10 154: Difference between revisions
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{{Template:Basic Knot Invariants|name=10_154}} |
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{{Knot Navigation Links|ext=gif}} |
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|{{Rolfsen Knot Site Links|n=10|k=154|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-9,10,-2,-4,8,9,-3,-6,7,-8,4,-5,6,-7,5/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=13.3333%><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=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=6.66667%>5</td ><td width=6.66667%>6</td ><td width=6.66667%>7</td ><td width=6.66667%>8</td ><td width=6.66667%>9</td ><td width=6.66667%>10</td ><td width=13.3333%>χ</td></tr> |
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<tr align=center><td>25</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>23</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>21</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>1</td><td> </td><td>0</td></tr> |
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<tr align=center><td>19</td><td> </td><td> </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>17</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>-1</td></tr> |
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<tr align=center><td>15</td><td> </td><td> </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>13</td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>11</td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>2</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>1</td><td bgcolor=yellow>1</td><td> </td><td> </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 bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow> </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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<tr align=center><td>5</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </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></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, 154]]</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, 154]]</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[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[9, 17, 10, 16], |
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X[17, 1, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], |
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X[15, 11, 16, 10], X[6, 12, 7, 11], X[2, 8, 3, 7]]</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, 154]]</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, -4, 8, 9, -3, -6, 7, -8, 4, -5, |
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6, -7, 5]</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, 154]]</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, -1, 2, 1, 3, 2, 2, 2, 3}]</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, 154]][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 3 |
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7 + t - - - 4 t + 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, 154]][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 + 5 z + 6 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, 154]}</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, 154]], KnotSignature[Knot[10, 154]]}</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>{13, 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, 154]][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> 3 6 7 8 9 10 11 12 |
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q + 2 q - 2 q + 2 q - 3 q + 2 q - 2 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, 154]}</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, 154]][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> 10 12 14 16 18 22 24 26 28 30 |
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q + q + q + 2 q + 2 q + q - q - q - 2 q - 2 q - |
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34 36 38 |
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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, 154]][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> 2 2 2 |
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-12 2 2 4 4 z 10 z 3 z 3 z 3 z 2 z 5 z |
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a + --- - -- - -- - --- - ---- - --- + --- + ---- + ---- - ---- + |
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10 8 6 13 11 9 7 14 12 10 |
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a a a a a a a a a a |
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2 2 3 3 3 3 4 4 4 |
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5 z 9 z 10 z 21 z 9 z 2 z 4 z z 7 z |
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---- + ---- + ----- + ----- + ---- - ---- - ---- - --- + ---- - |
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8 6 13 11 9 7 14 12 10 |
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a a a a a a a a a |
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4 4 5 5 5 6 6 6 6 7 |
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2 z 6 z 9 z 15 z 6 z z 3 z 5 z z 2 z |
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---- - ---- - ---- - ----- - ---- + --- - ---- - ---- + -- + ---- + |
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8 6 13 11 9 14 12 10 6 13 |
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a a a a a a a a a a |
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7 7 8 8 |
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3 z z z z |
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---- + -- + --- + --- |
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11 9 12 10 |
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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, 154]], Vassiliev[3][Knot[10, 154]]}</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, 9}</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, 154]][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 7 9 2 9 3 13 3 11 4 13 4 13 5 |
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q + q + q t + q t + q t + 2 q t + 2 q t + q t + |
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15 5 17 5 15 6 17 6 17 7 19 7 19 8 |
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2 q t + q t + 2 q t + q t + q t + 2 q t + q t + |
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21 8 21 9 23 9 25 10 |
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q t + q t + q t + q t</nowiki></pre></td></tr> |
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</table> |
Revision as of 20:45, 27 August 2005
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Visit 10 154's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 154's page at Knotilus! Visit 10 154's page at the original Knot Atlas! |
10 154 Quick Notes |
10 154 Further Notes and Views
Knot presentations
Planar diagram presentation | X4251 X8493 X12,6,13,5 X9,17,10,16 X17,1,18,20 X13,19,14,18 X19,15,20,14 X15,11,16,10 X6,12,7,11 X2837 |
Gauss code | 1, -10, 2, -1, 3, -9, 10, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5 |
Dowker-Thistlethwaite code | 4 8 12 2 -16 6 -18 -10 -20 -14 |
Conway Notation | [(21,2)-(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 |
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 |
.
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["10 154"];
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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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{ 13, 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: | (5, 9) |
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 10 154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | χ | |||||||||
25 | 1 | 1 | |||||||||||||||||||
23 | 1 | -1 | |||||||||||||||||||
21 | 1 | 1 | 0 | ||||||||||||||||||
19 | 2 | 1 | -1 | ||||||||||||||||||
17 | 1 | 1 | 1 | -1 | |||||||||||||||||
15 | 2 | 2 | 0 | ||||||||||||||||||
13 | 1 | 2 | 1 | 0 | |||||||||||||||||
11 | 2 | 2 | |||||||||||||||||||
9 | 1 | 1 | 0 | ||||||||||||||||||
7 | 1 | 1 | |||||||||||||||||||
5 | 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, 154]] |
Out[2]= | 10 |
In[3]:= | PD[Knot[10, 154]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[9, 17, 10, 16],X[17, 1, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14],X[15, 11, 16, 10], X[6, 12, 7, 11], X[2, 8, 3, 7]] |
In[4]:= | GaussCode[Knot[10, 154]] |
Out[4]= | GaussCode[1, -10, 2, -1, 3, -9, 10, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5] |
In[5]:= | BR[Knot[10, 154]] |
Out[5]= | BR[4, {1, 1, 2, -1, 2, 1, 3, 2, 2, 2, 3}] |
In[6]:= | alex = Alexander[Knot[10, 154]][t] |
Out[6]= | -3 4 3 |
In[7]:= | Conway[Knot[10, 154]][z] |
Out[7]= | 2 4 6 1 + 5 z + 6 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 154]} |
In[9]:= | {KnotDet[Knot[10, 154]], KnotSignature[Knot[10, 154]]} |
Out[9]= | {13, 4} |
In[10]:= | J=Jones[Knot[10, 154]][q] |
Out[10]= | 3 6 7 8 9 10 11 12 q + 2 q - 2 q + 2 q - 3 q + 2 q - 2 q + q |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[10, 154]} |
In[12]:= | A2Invariant[Knot[10, 154]][q] |
Out[12]= | 10 12 14 16 18 22 24 26 28 30 |
In[13]:= | Kauffman[Knot[10, 154]][a, z] |
Out[13]= | 2 2 2-12 2 2 4 4 z 10 z 3 z 3 z 3 z 2 z 5 z |
In[14]:= | {Vassiliev[2][Knot[10, 154]], Vassiliev[3][Knot[10, 154]]} |
Out[14]= | {0, 9} |
In[15]:= | Kh[Knot[10, 154]][q, t] |
Out[15]= | 5 7 9 2 9 3 13 3 11 4 13 4 13 5 |