10 18: Difference between revisions
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<tr><td>j</td><td> </td><td>\</td></tr> |
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<td width=6.66667%>-6</td ><td width=6.66667%>-5</td ><td width=6.66667%>-4</td ><td width=6.66667%>-3</td ><td width=6.66667%>-2</td ><td width=6.66667%>-1</td ><td width=6.66667%>0</td ><td width=6.66667%>1</td ><td width=6.66667%>2</td ><td width=6.66667%>3</td ><td width=6.66667%>4</td ><td width=13.3333%>χ</td></tr> |
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<tr align=center><td>7</td><td> </td><td> </td><td> </td><td> </td><td> </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> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>-1</td></tr> |
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<tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td>2</td></tr> |
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<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td> </td><td>-2</td></tr> |
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<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td>2</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>5</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </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>4</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-7</td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> |
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<tr align=center><td>-9</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> |
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<tr align=center><td>-11</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> |
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<tr align=center><td>-13</td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</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> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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</table> | |
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coloured_jones_2 = <math>q^{10}-2 q^9+6 q^7-8 q^6-3 q^5+19 q^4-16 q^3-14 q^2+38 q-18-32 q^{-1} +55 q^{-2} -14 q^{-3} -50 q^{-4} +63 q^{-5} -6 q^{-6} -57 q^{-7} +57 q^{-8} + q^{-9} -48 q^{-10} +38 q^{-11} +4 q^{-12} -29 q^{-13} +19 q^{-14} +3 q^{-15} -12 q^{-16} +7 q^{-17} + q^{-18} -3 q^{-19} + q^{-20} </math> | |
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coloured_jones_3 = <math>q^{21}-2 q^{20}+2 q^{18}+3 q^{17}-7 q^{16}-4 q^{15}+10 q^{14}+12 q^{13}-19 q^{12}-19 q^{11}+21 q^{10}+39 q^9-27 q^8-56 q^7+19 q^6+81 q^5-7 q^4-100 q^3-17 q^2+117 q+44-122 q^{-1} -77 q^{-2} +124 q^{-3} +106 q^{-4} -116 q^{-5} -136 q^{-6} +107 q^{-7} +158 q^{-8} -92 q^{-9} -176 q^{-10} +78 q^{-11} +182 q^{-12} -56 q^{-13} -186 q^{-14} +43 q^{-15} +168 q^{-16} -20 q^{-17} -150 q^{-18} +8 q^{-19} +119 q^{-20} +3 q^{-21} -89 q^{-22} -6 q^{-23} +61 q^{-24} +4 q^{-25} -38 q^{-26} -2 q^{-27} +25 q^{-28} -2 q^{-29} -15 q^{-30} +2 q^{-31} +11 q^{-32} -3 q^{-33} -7 q^{-34} +2 q^{-35} +3 q^{-36} + q^{-37} -3 q^{-38} + q^{-39} </math> | |
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coloured_jones_4 = <math>q^{36}-2 q^{35}+2 q^{33}-q^{32}+4 q^{31}-9 q^{30}+10 q^{28}-2 q^{27}+12 q^{26}-31 q^{25}-8 q^{24}+31 q^{23}+9 q^{22}+37 q^{21}-77 q^{20}-46 q^{19}+48 q^{18}+40 q^{17}+118 q^{16}-120 q^{15}-127 q^{14}+10 q^{13}+48 q^{12}+267 q^{11}-91 q^{10}-187 q^9-101 q^8-52 q^7+407 q^6+29 q^5-127 q^4-202 q^3-275 q^2+425 q+158+74 q^{-1} -195 q^{-2} -534 q^{-3} +307 q^{-4} +205 q^{-5} +328 q^{-6} -75 q^{-7} -732 q^{-8} +127 q^{-9} +168 q^{-10} +548 q^{-11} +84 q^{-12} -846 q^{-13} -43 q^{-14} +95 q^{-15} +696 q^{-16} +230 q^{-17} -874 q^{-18} -184 q^{-19} +2 q^{-20} +756 q^{-21} +355 q^{-22} -790 q^{-23} -273 q^{-24} -125 q^{-25} +683 q^{-26} +439 q^{-27} -574 q^{-28} -266 q^{-29} -253 q^{-30} +474 q^{-31} +422 q^{-32} -305 q^{-33} -143 q^{-34} -295 q^{-35} +217 q^{-36} +291 q^{-37} -109 q^{-38} +8 q^{-39} -225 q^{-40} +47 q^{-41} +130 q^{-42} -39 q^{-43} +83 q^{-44} -112 q^{-45} -5 q^{-46} +32 q^{-47} -33 q^{-48} +70 q^{-49} -36 q^{-50} +2 q^{-52} -28 q^{-53} +32 q^{-54} -8 q^{-55} +5 q^{-56} + q^{-57} -13 q^{-58} +7 q^{-59} -2 q^{-60} +3 q^{-61} + q^{-62} -3 q^{-63} + q^{-64} </math> | |
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coloured_jones_5 = | |
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coloured_jones_6 = | |
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coloured_jones_7 = | |
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computer_talk = |
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<table> |
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<tr valign=top> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
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<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> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 18]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[15, 20, 16, 1], |
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X[9, 17, 10, 16], X[7, 19, 8, 18], X[17, 9, 18, 8], X[19, 7, 20, 6], |
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X[13, 10, 14, 11], X[11, 2, 12, 3]]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 18]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, |
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6, -8, 4]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 18]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 12, 14, 18, 16, 2, 10, 20, 8, 6]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 18]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[5, {-1, -1, -1, -2, 1, -2, 3, -2, 3, 4, -3, 4}]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{5, 12}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 18]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>5</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 18]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:10_18_ML.gif]]</td></tr><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 18]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 2, 2, NotAvailable, 1}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 18]][t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 4 14 2 |
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-19 - -- + -- + 14 t - 4 t |
