5 1: Difference between revisions
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same_alexander = <nowiki>[[10_132]], </nowiki> | |
same_alexander = <nowiki>[[10_132]], </nowiki> | |
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same_jones = <nowiki>[[10_132]], </nowiki> | |
same_jones = <nowiki>[[10_132]], </nowiki> | |
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khovanov_table = <math>\textrm{WikiForm}(\textrm{<table border=1>$\backslash $n<tr align=center>$\backslash $n<td width=20.$\%$><table cellpadding=0 cellspacing=0>$\backslash $n <tr><td>$\backslash \backslash $</td><td>$\&$nbsp;</td><td>r</td></tr>$\backslash $n<tr><td>$\&$nbsp;</td><td>$\&$nbsp;$\backslash \backslash \&$nbsp;</td><td>$\&$nbsp;</td></tr>$\backslash $n<tr><td>j</td><td>$\&$nbsp;</td><td>$\backslash \backslash $</td></tr>$\backslash $n</table></td>$\backslash $n <td width=10.$\%$>-5</td ><td width=10.$\%$>-4</td ><td width=10.$\%$>-3</td ><td width=10.$\%$>-2</td ><td width=10.$\%$>-1</td ><td width=10.$\%$>0</td ><td width=20.$\%$>$\&$chi;</td></tr>$\backslash $n<tr align=center><td>-3</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>$\backslash $n<tr align=center><td>-5</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td bgcolor=yellow>$\&$nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>$\backslash $n<tr align=center><td>-7</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>$\&$nbsp;</td><td>$\&$nbsp;</td><td>1</td></tr>$\backslash $n<tr align=center><td>-9</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td bgcolor=yellow>$\&$nbsp;</td><td bgcolor=yellow>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>0</td></tr>$\backslash $n<tr align=center><td>-11</td><td>$\&$nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>0</td></tr>$\backslash $n<tr align=center><td>-13</td><td bgcolor=yellow>$\&$nbsp;</td><td bgcolor=yellow>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>0</td></tr>$\backslash $n<tr align=center><td>-15</td><td bgcolor=yellow>1</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>$\&$nbsp;</td><td>-1</td></tr>$\backslash $n</table>})</math> | |
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khovanov_table = <nowiki><table border=1> |
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<tr align=center> |
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<td width=20.%><table cellpadding=0 cellspacing=0> |
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<tr><td>\</td><td> </td><td>r</td></tr> |
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<tr><td> </td><td> \ </td><td> </td></tr> |
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<tr><td>j</td><td> </td><td>\</td></tr> |
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</table></td> |
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<td width=10.%>-5</td ><td width=10.%>-4</td ><td width=10.%>-3</td ><td width=10.%>-2</td ><td width=10.%>-1</td ><td width=10.%>0</td ><td width=20.%>χ</td></tr> |
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<tr align=center><td>-3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
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<tr align=center><td>-5</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
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<tr align=center><td>-7</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td>1</td></tr> |
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<tr align=center><td>-9</td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-11</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-13</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-15</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
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</table></nowiki> | |
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coloured_jones_2 = <math> q^{-4} + q^{-7} - q^{-9} + q^{-10} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} - q^{-18} + q^{-19} </math> | |
coloured_jones_2 = <math> q^{-4} + q^{-7} - q^{-9} + q^{-10} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} - q^{-18} + q^{-19} </math> | |
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coloured_jones_3 = <math> q^{-6} + q^{-10} - q^{-13} + q^{-14} - q^{-17} + q^{-18} - q^{-21} - q^{-25} + q^{-27} - q^{-29} + q^{-31} + q^{-35} - q^{-36} </math> | |
coloured_jones_3 = <math> q^{-6} + q^{-10} - q^{-13} + q^{-14} - q^{-17} + q^{-18} - q^{-21} - q^{-25} + q^{-27} - q^{-29} + q^{-31} + q^{-35} - q^{-36} </math> | |
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<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr> |
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr> |
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</table> |
</table> |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 2 |$\backslash $nin = <nowiki>PD[Knot[5, 1]]</nowiki> |$\backslash $nout = <nowiki>PD[X[1, 6, 2, 7], X[3, 8, 4, 9], X[5, 10, 6, 1], X[7, 2, 8, 3], $\backslash $n $\backslash $n X[9, 4, 10, 5]]</nowiki> $\}\}$})</math> |
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<nowiki></nowiki>{{InOut | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 3 |$\backslash $nin = <nowiki>GaussCode[Knot[5, 1]]</nowiki> |$\backslash $nout = <nowiki>GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]</nowiki> $\}\}$})</math> |
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n = 2 | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 4 |$\backslash $nin = <nowiki>DTCode[Knot[5, 1]]</nowiki> |$\backslash $nout = <nowiki>DTCode[6, 8, 10, 2, 4]</nowiki> $\}\}$})</math> |
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in = <nowiki>PD[Knot[5, 1]]</nowiki> | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 5 |$\backslash $nin = <nowiki>br = BR[Knot[5, 1]]</nowiki> |$\backslash $nout = <nowiki>BR[2, $\{$-1, -1, -1, -1, -1$\}$]</nowiki> $\}\}$})</math> |
