^{-5} -40 q^{-6} +94 q^{-7} -33 q^{-8} -93 q^{-9} +130 q^{-10} -13 q^{-11} -137 q^{-12} +136 q^{-13} +17 q^{-14} -147 q^{-15} +110 q^{-16} +37 q^{-17} -116 q^{-18} +63 q^{-19} +33 q^{-20} -62 q^{-21} +22 q^{-22} +16 q^{-23} -19 q^{-24} +5 q^{-25} +3 q^{-26} -3 q^{-27} + q^{-28} </math>|J3=<math>q^6-3 q^5+q^4+5 q^3+3 q^2-17 q-10+34 q^{-1} +35 q^{-2} -55 q^{-3} -80 q^{-4} +58 q^{-5} +163 q^{-6} -47 q^{-7} -245 q^{-8} -26 q^{-9} +349 q^{-10} +121 q^{-11} -403 q^{-12} -279 q^{-13} +450 q^{-14} +421 q^{-15} -418 q^{-16} -599 q^{-17} +383 q^{-18} +718 q^{-19} -285 q^{-20} -844 q^{-21} +193 q^{-22} +920 q^{-23} -83 q^{-24} -955 q^{-25} -34 q^{-26} +961 q^{-27} +131 q^{-28} -894 q^{-29} -238 q^{-30} +807 q^{-31} +286 q^{-32} -649 q^{-33} -329 q^{-34} +495 q^{-35} +304 q^{-36} -325 q^{-37} -261 q^{-38} +195 q^{-39} +193 q^{-40} -104 q^{-41} -118 q^{-42} +42 q^{-43} +70 q^{-44} -21 q^{-45} -31 q^{-46} +9 q^{-47} +13 q^{-48} -6 q^{-49} -3 q^{-50} + q^{-51} +3 q^{-52} -3 q^{-53} + q^{-54} </math>|J4=<math>q^{12}-3 q^{11}+q^{10}+5 q^9-3 q^8+3 q^7-20 q^6+2 q^5+36 q^4+7 q^3+15 q^2-109 q-57+109 q^{-1} +125 q^{-2} +177 q^{-3} -280 q^{-4} -366 q^{-5} -17 q^{-6} +313 q^{-7} +815 q^{-8} -139 q^{-9} -853 q^{-10} -770 q^{-11} -20 q^{-12} +1774 q^{-13} +855 q^{-14} -712 q^{-15} -1910 q^{-16} -1505 q^{-17} +2073 q^{-18} +2368 q^{-19} +746 q^{-20} -2366 q^{-21} -3698 q^{-22} +1000 q^{-23} +3274 q^{-24} +3057 q^{-25} -1488 q^{-26} -5467 q^{-27} -997 q^{-28} +3002 q^{-29} +5172 q^{-30} +266 q^{-31} -6245 q^{-32} -3014 q^{-33} +1950 q^{-34} +6557 q^{-35} +2109 q^{-36} -6208 q^{-37} -4600 q^{-38} +638 q^{-39} +7197 q^{-40} +3709 q^{-41} -5511 q^{-42} -5652 q^{-43} -828 q^{-44} +6975 q^{-45} +4939 q^{-46} -4023 q^{-47} -5826 q^{-48} -2340 q^{-49} +5546 q^{-50} +5354 q^{-51} -1880 q^{-52} -4700 q^{-53} -3280 q^{-54} +3163 q^{-55} +4448 q^{-56} -5 q^{-57} -2607 q^{-58} -2975 q^{-59} +978 q^{-60} +2591 q^{-61} +710 q^{-62} -758 q^{-63} -1754 q^{-64} -46 q^{-65} +964 q^{-66} +474 q^{-67} +48 q^{-68} -664 q^{-69} -147 q^{-70} +216 q^{-71} +121 q^{-72} +121 q^{-73} -170 q^{-74} -38 q^{-75} +36 q^{-76} -3 q^{-77} +47 q^{-78} -36 q^{-79} -2 q^{-80} +10 q^{-81} -9 q^{-82} +10 q^{-83} -7 q^{-84} + q^{-85} +3 q^{-86} -3 q^{-87} + q^{-88} </math>|J5=Not Available|J6=Not Available|J7=Not Available}}

{{Computer Talk Header}}
{{Computer Talk Header}}


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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</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>