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2 t |
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t</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 18]][z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 |
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1 - 2 z - 4 z</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 18], Knot[10, 24]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 18]], KnotSignature[Knot[10, 18]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{55, -2}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 18]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -7 3 5 7 9 9 8 2 3 |
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-6 + q - -- + -- - -- + -- - -- + - + 4 q - 2 q + q |
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6 5 4 3 2 q |
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q q q q q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 18]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 18]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -22 -20 -18 2 -14 -12 -10 -8 -6 2 -2 |
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q - q - q + --- - q + q + q - q + q - -- + q - |
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16 4 |
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q q |
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2 4 10 |
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q + 2 q + q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 18]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 |
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-2 2 4 2 z 2 2 6 2 4 2 4 4 4 |
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a - a + a - z + -- - 3 a z + a z - z - 2 a z - a z |
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2 |
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a</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 18]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 |
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-2 2 4 2 z 3 5 2 4 z 2 2 |
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-a + a + a - --- - 4 a z - 4 a z - 2 a z + z + ---- - 8 a z - |
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a 2 |
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a |
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3 |
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4 2 6 2 8 2 6 z 3 3 3 5 3 |
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3 a z + a z - a z + ---- + 11 a z + 14 a z + 5 a z - |
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a |
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4 5 |
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7 3 4 4 z 2 4 4 4 6 4 8 4 7 z |
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4 a z + z - ---- + 17 a z + 6 a z - 5 a z + a z - ---- - |
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2 a |
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a |
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6 |
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5 3 5 5 5 7 5 6 z 2 6 |
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10 a z - 12 a z - 6 a z + 3 a z - 5 z + -- - 15 a z - |
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2 |
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a |
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7 |
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4 6 6 6 2 z 7 3 7 5 7 8 |
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5 a z + 4 a z + ---- + a z + 3 a z + 4 a z + 2 z + |
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a |
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2 8 4 8 9 3 9 |
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5 a z + 3 a z + a z + a z</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 18]], Vassiliev[3][Knot[10, 18]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{-2, 1}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 18]][q, t]</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4 5 1 2 1 3 2 4 3 |
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-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + |
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3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 |
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q q t q t q t q t q t q t q t |
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5 4 4 5 3 t 2 3 2 3 3 |
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----- + ----- + ---- + ---- + --- + 3 q t + q t + 3 q t + q t + |
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7 2 5 2 5 3 q |
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q t q t q t q t |
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5 3 7 4 |
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q t + q t</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 18], 2][q]</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -20 3 -18 7 12 3 19 29 4 38 |
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-18 + q - --- + q + --- - --- + --- + --- - --- + --- + --- - |
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19 17 16 15 14 13 12 11 |
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q q q q q q q q |
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48 -9 57 57 6 63 50 14 55 32 2 |
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--- + q + -- - -- - -- + -- - -- - -- + -- - -- + 38 q - 14 q - |
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10 8 7 6 5 4 3 2 q |
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q q q q q q q q |
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3 4 5 6 7 9 10 |
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16 q + 19 q - 3 q - 8 q + 6 q - 2 q + q</nowiki></code></td></tr> |
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</table> }} |
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Revision as of 15:08, 10 April 2007
{{Rolfsen Knot Page| n = 10 | k = 18 | KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,-3,8,-6,7,-5,9,-10,2,-9,3,-4,5,-7,6,-8,4/goTop.html |
braid_table =
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braid_crossings = 12 | braid_width = 5 | braid_index = 5 | same_alexander = 10_24, | same_jones = |
khovanov_table =
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