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out = <nowiki>PD[X[1, 6, 2, 7], X[3, 8, 4, 9], X[5, 10, 6, 1], X[7, 2, 8, 3], |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 6 |$\backslash $nin = <nowiki>$\{$First[br], Crossings[br]$\}$</nowiki> |$\backslash $nout = <nowiki>$\{$2, 5$\}$</nowiki> $\}\}$})</math> |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 7 |$\backslash $nin = <nowiki>BraidIndex[Knot[5, 1]]</nowiki> |$\backslash $nout = <nowiki>2</nowiki> $\}\}$})</math> |
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X[9, 4, 10, 5]]</nowiki> }}<nowiki></nowiki> |
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<math>\textrm{WikiForm}(\textrm{<tr valign=top><td><pre style=$\texttt{"}$color: blue; border: 0px; padding: 0em$\texttt{"}$><nowiki>In[8]:=</nowiki></pre></td><td><pre style=$\texttt{"}$color: red; border: 0px; padding: 0em$\texttt{"}$><nowiki>Show[DrawMorseLink[Knot[5, 1]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:5$\_$1$\_$ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr><nowiki>})</math> |
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<nowiki></nowiki>{{InOut | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 9 |$\backslash $nin = <nowiki> ($\#$[Knot[5, 1]]$\&$) /@ $\{\backslash $n SymmetryType, UnknottingNumber, ThreeGenus,$\backslash $n BridgeIndex, SuperBridgeIndex, NakanishiIndex$\backslash $n $\}$</nowiki> |$\backslash $nout = <nowiki>$\{$Reversible, 2, 2, 2, 3, 1$\}$</nowiki> $\}\}$})</math> |
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n = 3 | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 10 |$\backslash $nin = <nowiki>alex = Alexander[Knot[5, 1]][t]</nowiki> |$\backslash $nout = <nowiki> -2 1 2$\backslash $n1 + t - - - t + t$\backslash $n t</nowiki> $\}\}$})</math> |
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in = <nowiki>GaussCode[Knot[5, 1]]</nowiki> | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 11 |$\backslash $nin = <nowiki>Conway[Knot[5, 1]][z]</nowiki> |$\backslash $nout = <nowiki> 2 4$\backslash $n1 + 3 z + z</nowiki> $\}\}$})</math> |
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out = <nowiki>GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]</nowiki> }}<nowiki></nowiki> |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 12 |$\backslash $nin = <nowiki>Select[AllKnots[], (alex === Alexander[$\#$][t])$\&$]</nowiki> |$\backslash $nout = <nowiki>$\{$Knot[5, 1], Knot[10, 132]$\}$</nowiki> $\}\}$})</math> |
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<nowiki></nowiki>{{InOut | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 13 |$\backslash $nin = <nowiki>$\{$KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]$\}$</nowiki> |$\backslash $nout = <nowiki>$\{$5, -4$\}$</nowiki> $\}\}$})</math> |
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n = 4 | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 14 |$\backslash $nin = <nowiki>Jones[Knot[5, 1]][q]</nowiki> |$\backslash $nout = <nowiki> -7 -6 -5 -4 -2$\backslash $n-q + q - q + q + q</nowiki> $\}\}$})</math> |
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in = <nowiki>DTCode[Knot[5, 1]]</nowiki> | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 15 |$\backslash $nin = <nowiki>Select[AllKnots[], (J === Jones[$\#$][q] || (J /. q-> 1/q) === Jones[$\#$][q])$\&$]</nowiki> |$\backslash $nout = <nowiki>$\{$Knot[5, 1], Knot[10, 132]$\}$</nowiki> $\}\}$})</math> |
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out = <nowiki>DTCode[6, 8, 10, 2, 4]</nowiki> }}<nowiki></nowiki> |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 16 |$\backslash $nin = <nowiki>A2Invariant[Knot[5, 1]][q]</nowiki> |$\backslash $nout = <nowiki> -22 -20 -18 -14 -12 2 -8 -6$\backslash $n-q - q - q + q + q + --- + q + q$\backslash $n 10$\backslash $n q</nowiki> $\}\}$})</math> |
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<nowiki></nowiki>{{InOut | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 17 |$\backslash $nin = <nowiki>HOMFLYPT[Knot[5, 1]][a, z]</nowiki> |$\backslash $nout = <nowiki> 4 6 4 2 6 2 4 4$\backslash $n3 a - 2 a + 4 a z - a z + a z</nowiki> $\}\}$})</math> |
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n = 5 | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 18 |$\backslash $nin = <nowiki>Kauffman[Knot[5, 1]][a, z]</nowiki> |$\backslash $nout = <nowiki> 4 6 5 7 9 4 2 6 2 8 2$\backslash $n3 a + 2 a - 2 a z - a z + a z - 4 a z - 3 a z + a z + $\backslash $n $\backslash $n 5 3 7 3 4 4 6 4$\backslash $n a z + a z + a z + a z</nowiki> $\}\}$})</math> |
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in = <nowiki>br = BR[Knot[5, 1]]</nowiki> | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 19 |$\backslash $nin = <nowiki>$\{$Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]$\}$</nowiki> |$\backslash $nout = <nowiki>$\{$3, -5$\}$</nowiki> $\}\}$})</math> |
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out = <nowiki>BR[2, {-1, -1, -1, -1, -1}]</nowiki> }}<nowiki></nowiki> |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 20 |$\backslash $nin = <nowiki>Kh[Knot[5, 1]][q, t]</nowiki> |$\backslash $nout = <nowiki> -5 -3 1 1 1 1$\backslash $nq + q + ------ + ------ + ------ + -----$\backslash $n 15 5 11 4 11 3 7 2$\backslash $n q t q t q t q t</nowiki> $\}\}$})</math> |
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<nowiki></nowiki>{{InOut | |
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<math>\textrm{WikiForm}(\textrm{$\{\{$InOut |$\backslash $nn = 21 |$\backslash $nin = <nowiki>ColouredJones[Knot[5, 1], 2][q]</nowiki> |$\backslash $nout = <nowiki> -19 -18 -16 2 -13 -12 -10 -9 -7 -4$\backslash $nq - q + q - --- + q - q + q - q + q + q$\backslash $n 15$\backslash $n q</nowiki> $\}\}$})</math> }} |