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[10, 98]]</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>10</nowiki></pre></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, 98]]</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>PD[Knot[10, 98]]</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, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9],
<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>PD[X[1, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9],
X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4],
X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4],
X[13, 20, 14, 1], X[19, 12, 20, 13]]</nowiki></pre></td></tr>
X[13, 20, 14, 1], X[19, 12, 20, 13]]</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>GaussCode[Knot[10, 98]]</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>GaussCode[-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4,
<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, 98]]</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, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4,
3, -10, 9]</nowiki></pre></td></tr>
3, -10, 9]</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[Knot[10, 98]]</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, 98]]</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[6, 10, 14, 18, 2, 16, 20, 4, 8, 12]</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 98]]</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[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]</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[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]</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>alex = Alexander[Knot[10, 98]][t]</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> 2 9 18 2 3
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr>

<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, 98]]</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>4</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, 98]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_98_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, 98]]&) /@ {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>{Chiral, 2, 3, 3, NotAvailable, 2}</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, 98]][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> 2 9 18 2 3
23 - -- + -- - -- - 18 t + 9 t - 2 t
23 - -- + -- - -- - 18 t + 9 t - 2 t
3 2 t
3 2 t
t t</nowiki></pre></td></tr>
t t</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>Conway[Knot[10, 98]][z]</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> 4 6
<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, 98]][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> 4 6
1 - 3 z - 2 z</nowiki></pre></td></tr>
1 - 3 z - 2 z</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>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58],
<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, 87], Knot[10, 98], Knot[11, Alternating, 58],
Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}</nowiki></pre></td></tr>
Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}</nowiki></pre></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>{KnotDet[Knot[10, 98]], KnotSignature[Knot[10, 98]]}</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>{81, -4}</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, 98]], KnotSignature[Knot[10, 98]]}</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>J=Jones[Knot[10, 98]][q]</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>{81, -4}</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> -10 3 7 11 12 14 13 9 7 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, 98]][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 7 11 12 14 13 9 7 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[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
<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>{Knot[10, 98]}</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, 98]}</nowiki></pre></td></tr>
<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 98]][q]</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> -30 -28 2 2 3 5 -16 -14 5 -8
<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, 98]][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> -30 -28 2 2 3 5 -16 -14 5 -8
1 + q - q + --- + --- - --- - --- - q + q + --- - q +
1 + q - q + --- + --- - --- - --- - q + q + --- - q +
26 24 22 18 10
26 24 22 18 10
Line 96: Line 151:
6
6
q</nowiki></pre></td></tr>
q</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>Kauffman[Knot[10, 98]][a, z]</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> 2 4 6 8 5 7 9 2 2
<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, 98]][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 8 2 2 4
a + 3 a - 5 a + 2 a + 2 a z + a z - 5 a z + 2 a z + a z -
4 4 6 4 8 4 4 6 6 6
2 a z - 3 a z + a z - 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, 98]][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 5 7 9 2 2
-a + 3 a + 5 a + 2 a - 6 a z - 12 a z - 6 a z + 3 a z -
-a + 3 a + 5 a + 2 a - 6 a z - 12 a z - 6 a z + 3 a z -
Line 117: Line 180:
6 8 8 8 5 9 7 9
6 8 8 8 5 9 7 9
10 a z + 6 a z + 2 a z + 2 a z</nowiki></pre></td></tr>
10 a z + 6 a z + 2 a z + 2 a z</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>{Vassiliev[2][Knot[10, 98]], Vassiliev[3][Knot[10, 98]]}</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>{0, 3}</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, 98]], Vassiliev[3][Knot[10, 98]]}</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>Kh[Knot[10, 98]][q, t]</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>{0, 3}</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>3 5 1 2 1 5 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, 98]][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 5 1 2 1 5 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 134: Line 199:
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, 98], 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 3 5 19 16 22 62 33 63 116
1 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- +
27 26 25 24 23 22 21 20 19 18
q q q q q q q q q q
37 110 147 17 136 137 13 130 93 33 94 40
--- + --- - --- + --- + --- - --- - --- + --- - -- - -- + -- - -- -
17 16 15 14 13 12 11 10 9 8 7 6
q q q q q q q q q q q q
34 45 7 17 11 2
-- + -- - -- - -- + -- - 3 q + q
5 4 3 2 q
q q q q</nowiki></pre></td></tr>

</table>
</table>

See/edit the [[Rolfsen_Splice_Template]].


[[Category:Knot Page]]
[[Category:Knot Page]]

Revision as of 18:02, 29 August 2005

10 97.gif

10_97

10 99.gif

10_99

10 98.gif Visit 10 98's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 98's page at Knotilus!

Visit 10 98's page at the original Knot Atlas!

