10 127: Difference between revisions
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{{Template:Basic Knot Invariants|name=10_127}} |
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{{Knot Navigation Links|ext=gif}} |
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|{{Rolfsen Knot Site Links|n=10|k=127|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,3,-9,-10,2,-5,7,-6,8,9,-3,-4,5,-7,6,-8,4/goTop.html}} |
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|{{:{{PAGENAME}} Quick Notes}} |
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<br style="clear:both" /> |
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{{:{{PAGENAME}} Further Notes and Views}} |
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{{Knot Presentations}} |
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{{3D Invariants}} |
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{{4D Invariants}} |
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{{Polynomial Invariants}} |
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{{Vassiliev Invariants}} |
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===[[Khovanov Homology]]=== |
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The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>. |
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<center><table border=1> |
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<tr align=center> |
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<td width=15.3846%><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=7.69231%>-8</td ><td width=7.69231%>-7</td ><td width=7.69231%>-6</td ><td width=7.69231%>-5</td ><td width=7.69231%>-4</td ><td width=7.69231%>-3</td ><td width=7.69231%>-2</td ><td width=7.69231%>-1</td ><td width=7.69231%>0</td ><td width=15.3846%>χ</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>2</td><td>2</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 bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</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 bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td>2</td></tr> |
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<tr align=center><td>-9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td>-1</td></tr> |
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<tr align=center><td>-11</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-13</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-15</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> |
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<tr align=center><td>-17</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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<tr align=center><td>-19</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
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<tr align=center><td>-21</td><td bgcolor=yellow>1</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></center> |
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{{Computer Talk Header}} |
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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><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</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, 127]]</nowiki></pre></td></tr> |
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<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>10</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 127]]</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>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1], |
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X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], |
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X[19, 12, 20, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]]</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, 127]]</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>GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, |
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6, -8, 4]</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, 127]]</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[3, {-1, -1, -1, -1, -1, -2, 1, 1, -2, -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, 127]][t]</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> -3 4 6 2 3 |
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7 - t + -- - - - 6 t + 4 t - t |
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2 t |
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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, 127]][z]</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> 2 4 6 |
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1 + z - 2 z - 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 style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}</nowiki></pre></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>{KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]}</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>{29, -4}</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>J=Jones[Knot[10, 127]][q]</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> -10 2 3 5 5 5 4 2 2 |
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q - -- + -- - -- + -- - -- + -- - -- + -- |
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9 8 7 6 5 4 3 2 |
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q q q q 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[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 style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 127]}</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, 127]][q]</nowiki></pre></td></tr> |
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<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> -30 -26 2 -20 3 -12 3 -8 2 |
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q + q - --- - q - --- + q + --- + q + -- |
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22 18 10 6 |
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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, 127]][a, z]</nowiki></pre></td></tr> |
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<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> 4 6 8 5 7 9 11 4 2 |
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5 a + 6 a + 2 a - 5 a z - 8 a z - 2 a z + a z - 9 a z - |
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6 2 8 2 10 2 12 2 5 3 7 3 |
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14 a z - 2 a z + a z - 2 a z + 5 a z + 16 a z + |
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9 3 11 3 4 4 6 4 8 4 10 4 |
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7 a z - 4 a z + 3 a z + 11 a z + 4 a z - 3 a z + |
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12 4 5 5 7 5 9 5 11 5 6 6 |
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a z - 3 a z - 10 a z - 5 a z + 2 a z - 4 a z - |
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8 6 10 6 5 7 7 7 9 7 6 8 8 8 |
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2 a z + 2 a z + a z + 3 a z + 2 a z + a z + a z</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, 127]], Vassiliev[3][Knot[10, 127]]}</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>{0, 1}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 127]][q, t]</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> -5 2 1 1 1 2 1 3 |
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q + -- + ------ + ------ + ------ + ------ + ------ + ------ + |
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3 21 8 19 7 17 7 17 6 15 6 15 5 |
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q q t q t q t q t q t q t |
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2 2 3 3 2 1 3 1 |
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------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + |
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13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 |
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q t q t q t q t q t q t q t q t |
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1 |
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---- |
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5 |
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q t</nowiki></pre></td></tr> |
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</table> |
Revision as of 20:46, 27 August 2005
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Visit 10 127's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 127's page at Knotilus! Visit 10 127's page at the original Knot Atlas! |
10 127 Quick Notes |
10 127 Further Notes and Views
Knot presentations
Planar diagram presentation | X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X6,14,7,13 X7283 |
Gauss code | -1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4 |
Dowker-Thistlethwaite code | 4 8 -14 2 16 18 -6 20 10 12 |
Conway Notation | [41,21,2-] |
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 | |
5 | |
6 |
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 | |
1,0,1 |
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 |
.
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["10 127"];
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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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{ 29, -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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Vassiliev invariants
V2 and V3: | (1, 1) |
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 10 127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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-8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | χ | |||||||||
-3 | 2 | 2 | |||||||||||||||||
-5 | 1 | 1 | 0 | ||||||||||||||||
-7 | 3 | 1 | 2 | ||||||||||||||||
-9 | 2 | 1 | -1 | ||||||||||||||||
-11 | 3 | 3 | 0 | ||||||||||||||||
-13 | 2 | 2 | 0 | ||||||||||||||||
-15 | 1 | 3 | -2 | ||||||||||||||||
-17 | 1 | 2 | 1 | ||||||||||||||||
-19 | 1 | -1 | |||||||||||||||||
-21 | 1 | 1 |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 17, 2005, 14:44:34)... | |
In[2]:= | Crossings[Knot[10, 127]] |
Out[2]= | 10 |
In[3]:= | PD[Knot[10, 127]] |
Out[3]= | PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1],X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11],X[19, 12, 20, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]] |
In[4]:= | GaussCode[Knot[10, 127]] |
Out[4]= | GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4] |
In[5]:= | BR[Knot[10, 127]] |
Out[5]= | BR[3, {-1, -1, -1, -1, -1, -2, 1, 1, -2, -2}] |
In[6]:= | alex = Alexander[Knot[10, 127]][t] |
Out[6]= | -3 4 6 2 3 |
In[7]:= | Conway[Knot[10, 127]][z] |
Out[7]= | 2 4 6 1 + z - 2 z - z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]} |
In[9]:= | {KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]} |
Out[9]= | {29, -4} |
In[10]:= | J=Jones[Knot[10, 127]][q] |
Out[10]= | -10 2 3 5 5 5 4 2 2 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[10, 127]} |
In[12]:= | A2Invariant[Knot[10, 127]][q] |
Out[12]= | -30 -26 2 -20 3 -12 3 -8 2 |
In[13]:= | Kauffman[Knot[10, 127]][a, z] |
Out[13]= | 4 6 8 5 7 9 11 4 2 |
In[14]:= | {Vassiliev[2][Knot[10, 127]], Vassiliev[3][Knot[10, 127]]} |
Out[14]= | {0, 1} |
In[15]:= | Kh[Knot[10, 127]][q, t] |
Out[15]= | -5 2 1 1 1 2 1 3 |