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n = 6 | |
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in = <nowiki>{First[br], Crossings[br]}</nowiki> | |
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out = <nowiki>{2, 5}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 7 | |
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in = <nowiki>BraidIndex[Knot[5, 1]]</nowiki> | |
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out = <nowiki>2</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki><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[5, 1]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:5_1_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><nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 9 | |
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in = <nowiki> (#[Knot[5, 1]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki> | |
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out = <nowiki>{Reversible, 2, 2, 2, 3, 1}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 10 | |
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in = <nowiki>alex = Alexander[Knot[5, 1]][t]</nowiki> | |
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out = <nowiki> -2 1 2 |
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1 + t - - - t + t |
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t</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 11 | |
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in = <nowiki>Conway[Knot[5, 1]][z]</nowiki> | |
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out = <nowiki> 2 4 |
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1 + 3 z + z</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 12 | |
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in = <nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki> | |
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out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 13 | |
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in = <nowiki>{KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]}</nowiki> | |
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out = <nowiki>{5, -4}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 14 | |
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in = <nowiki>Jones[Knot[5, 1]][q]</nowiki> | |
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out = <nowiki> -7 -6 -5 -4 -2 |
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-q + q - q + q + q</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 15 | |
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in = <nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki> | |
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out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 16 | |
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in = <nowiki>A2Invariant[Knot[5, 1]][q]</nowiki> | |
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out = <nowiki> -22 -20 -18 -14 -12 2 -8 -6 |
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-q - q - q + q + q + --- + q + q |
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10 |
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q</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 17 | |
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in = <nowiki>HOMFLYPT[Knot[5, 1]][a, z]</nowiki> | |
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out = <nowiki> 4 6 4 2 6 2 4 4 |
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3 a - 2 a + 4 a z - a z + a z</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 18 | |
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in = <nowiki>Kauffman[Knot[5, 1]][a, z]</nowiki> | |
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out = <nowiki> 4 6 5 7 9 4 2 6 2 8 2 |
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3 a + 2 a - 2 a z - a z + a z - 4 a z - 3 a z + a z + |
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5 3 7 3 4 4 6 4 |
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a z + a z + a z + a z</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 19 | |
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in = <nowiki>{Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]}</nowiki> | |
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out = <nowiki>{3, -5}</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 20 | |
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in = <nowiki>Kh[Knot[5, 1]][q, t]</nowiki> | |
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out = <nowiki> -5 -3 1 1 1 1 |
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q + q + ------ + ------ + ------ + ----- |
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15 5 11 4 11 3 7 2 |
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q t q t q t q t</nowiki> }}<nowiki></nowiki> |
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<nowiki></nowiki>{{InOut | |
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n = 21 | |
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in = <nowiki>ColouredJones[Knot[5, 1], 2][q]</nowiki> | |
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out = <nowiki> -19 -18 -16 2 -13 -12 -10 -9 -7 -4 |
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q - q + q - --- + q - q + q - q + q + q |
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15 |
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q</nowiki> }}<nowiki></nowiki> }} |
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Revision as of 16:11, 1 September 2005