10 98 Quick Notes


10 98 Further Notes and Views

Knot presentations

Planar diagram presentation X1627 X3,10,4,11 X7,18,8,19 X17,8,18,9 X9,2,10,3 X11,16,12,17 X5,15,6,14 X15,5,16,4 X13,20,14,1 X19,12,20,13
Gauss code -1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, 9
Dowker-Thistlethwaite code 6 10 14 18 2 16 20 4 8 12
Conway Notation [.2.2.2.20]

Minimum Braid Representative:

BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif

Length is 11, width is 4.

Braid index is 4.

A Morse Link Presentation:

10 98 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [-13][1]
Hyperbolic Volume 14.4129
A-Polynomial See Data:10 98/A-polynomial

[edit Notes for 10 98's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 98's four dimensional invariants]

Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6-3 z^4+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{t^2-t+1\right\}}
Determinant and Signature { 81, -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} +7 q^{-2} -9 q^{-3} +13 q^{-4} -14 q^{-5} +12 q^{-6} -11 q^{-7} +7 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 z^4 a^8+2 z^2 a^8+2 a^8-z^6 a^6-3 z^4 a^6-5 z^2 a^6-5 a^6-z^6 a^4-2 z^4 a^4+z^2 a^4+3 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}-2 z^3 a^{11}+6 z^6 a^{10}-7 z^4 a^{10}+4 z^2 a^{10}+8 z^7 a^9-14 z^5 a^9+14 z^3 a^9-6 z a^9+6 z^8 a^8-7 z^6 a^8+2 z^4 a^8+2 a^8+2 z^9 a^7+8 z^7 a^7-26 z^5 a^7+25 z^3 a^7-12 z a^7+10 z^8 a^6-23 z^6 a^6+17 z^4 a^6-10 z^2 a^6+5 a^6+2 z^9 a^5+3 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+4 z^8 a^4-9 z^6 a^4+4 z^4 a^4-2 z^2 a^4+3 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 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^{30}-q^{28}+2 q^{26}+2 q^{24}-3 q^{22}-5 q^{18}-q^{16}+q^{14}+5 q^{10}-q^8+2 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}+6 q^{154}-5 q^{152}+10 q^{148}-21 q^{146}+31 q^{144}-39 q^{142}+33 q^{140}-18 q^{138}-12 q^{136}+58 q^{134}-94 q^{132}+117 q^{130}-107 q^{128}+56 q^{126}+17 q^{124}-112 q^{122}+185 q^{120}-203 q^{118}+154 q^{116}-41 q^{114}-84 q^{112}+183 q^{110}-192 q^{108}+134 q^{106}-19 q^{104}-110 q^{102}+175 q^{100}-143 q^{98}+34 q^{96}+116 q^{94}-230 q^{92}+256 q^{90}-161 q^{88}-7 q^{86}+163 q^{84}-297 q^{82}+322 q^{80}-242 q^{78}+78 q^{76}+92 q^{74}-232 q^{72}+288 q^{70}-230 q^{68}+90 q^{66}+43 q^{64}-158 q^{62}+191 q^{60}-129 q^{58}+8 q^{56}+128 q^{54}-194 q^{52}+181 q^{50}-68 q^{48}-83 q^{46}+204 q^{44}-244 q^{42}+197 q^{40}-81 q^{38}-51 q^{36}+157 q^{34}-188 q^{32}+162 q^{30}-84 q^{28}+3 q^{26}+51 q^{24}-78 q^{22}+69 q^{20}-44 q^{18}+20 q^{16}+2 q^{14}-11 q^{12}+13 q^{10}-10 q^8+6 q^6-2 q^4+q^2}
Further Quantum Invariants
Further quantum knot invariants for 10_98.

A1 Invariants.

