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{{Knot Presentations}} |
{{Knot Presentations}} |
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<center><table border=1 cellpadding=10><tr align=center valign=top> |
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<td> |
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[[Braid Representatives|Minimum Braid Representative]]: |
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<table cellspacing=0 cellpadding=0 border=0> |
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<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]]</td></tr> |
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</table> |
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[[Invariants from Braid Theory|Length]] is 11, width is 4. |
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[[Invariants from Braid Theory|Braid index]] is 4. |
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</td> |
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<td> |
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[[Lightly Documented Features|A Morse Link Presentation]]: |
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[[Image:{{PAGENAME}}_ML.gif]] |
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</td> |
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</tr></table></center> |
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{{3D Invariants}} |
{{3D Invariants}} |
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{{4D Invariants}} |
{{4D Invariants}} |
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{{Polynomial Invariants}} |
{{Polynomial Invariants}} |
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=== "Similar" Knots (within the Atlas) === |
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Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
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{...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{...} |
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{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
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<tr align=center><td>-1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>-1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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</table>}} |
</table>}} |
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{{Display Coloured Jones|J2=<math>q^{28}-3 q^{27}+2 q^{26}+5 q^{25}-15 q^{24}+13 q^{23}+16 q^{22}-49 q^{21}+31 q^{20}+47 q^{19}-98 q^{18}+37 q^{17}+88 q^{16}-129 q^{15}+22 q^{14}+114 q^{13}-122 q^{12}-4 q^{11}+113 q^{10}-86 q^9-27 q^8+86 q^7-39 q^6-32 q^5+44 q^4-7 q^3-17 q^2+11 q+1-3 q^{-1} + q^{-2} </math>|J3=<math>q^{54}-3 q^{53}+2 q^{52}+q^{51}-4 q^{49}+6 q^{48}+3 q^{47}-18 q^{46}-2 q^{45}+44 q^{44}+7 q^{43}-92 q^{42}-30 q^{41}+161 q^{40}+86 q^{39}-244 q^{38}-173 q^{37}+306 q^{36}+307 q^{35}-358 q^{34}-437 q^{33}+356 q^{32}+571 q^{31}-324 q^{30}-668 q^{29}+256 q^{28}+733 q^{27}-175 q^{26}-752 q^{25}+78 q^{24}+737 q^{23}+24 q^{22}-696 q^{21}-119 q^{20}+615 q^{19}+220 q^{18}-526 q^{17}-280 q^{16}+389 q^{15}+338 q^{14}-270 q^{13}-328 q^{12}+131 q^{11}+301 q^{10}-38 q^9-222 q^8-38 q^7+155 q^6+53 q^5-78 q^4-53 q^3+34 q^2+34 q-10-17 q^{-1} +3 q^{-2} +5 q^{-3} + q^{-4} -3 q^{-5} + q^{-6} </math>|J4=<math>q^{88}-3 q^{87}+2 q^{86}+q^{85}-4 q^{84}+11 q^{83}-11 q^{82}+4 q^{81}-5 q^{80}-19 q^{79}+53 q^{78}-14 q^{77}+2 q^{76}-53 q^{75}-89 q^{74}+171 q^{73}+87 q^{72}+50 q^{71}-249 q^{70}-426 q^{69}+308 q^{68}+509 q^{67}+495 q^{66}-514 q^{65}-1405 q^{64}-34 q^{63}+1157 q^{62}+1830 q^{61}-203 q^{60}-2854 q^{59}-1404 q^{58}+1226 q^{57}+3742 q^{56}+1236 q^{55}-3775 q^{54}-3345 q^{53}+164 q^{52}+5079 q^{51}+3213 q^{50}-3517 q^{49}-4688 q^{48}-1478 q^{47}+5204 q^{46}+4652 q^{45}-2484 q^{44}-4930 q^{43}-2834 q^{42}+4445 q^{41}+5178 q^{40}-1267 q^{39}-4377 q^{38}-3697 q^{37}+3230 q^{36}+5058 q^{35}+11 q^{34}-3336 q^{33}-4202 q^{32}+1649 q^{31}+4390 q^{30}+1312 q^{29}-1798 q^{28}-4177 q^{27}-124 q^{26}+3007 q^{25}+2152 q^{24}+30 q^{23}-3224 q^{22}-1407 q^{21}+1115 q^{20}+1924 q^{19}+1354 q^{18}-1546 q^{17}-1508 q^{16}-361 q^{15}+834 q^{14}+1466 q^{13}-152 q^{12}-708 q^{11}-711 q^{10}-81 q^9+753 q^8+266 q^7-30 q^6-343 q^5-261 q^4+172 q^3+120 q^2+105 q-55-107 q^{-1} +14 q^{-2} +7 q^{-3} +36 q^{-4} +2 q^{-5} -20 q^{-6} +3 q^{-7} -3 q^{-8} +5 q^{-9} + q^{-10} -3 q^{-11} + q^{-12} </math>|J5=Not Available|J6=Not Available|J7=Not Available}} |
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{{Computer Talk Header}} |
{{Computer Talk Header}} |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
</tr> |
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<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August |
<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> |
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<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, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 15, 7], X[8, 14, 9, 13], |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 15, 7], X[8, 14, 9, 13], |
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X[2, 10, 3, 9], X[18, 12, 19, 11], X[16, 5, 17, 6], X[4, 17, 5, 18], |