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
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An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2). When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1 |
 The VISA Interlink Logo [1] |
.png) Version of the US bicentennial emblem | |
 A pentagonal table by Bob Mackay [2] |
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 Partial view of US bicentennial logo on a shirt seen in Lisboa [3] | ||
This sentence was last edited by Dror. Sometime later, Scott added this sentence.
Knot presentations
| Planar diagram presentation | X1627 X3849 X5,10,6,1 X7283 X9,4,10,5 |
| Gauss code | -1, 4, -2, 5, -3, 1, -4, 2, -5, 3 |
| Dowker-Thistlethwaite code | 6 8 10 2 4 |
| Conway Notation | [5] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||
Length is 5, width is 2, Braid index is 2 |
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 [{7, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 1}] |
[edit Notes on presentations of 5 1]
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["5 1"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1627 X3849 X5,10,6,1 X7283 X9,4,10,5 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -2, 5, -3, 1, -4, 2, -5, 3 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 8 10 2 4 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[5] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 2, 5, 2 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{7, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | |
| Conway polynomial | |
| 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 \{1\}} |
| Determinant and Signature | { 5, -4 } |
| Jones polynomial | |
| HOMFLY-PT polynomial (db, data sources) | |
| Kauffman polynomial (db, data sources) | |
| 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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6} |
| 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^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}} |
Further Quantum Invariants
Further quantum knot invariants for 5_1.
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^{15}+q^7+q^5+q^3} |
| 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^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^8+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^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9} |
| 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^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}} |
| 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 -q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}} |
| 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 q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}} |
| 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 q^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}} |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | |
| 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^{60}-2 q^{36}-2 q^{34}-4 q^{32}-4 q^{30}-3 q^{28}+2 q^{24}+4 q^{22}+5 q^{20}+4 q^{18}+4 q^{16}+2 q^{14}+q^{12}} |
| 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^{54}+q^{52}+2 q^{50}+q^{48}-2 q^{44}-3 q^{42}-3 q^{40}-3 q^{38}-2 q^{36}-q^{34}+q^{28}+q^{26}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}} |
| 3,0 | 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^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18}} |
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^{50}-q^{36}-2 q^{34}-3 q^{32}-3 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+3 q^{16}+q^{14}+q^{12}} |
| 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^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9} |
| 1,0,1 | 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^{80}+q^{58}+q^{56}+3 q^{54}+3 q^{52}+2 q^{50}-q^{48}-5 q^{46}-9 q^{44}-12 q^{42}-12 q^{40}-9 q^{38}-3 q^{36}+2 q^{34}+7 q^{32}+10 q^{30}+11 q^{28}+10 q^{26}+7 q^{24}+5 q^{22}+2 q^{20}+q^{18}} |
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^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2 q^{48}-4 q^{46}-5 q^{44}-6 q^{42}-6 q^{40}-4 q^{38}-q^{36}+2 q^{34}+4 q^{32}+7 q^{30}+6 q^{28}+6 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}} |
| 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^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}} |
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^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{12}} |
| 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^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2 q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+q^{22}+q^{18}} |
B3 Invariants.
| Weight | Invariant |
|---|---|
| 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^{120}-q^{86}-q^{82}-q^{80}-2 q^{78}-q^{76}-2 q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2 q^{54}+q^{52}+3 q^{50}+q^{48}+3 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{34}+q^{30}} |
B4 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^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42}} |
C3 Invariants.
| Weight | Invariant |
|---|---|
| 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^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2 q^{30}+2 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+q^{18}} |
C4 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^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24}} |
D4 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^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30}} |
| 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^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18}} |
G2 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^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54}} |
| 1,0 |
.