Weight Invariant
1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-2 q^{19}+4 q^{17}-4 q^{15}+q^{13}-2 q^{11}-q^9+4 q^7-2 q^5+4 q^3-2 q+ q^{-1} }
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{58}-2 q^{56}+q^{54}+5 q^{52}-11 q^{50}+2 q^{48}+19 q^{46}-24 q^{44}-7 q^{42}+34 q^{40}-20 q^{38}-16 q^{36}+31 q^{34}-20 q^{30}+6 q^{28}+16 q^{26}-14 q^{24}-20 q^{22}+24 q^{20}+4 q^{18}-32 q^{16}+21 q^{14}+20 q^{12}-29 q^{10}+4 q^8+21 q^6-13 q^4-5 q^2+9- q^{-2} -2 q^{-4} + q^{-6} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{111}-2 q^{109}+q^{107}+2 q^{105}-2 q^{103}-5 q^{101}+5 q^{99}+13 q^{97}-15 q^{95}-30 q^{93}+27 q^{91}+60 q^{89}-27 q^{87}-110 q^{85}+13 q^{83}+166 q^{81}+23 q^{79}-198 q^{77}-87 q^{75}+213 q^{73}+145 q^{71}-179 q^{69}-197 q^{67}+115 q^{65}+206 q^{63}-39 q^{61}-194 q^{59}-40 q^{57}+164 q^{55}+103 q^{53}-111 q^{51}-152 q^{49}+75 q^{47}+186 q^{45}-16 q^{43}-218 q^{41}-28 q^{39}+217 q^{37}+84 q^{35}-213 q^{33}-146 q^{31}+174 q^{29}+189 q^{27}-111 q^{25}-212 q^{23}+41 q^{21}+199 q^{19}+31 q^{17}-155 q^{15}-71 q^{13}+94 q^{11}+86 q^9-42 q^7-66 q^5+4 q^3+42 q+10 q^{-1} -19 q^{-3} -8 q^{-5} +6 q^{-7} +4 q^{-9} - q^{-11} -2 q^{-13} + q^{-15} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{180}-2 q^{178}+q^{176}+2 q^{174}-5 q^{172}+4 q^{170}-2 q^{168}+5 q^{166}+2 q^{164}-27 q^{162}+10 q^{160}+16 q^{158}+42 q^{156}+6 q^{154}-128 q^{152}-54 q^{150}+70 q^{148}+250 q^{146}+141 q^{144}-353 q^{142}-426 q^{140}-73 q^{138}+675 q^{136}+776 q^{134}-314 q^{132}-1120 q^{130}-884 q^{128}+743 q^{126}+1767 q^{124}+546 q^{122}-1303 q^{120}-2018 q^{118}-161 q^{116}+2024 q^{114}+1719 q^{112}-374 q^{110}-2249 q^{108}-1343 q^{106}+1040 q^{104}+1980 q^{102}+854 q^{100}-1289 q^{98}-1704 q^{96}-275 q^{94}+1237 q^{92}+1411 q^{90}-77 q^{88}-1307 q^{86}-1085 q^{84}+381 q^{82}+1433 q^{80}+736 q^{78}-864 q^{76}-1504 q^{74}-192 q^{72}+1394 q^{70}+1357 q^{68}-486 q^{66}-1871 q^{64}-819 q^{62}+1198 q^{60}+1976 q^{58}+222 q^{56}-1893 q^{54}-1621 q^{52}+376 q^{50}+2145 q^{48}+1267 q^{46}-1044 q^{44}-1950 q^{42}-877 q^{40}+1316 q^{38}+1772 q^{36}+314 q^{34}-1199 q^{32}-1498 q^{30}-13 q^{28}+1127 q^{26}+986 q^{24}-8 q^{22}-967 q^{20}-634 q^{18}+119 q^{16}+606 q^{14}+465 q^{12}-173 q^{10}-361 q^8-235 q^6+74 q^4+245 q^2+83-35 q^{-2} -108 q^{-4} -49 q^{-6} +40 q^{-8} +28 q^{-10} +18 q^{-12} -13 q^{-14} -14 q^{-16} +3 q^{-18} + q^{-20} +4 q^{-22} - q^{-24} -2 q^{-26} + q^{-28} }

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{30}-q^{28}+2 q^{26}+2 q^{24}-3 q^{22}-5 q^{18}-q^{16}+q^{14}+5 q^{10}-q^8+2 q^6+q^4-q^2+1}
1,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+40 q^{76}-72 q^{74}+114 q^{72}-174 q^{70}+266 q^{68}-376 q^{66}+494 q^{64}-632 q^{62}+766 q^{60}-860 q^{58}+866 q^{56}-786 q^{54}+593 q^{52}-272 q^{50}-140 q^{48}+616 q^{46}-1070 q^{44}+1472 q^{42}-1744 q^{40}+1876 q^{38}-1845 q^{36}+1644 q^{34}-1308 q^{32}+856 q^{30}-396 q^{28}-82 q^{26}+494 q^{24}-800 q^{22}+968 q^{20}-996 q^{18}+934 q^{16}-780 q^{14}+598 q^{12}-416 q^{10}+268 q^8-150 q^6+80 q^4-36 q^2+14-4 q^{-2} + q^{-4} }
2,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{76}-q^{74}+q^{72}+2 q^{70}-q^{68}-4 q^{66}+2 q^{64}+7 q^{62}-6 q^{60}-13 q^{58}+q^{56}+8 q^{54}-11 q^{52}-5 q^{50}+19 q^{48}+18 q^{46}-2 q^{44}+10 q^{40}-8 q^{38}-15 q^{36}-q^{34}-6 q^{32}-14 q^{30}+3 q^{28}+10 q^{26}-6 q^{24}-6 q^{22}+14 q^{20}+9 q^{18}-10 q^{16}-4 q^{14}+13 q^{12}+3 q^{10}-7 q^8-q^6+5 q^4+3 q^2-2- q^{-2} + q^{-4} }