X[2, 10, 3, 9], X[18, 12, 19, 11], X[16, 5, 17, 6], X[4, 17, 5, 18], |
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X[20, 16, 1, 15], X[12, 20, 13, 19]]</nowiki></pre></td></tr> |
X[20, 16, 1, 15], X[12, 20, 13, 19]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </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, 4, -3, 9, -7, 8, |
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-6, 10, -9]</nowiki></pre></td></tr> |
-6, 10, -9]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[10, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[10, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[6, 10, 16, 14, 2, 18, 8, 20, 4, 12]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {1, 1, 2, 2, -3, 2, 2, -1, 2, -3, 2}]</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {1, 1, 2, 2, -3, 2, 2, -1, 2, -3, 2}]</nowiki></pre></td></tr> |
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<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, 111]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr> |
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<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, 111]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr> |
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<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, 111]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_111_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> |
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<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, 111]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr> |
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<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, 111]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 9 17 2 3 |
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21 - -- + -- - -- - 17 t + 9 t - 2 t |
21 - -- + -- - -- - 17 t + 9 t - 2 t |
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3 2 t |
3 2 t |
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t t</nowiki></pre></td></tr> |
t t</nowiki></pre></td></tr> |
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<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, 111]][z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]][z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 |
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1 + z - 3 z - 2 z</nowiki></pre></td></tr> |
1 + z - 3 z - 2 z</nowiki></pre></td></tr> |
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<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> |
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<tr valign=top><td><pre |
<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> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 111]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{77, 4}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<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, 111]], KnotSignature[Knot[10, 111]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{77, 4}</nowiki></pre></td></tr> |
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<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, 111]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 3 4 5 6 7 8 9 |
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1 - 3 q + 7 q - 9 q + 12 q - 13 q + 12 q - 10 q + 6 q - 3 q + |
1 - 3 q + 7 q - 9 q + 12 q - 13 q + 12 q - 10 q + 6 q - 3 q + |
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10 |
10 |
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q</nowiki></pre></td></tr> |
q</nowiki></pre></td></tr> |
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<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> |
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<tr valign=top><td><pre |
<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> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 111]}</nowiki></pre></td></tr> |
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<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math> |
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<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, 111]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 8 10 12 14 18 20 22 |
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1 - q + q + 2 q - q + 4 q - q + q - 3 q + q - 3 q + |
1 - q + q + 2 q - q + 4 q - q + q - 3 q + q - 3 q + |
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24 26 28 30 |
24 26 28 30 |
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q + q - q + q</nowiki></pre></td></tr> |
q + q - q + q</nowiki></pre></td></tr> |
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<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, 111]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 2 4 4 4 4 |
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-8 3 2 -2 2 z 4 z z 2 z z 3 z 2 z z |
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a - -- + -- + a + ---- - ---- + -- + ---- + -- - ---- - ---- + -- - |
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6 4 8 6 4 2 8 6 4 2 |
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a a a a a a a a a a |
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6 6 |