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["5 1"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
In[6]:=
|
Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 5, -4 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {[[10_132]], }
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {[[10_132]], }
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["5 1"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ 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^2+ t^{-2} -t- t^{-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^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} } } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{[[10_132]], } |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{[[10_132]], } |
Vassiliev invariants
| V2 and V3: | (3, -5) |
| V2,1 through 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
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^{-4} + q^{-7} - q^{-9} + q^{-10} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} - q^{-18} + q^{-19} } |
| 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} + q^{-10} - q^{-13} + q^{-14} - q^{-17} + q^{-18} - q^{-21} - q^{-25} + q^{-27} - q^{-29} + q^{-31} + q^{-35} - q^{-36} } |
| 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^{-8} + q^{-13} - q^{-17} + q^{-18} - q^{-22} + q^{-23} - q^{-27} + q^{-28} - q^{-29} - q^{-32} + q^{-33} - q^{-34} + q^{-36} - q^{-37} + q^{-38} - q^{-39} + q^{-41} - q^{-42} + q^{-43} - q^{-44} + q^{-45} + q^{-46} - q^{-47} + q^{-48} - q^{-49} + q^{-51} - q^{-52} + q^{-53} - q^{-54} - q^{-57} + q^{-58} } |
| 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 q^{-10} + q^{-16} - q^{-21} + q^{-22} - q^{-27} + q^{-28} - q^{-33} + q^{-34} - q^{-36} - q^{-39} + q^{-40} - q^{-42} + q^{-46} - q^{-48} + q^{-52} - q^{-54} + q^{-57} + q^{-58} - q^{-60} + q^{-63} - q^{-66} + q^{-69} - q^{-72} - q^{-73} + q^{-75} - q^{-79} + q^{-81} + q^{-84} - q^{-85} } |
| 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 q^{-12} + q^{-19} - q^{-25} + q^{-26} - q^{-32} + q^{-33} - q^{-39} + q^{-40} - q^{-43} - q^{-46} + q^{-47} - q^{-50} - q^{-53} +2 q^{-54} - q^{-57} - q^{-60} +2 q^{-61} - q^{-64} - q^{-67} +2 q^{-68} + q^{-69} - q^{-71} - q^{-74} +2 q^{-75} + q^{-76} -2 q^{-78} - q^{-81} +2 q^{-82} + q^{-83} -2 q^{-85} - q^{-88} +2 q^{-89} -2 q^{-92} - q^{-95} +2 q^{-96} + q^{-97} -2 q^{-99} - q^{-102} +2 q^{-103} + q^{-104} - q^{-106} - q^{-109} +2 q^{-110} - q^{-113} - q^{-116} + q^{-117} } |
| 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 q^{-14} + q^{-22} - q^{-29} + q^{-30} - q^{-37} + q^{-38} - q^{-45} + q^{-46} - q^{-50} - q^{-53} + q^{-54} - q^{-58} - q^{-61} + q^{-62} + q^{-63} - q^{-66} - q^{-69} + q^{-70} + q^{-71} - q^{-74} - q^{-77} + q^{-78} + q^{-79} + q^{-81} - q^{-82} - q^{-85} + q^{-86} + q^{-87} + q^{-89} - q^{-90} - q^{-92} - q^{-93} + q^{-94} + q^{-95} + q^{-97} - q^{-98} - q^{-100} - q^{-101} + q^{-102} + q^{-103} + q^{-105} - q^{-106} - q^{-107} - q^{-108} - q^{-109} + q^{-110} + q^{-111} + q^{-113} - q^{-114} - q^{-115} - q^{-117} + q^{-118} + q^{-119} + q^{-121} - q^{-122} - q^{-123} - q^{-125} + q^{-126} + q^{-127} + q^{-128} + q^{-129} - q^{-130} - q^{-131} - q^{-133} + q^{-134} + q^{-136} + q^{-137} - q^{-138} - q^{-139} - q^{-141} + q^{-142} + q^{-145} - q^{-146} - q^{-147} + q^{-150} + q^{-153} - q^{-154} } |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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