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{68}-2 q^{66}+6 q^{62}-7 q^{60}-4 q^{58}+17 q^{56}-13 q^{54}-13 q^{52}+24 q^{50}-14 q^{48}-14 q^{46}+27 q^{44}-5 q^{40}+16 q^{38}+4 q^{36}-8 q^{34}-17 q^{32}-q^{28}-26 q^{26}+10 q^{24}+19 q^{22}-19 q^{20}+11 q^{18}+18 q^{16}-17 q^{14}+8 q^{12}+8 q^{10}-8 q^8+4 q^6+2 q^4-2 q^2+1}
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{39}-q^{37}+3 q^{35}+3 q^{31}-3 q^{29}-6 q^{25}-3 q^{23}-2 q^{21}+3 q^{17}+q^{15}+5 q^{13}-q^{11}+3 q^9-q^7+2 q^5-q^3+q}

A4 Invariants.

Weight Invariant
0,1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{86}-q^{84}-2 q^{82}+5 q^{80}+4 q^{78}-8 q^{76}-2 q^{74}+12 q^{72}-q^{70}-20 q^{68}-3 q^{66}+10 q^{64}-15 q^{62}-17 q^{60}+22 q^{58}+22 q^{56}-3 q^{54}+21 q^{52}+33 q^{50}-3 q^{48}-18 q^{46}+4 q^{44}-15 q^{42}-43 q^{40}-15 q^{38}+10 q^{36}-16 q^{34}-15 q^{32}+25 q^{30}+16 q^{28}-8 q^{26}+4 q^{24}+18 q^{22}+q^{20}-6 q^{18}+6 q^{16}+7 q^{14}-3 q^{12}-q^{10}+4 q^8-q^4+q^2}
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-q^{46}+3 q^{44}+q^{42}+q^{40}+3 q^{38}-3 q^{36}-6 q^{32}-4 q^{30}-4 q^{28}-2 q^{26}+2 q^{22}+4 q^{20}+q^{18}+5 q^{16}-q^{14}+3 q^{12}+2 q^6-q^4+q^2}

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{68}-2 q^{66}+4 q^{64}-8 q^{62}+13 q^{60}-18 q^{58}+25 q^{56}-29 q^{54}+31 q^{52}-28 q^{50}+22 q^{48}-12 q^{46}-q^{44}+16 q^{42}-33 q^{40}+44 q^{38}-56 q^{36}+58 q^{34}-59 q^{32}+50 q^{30}-39 q^{28}+24 q^{26}-8 q^{24}-5 q^{22}+19 q^{20}-25 q^{18}+32 q^{16}-29 q^{14}+28 q^{12}-22 q^{10}+16 q^8-10 q^6+6 q^4-2 q^2+1}
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+7 q^{100}+2 q^{98}-10 q^{96}-11 q^{94}+5 q^{92}+21 q^{90}+6 q^{88}-23 q^{86}-22 q^{84}+12 q^{82}+30 q^{80}+q^{78}-31 q^{76}-15 q^{74}+24 q^{72}+25 q^{70}-10 q^{68}-22 q^{66}+8 q^{64}+27 q^{62}+6 q^{60}-22 q^{58}-8 q^{56}+16 q^{54}+8 q^{52}-21 q^{50}-19 q^{48}+10 q^{46}+16 q^{44}-13 q^{42}-29 q^{40}+2 q^{38}+32 q^{36}+15 q^{34}-25 q^{32}-24 q^{30}+16 q^{28}+34 q^{26}+2 q^{24}-25 q^{22}-13 q^{20}+17 q^{18}+17 q^{16}-4 q^{14}-12 q^{12}-2 q^{10}+7 q^8+4 q^6-2 q^4-2 q^2+ q^{-2} }

D4 Invariants.