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z z |
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-- - -- |
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6 4 |
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a a</nowiki></pre></td></tr> |
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<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, 111]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 |
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-8 3 2 -2 4 z 10 z 7 z z z z 3 z |
-8 3 2 -2 4 z 10 z 7 z z z z 3 z |
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a + -- + -- - a - --- - ---- - --- - -- - --- + --- - ---- - |
a + -- + -- - a - --- - ---- - --- - -- - --- + --- - ---- - |
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8 6 4 7 5 |
8 6 4 7 5 |
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a a a a a</nowiki></pre></td></tr> |
a a a a a</nowiki></pre></td></tr> |
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<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, 111]], Vassiliev[3][Knot[10, 111]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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, 111]], Vassiliev[3][Knot[10, 111]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{1, 0}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 |
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<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, 111]][q, t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 |
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3 5 1 2 q q 5 7 7 2 9 2 |
3 5 1 2 q q 5 7 7 2 9 2 |
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5 q + 3 q + ---- + --- + -- + 5 q t + 4 q t + 7 q t + 5 q t + |
5 q + 3 q + ---- + --- + -- + 5 q t + 4 q t + 7 q t + 5 q t + |
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Line 135: | Line 206: | ||
15 6 17 6 17 7 19 7 21 8 |
15 6 17 6 17 7 19 7 21 8 |
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2 q t + 4 q t + q t + 2 q t + q t</nowiki></pre></td></tr> |
2 q t + 4 q t + q t + 2 q t + q t</nowiki></pre></td></tr> |
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<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, 111], 2][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 3 2 3 4 5 6 7 |
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1 + q - - + 11 q - 17 q - 7 q + 44 q - 32 q - 39 q + 86 q - |
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q |
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8 9 10 11 12 13 14 |
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27 q - 86 q + 113 q - 4 q - 122 q + 114 q + 22 q - |
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15 16 17 18 19 20 21 |
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129 q + 88 q + 37 q - 98 q + 47 q + 31 q - 49 q + |
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22 23 24 25 26 27 28 |
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16 q + 13 q - 15 q + 5 q + 2 q - 3 q + q</nowiki></pre></td></tr> |
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</table> |
</table> |
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See/edit the [[Rolfsen_Splice_Template]]. |
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[[Category:Knot Page]] |
[[Category:Knot Page]] |
Revision as of 17:22, 29 August 2005
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Visit 10 111's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 111's page at Knotilus! Visit 10 111's page at the original Knot Atlas! |
10 111 Further Notes and Views
Knot presentations
Planar diagram presentation | X6271 X10,4,11,3 X14,8,15,7 X8,14,9,13 X2,10,3,9 X18,12,19,11 X16,5,17,6 X4,17,5,18 X20,16,1,15 X12,20,13,19 |
Gauss code | 1, -5, 2, -8, 7, -1, 3, -4, 5, -2, 6, -10, 4, -3, 9, -7, 8, -6, 10, -9 |
Dowker-Thistlethwaite code | 6 10 16 14 2 18 8 20 4 12 |
Conway Notation | [2.2.20.2] |
Length is 11, width is 4. Braid index is 4. |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
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1 | |
2 | |
3 | |
4 |
A2 Invariants.
Weight | Invariant |
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1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
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0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
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1,0,0,0 |
G2 Invariants.
Weight | Invariant |
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1,0 |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 111"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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Out[5]=
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In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 77, 4 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {...}
Same Jones Polynomial (up to mirroring, ): {...}
Vassiliev invariants
V2 and V3: | (1, 0) |
V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 4 is the signature of 10 111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
2 | |
3 | |
4 | |
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.
See/edit the Rolfsen_Splice_Template.