Weight Invariant
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{94}-2 q^{92}+2 q^{90}-3 q^{88}+7 q^{86}-10 q^{84}+10 q^{82}-13 q^{80}+21 q^{78}-23 q^{76}+19 q^{74}-25 q^{72}+24 q^{70}-21 q^{68}+12 q^{66}-11 q^{64}+6 q^{62}+14 q^{60}-9 q^{58}+28 q^{56}-26 q^{54}+44 q^{52}-41 q^{50}+41 q^{48}-55 q^{46}+34 q^{44}-47 q^{42}+24 q^{40}-34 q^{38}+13 q^{36}-6 q^{34}+3 q^{32}+11 q^{30}-10 q^{28}+26 q^{26}-19 q^{24}+26 q^{22}-23 q^{20}+25 q^{18}-18 q^{16}+17 q^{14}-12 q^{12}+10 q^{10}-5 q^8+4 q^6-2 q^4+q^2}

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+6 q^{154}-5 q^{152}+10 q^{148}-21 q^{146}+31 q^{144}-39 q^{142}+33 q^{140}-18 q^{138}-12 q^{136}+58 q^{134}-94 q^{132}+117 q^{130}-107 q^{128}+56 q^{126}+17 q^{124}-112 q^{122}+185 q^{120}-203 q^{118}+154 q^{116}-41 q^{114}-84 q^{112}+183 q^{110}-192 q^{108}+134 q^{106}-19 q^{104}-110 q^{102}+175 q^{100}-143 q^{98}+34 q^{96}+116 q^{94}-230 q^{92}+256 q^{90}-161 q^{88}-7 q^{86}+163 q^{84}-297 q^{82}+322 q^{80}-242 q^{78}+78 q^{76}+92 q^{74}-232 q^{72}+288 q^{70}-230 q^{68}+90 q^{66}+43 q^{64}-158 q^{62}+191 q^{60}-129 q^{58}+8 q^{56}+128 q^{54}-194 q^{52}+181 q^{50}-68 q^{48}-83 q^{46}+204 q^{44}-244 q^{42}+197 q^{40}-81 q^{38}-51 q^{36}+157 q^{34}-188 q^{32}+162 q^{30}-84 q^{28}+3 q^{26}+51 q^{24}-78 q^{22}+69 q^{20}-44 q^{18}+20 q^{16}+2 q^{14}-11 q^{12}+13 q^{10}-10 q^8+6 q^6-2 q^4+q^2}

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

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

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

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

In[3]:= K = Knot["10 98"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[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 -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }
In[5]:= Conway[K][z]
Out[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 -2 z^6-3 z^4+1}
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= 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 \left\{t^2-t+1\right\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 81, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[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 1-3 q^{-1} +7 q^{-2} -9 q^{-3} +13 q^{-4} -14 q^{-5} +12 q^{-6} -11 q^{-7} +7 q^{-8} -3 q^{-9} + q^{-10} }
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[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 z^4 a^8+2 z^2 a^8+2 a^8-z^6 a^6-3 z^4 a^6-5 z^2 a^6-5 a^6-z^6 a^4-2 z^4 a^4+z^2 a^4+3 a^4+z^4 a^2+2 z^2 a^2+a^2}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[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 z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-2 z^3 a^{11}+6 z^6 a^{10}-7 z^4 a^{10}+4 z^2 a^{10}+8 z^7 a^9-14 z^5 a^9+14 z^3 a^9-6 z a^9+6 z^8 a^8-7 z^6 a^8+2 z^4 a^8+2 a^8+2 z^9 a^7+8 z^7 a^7-26 z^5 a^7+25 z^3 a^7-12 z a^7+10 z^8 a^6-23 z^6 a^6+17 z^4 a^6-10 z^2 a^6+5 a^6+2 z^9 a^5+3 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+4 z^8 a^4-9 z^6 a^4+4 z^4 a^4-2 z^2 a^4+3 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_87, K11a58, K11a165, K11n72, ...}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}

Vassiliev invariants

V2 and V3: (0, 3)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
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 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 24} 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 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 24} 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 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 48} 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 -64} 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 -136} 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 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 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 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 904} 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{920}{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 1160} 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 184} 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}

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 ). 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 98. 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        51 4
-5       53  -2
-7      84   4
-9     65    -1
-11    68     -2
-13   56      1
-15  26       -4
-17 15        4
-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}^{2}\oplus{\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}}
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}^{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^{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=-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}^{6}\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}^{8}\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=-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^{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}^{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 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}^{4}\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^{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}}
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

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^2-3 q+1+11 q^{-1} -17 q^{-2} -7 q^{-3} +45 q^{-4} -34 q^{-5} -40 q^{-6} +94 q^{-7} -33 q^{-8} -93 q^{-9} +130 q^{-10} -13 q^{-11} -137 q^{-12} +136 q^{-13} +17 q^{-14} -147 q^{-15} +110 q^{-16} +37 q^{-17} -116 q^{-18} +63 q^{-19} +33 q^{-20} -62 q^{-21} +22 q^{-22} +16 q^{-23} -19 q^{-24} +5 q^{-25} +3 q^{-26} -3 q^{-27} + q^{-28} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^6-3 q^5+q^4+5 q^3+3 q^2-17 q-10+34 q^{-1} +35 q^{-2} -55 q^{-3} -80 q^{-4} +58 q^{-5} +163 q^{-6} -47 q^{-7} -245 q^{-8} -26 q^{-9} +349 q^{-10} +121 q^{-11} -403 q^{-12} -279 q^{-13} +450 q^{-14} +421 q^{-15} -418 q^{-16} -599 q^{-17} +383 q^{-18} +718 q^{-19} -285 q^{-20} -844 q^{-21} +193 q^{-22} +920 q^{-23} -83 q^{-24} -955 q^{-25} -34 q^{-26} +961 q^{-27} +131 q^{-28} -894 q^{-29} -238 q^{-30} +807 q^{-31} +286 q^{-32} -649 q^{-33} -329 q^{-34} +495 q^{-35} +304 q^{-36} -325 q^{-37} -261 q^{-38} +195 q^{-39} +193 q^{-40} -104 q^{-41} -118 q^{-42} +42 q^{-43} +70 q^{-44} -21 q^{-45} -31 q^{-46} +9 q^{-47} +13 q^{-48} -6 q^{-49} -3 q^{-50} + q^{-51} +3 q^{-52} -3 q^{-53} + q^{-54} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-3 q^{11}+q^{10}+5 q^9-3 q^8+3 q^7-20 q^6+2 q^5+36 q^4+7 q^3+15 q^2-109 q-57+109 q^{-1} +125 q^{-2} +177 q^{-3} -280 q^{-4} -366 q^{-5} -17 q^{-6} +313 q^{-7} +815 q^{-8} -139 q^{-9} -853 q^{-10} -770 q^{-11} -20 q^{-12} +1774 q^{-13} +855 q^{-14} -712 q^{-15} -1910 q^{-16} -1505 q^{-17} +2073 q^{-18} +2368 q^{-19} +746 q^{-20} -2366 q^{-21} -3698 q^{-22} +1000 q^{-23} +3274 q^{-24} +3057 q^{-25} -1488 q^{-26} -5467 q^{-27} -997 q^{-28} +3002 q^{-29} +5172 q^{-30} +266 q^{-31} -6245 q^{-32} -3014 q^{-33} +1950 q^{-34} +6557 q^{-35} +2109 q^{-36} -6208 q^{-37} -4600 q^{-38} +638 q^{-39} +7197 q^{-40} +3709 q^{-41} -5511 q^{-42} -5652 q^{-43} -828 q^{-44} +6975 q^{-45} +4939 q^{-46} -4023 q^{-47} -5826 q^{-48} -2340 q^{-49} +5546 q^{-50} +5354 q^{-51} -1880 q^{-52} -4700 q^{-53} -3280 q^{-54} +3163 q^{-55} +4448 q^{-56} -5 q^{-57} -2607 q^{-58} -2975 q^{-59} +978 q^{-60} +2591 q^{-61} +710 q^{-62} -758 q^{-63} -1754 q^{-64} -46 q^{-65} +964 q^{-66} +474 q^{-67} +48 q^{-68} -664 q^{-69} -147 q^{-70} +216 q^{-71} +121 q^{-72} +121 q^{-73} -170 q^{-74} -38 q^{-75} +36 q^{-76} -3 q^{-77} +47 q^{-78} -36 q^{-79} -2 q^{-80} +10 q^{-81} -9 q^{-82} +10 q^{-83} -7 q^{-84} + q^{-85} +3 q^{-86} -3 q^{-87} + q^{-88} }
5 Not Available
6 Not Available
7 Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session. 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=
PD[Knot[10, 98]]
Out[2]=  
PD[X[1, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9], 

 X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4], 



  X[13, 20, 14, 1], X[19, 12, 20, 13]]
In[3]:=
GaussCode[Knot[10, 98]]
Out[3]=  
GaussCode[-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 
  3, -10, 9]
In[4]:=
DTCode[Knot[10, 98]]
Out[4]=  
DTCode[6, 10, 14, 18, 2, 16, 20, 4, 8, 12]
In[5]:=
br = BR[Knot[10, 98]]
Out[5]=  
BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=  
{4, 11}
In[7]:=
BraidIndex[Knot[10, 98]]
Out[7]=  
4
In[8]:=
Show[DrawMorseLink[Knot[10, 98]]]
10 98 ML.gif
Out[8]=-Graphics-
In[9]:=
(#[Knot[10, 98]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=  
{Chiral, 2, 3, 3, NotAvailable, 2}
In[10]:=
alex = Alexander[Knot[10, 98]][t]
Out[10]=  
     2    9    18             2      3

23 - -- + -- - -- - 18 t + 9 t  - 2 t



     3    2   t


     t    t
In[11]:=
Conway[Knot[10, 98]][z]
Out[11]=  
       4      6
1 - 3 z  - 2 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=  
{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], 
  Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[13]:=
{KnotDet[Knot[10, 98]], KnotSignature[Knot[10, 98]]}
Out[13]=  
{81, -4}
In[14]:=
Jones[Knot[10, 98]][q]
Out[14]=  
     -10   3    7    11   12   14   13   9    7    3

1 + q    - -- + -- - -- + -- - -- + -- - -- + -- - -



           9    8    7    6    5    4    3    2   q


           q    q    q    q    q    q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=  
{Knot[10, 98]}
In[16]:=
A2Invariant[Knot[10, 98]][q]
Out[16]=  
     -30    -28    2     2     3     5     -16    -14    5     -8

1 + q    - q    + --- + --- - --- - --- - q    + q    + --- - q   + 



                  26    24    22    18                  10
                 q     q     q     q                   q

 2     -4    -2
 -- + q   - q
  6


  q
In[17]:=
HOMFLYPT[Knot[10, 98]][a, z]
Out[17]=  
 2      4      6      8      2  2    4  2      6  2      8  2    2  4

a  + 3 a  - 5 a  + 2 a  + 2 a  z  + a  z  - 5 a  z  + 2 a  z  + a  z  - 



    4  4      6  4    8  4    4  6    6  6


  2 a  z  - 3 a  z  + a  z  - a  z  - a  z
In[18]:=
Kauffman[Knot[10, 98]][a, z]
Out[18]=  
  2      4      6      8      5         7        9        2  2

-a  + 3 a  + 5 a  + 2 a  - 6 a  z - 12 a  z - 6 a  z + 3 a  z  - 



    4  2       6  2      10  2    12  2      3  3       5  3
 2 a  z  - 10 a  z  + 4 a   z  - a   z  + 5 a  z  + 14 a  z  + 

     7  3       9  3      11  3      2  4      4  4       6  4
 25 a  z  + 14 a  z  - 2 a   z  - 3 a  z  + 4 a  z  + 17 a  z  + 

    8  4      10  4    12  4      3  5       5  5       7  5
 2 a  z  - 7 a   z  + a   z  - 8 a  z  - 17 a  z  - 26 a  z  - 

     9  5      11  5    2  6      4  6       6  6      8  6
 14 a  z  + 3 a   z  + a  z  - 9 a  z  - 23 a  z  - 7 a  z  + 

    10  6      3  7      5  7      7  7      9  7      4  8
 6 a   z  + 3 a  z  + 3 a  z  + 8 a  z  + 8 a  z  + 4 a  z  + 

     6  8      8  8      5  9      7  9


  10 a  z  + 6 a  z  + 2 a  z  + 2 a  z
In[19]:=
{Vassiliev[2][Knot[10, 98]], Vassiliev[3][Knot[10, 98]]}
Out[19]=  
{0, 3}
In[20]:=
Kh[Knot[10, 98]][q, t]
Out[20]=  
3    5      1        2        1        5        2        6

-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 



5    3    21  8    19  7    17  7    17  6    15  6    15  5



q    q    q   t    q   t    q   t    q   t    q   t    q   t



   5        6        6        8        6       5       8      4
 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 
  13  5    13  4    11  4    11  3    9  3    9  2    7  2    7
 q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t

  5     t    2 t      2
 ---- + -- + --- + q t
  5      3    q


  q  t   q
In[21]:=
ColouredJones[Knot[10, 98], 2][q]
Out[21]=  
     -28    3     3     5    19    16    22    62    33    63    116

1 + q    - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + 



           27    26    25    24    23    22    21    20    19    18
          q     q     q     q     q     q     q     q     q     q

 37    110   147   17    136   137   13    130   93   33   94   40
 --- + --- - --- + --- + --- - --- - --- + --- - -- - -- + -- - -- - 
  17    16    15    14    13    12    11    10    9    8    7    6
 q     q     q     q     q     q     q     q     q    q    q    q

 34   45   7    17   11          2
 -- + -- - -- - -- + -- - 3 q + q
  5    4    3    2   q


  q    q    q    q

See/edit the Rolfsen_Splice_Template.

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