T(23,2): Difference between revisions
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{{Knot Navigation Links|prev=T(11,3).jpg|next=T(6,5).jpg}} |
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{{Torus Knot Page| |
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Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/23.2.html T(23,2)'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]! |
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m = 23 | |
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n = 2 | |
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===Knot presentations=== |
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braid_table = <table cellspacing=0 cellpadding=0 border=0> |
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{| |
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<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]]</td></tr> |
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|'''[[Planar Diagrams|Planar diagram presentation]]''' |
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<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]]</td></tr> |
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|style="padding-left: 1em;" | X<sub>9,33,10,32</sub> X<sub>33,11,34,10</sub> X<sub>11,35,12,34</sub> X<sub>35,13,36,12</sub> X<sub>13,37,14,36</sub> X<sub>37,15,38,14</sub> X<sub>15,39,16,38</sub> X<sub>39,17,40,16</sub> X<sub>17,41,18,40</sub> X<sub>41,19,42,18</sub> X<sub>19,43,20,42</sub> X<sub>43,21,44,20</sub> X<sub>21,45,22,44</sub> X<sub>45,23,46,22</sub> X<sub>23,1,24,46</sub> X<sub>1,25,2,24</sub> X<sub>25,3,26,2</sub> X<sub>3,27,4,26</sub> X<sub>27,5,28,4</sub> X<sub>5,29,6,28</sub> X<sub>29,7,30,6</sub> X<sub>7,31,8,30</sub> X<sub>31,9,32,8</sub> |
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|- |
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same_alexander = | |
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|'''[[Gauss Codes|Gauss code]]''' |
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same_jones = | |
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|style="padding-left: 1em;" | {-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15} |
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|- |
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|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]''' |
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|style="padding-left: 1em;" | 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22 |
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|} |
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===Polynomial invariants=== |
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{{Polynomial Invariants|name=T(23,2)}} |
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===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]=== |
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{| style="margin-left: 1em;" |
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|- |
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|'''V<sub>2</sub> and V<sub>3</sub>''' |
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|style="padding-left: 1em;" | {0, 506}) |
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|} |
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[[Khovanov Homology]]. 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>22 is the signature of T(23,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>. |
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<tr align=center> |
<tr align=center> |
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<td width=7.14286%><table cellpadding=0 cellspacing=0> |
<td width=7.14286%><table cellpadding=0 cellspacing=0> |
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<tr><td>\</td><td> </td><td>r</td></tr> |
<tr><td>\</td><td> </td><td>r</td></tr> |
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<tr><td> </td><td> \ </td><td> </td></tr> |
<tr><td> </td><td> \ </td><td> </td></tr> |
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<tr><td>j</td><td> </td><td>\</td></tr> |
<tr><td>j</td><td> </td><td>\</td></tr> |
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</table></td> |
</table></td> |
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<td width=3.57143%>0</td ><td width=3.57143%>1</td ><td width=3.57143%>2</td ><td width=3.57143%>3</td ><td width=3.57143%>4</td ><td width=3.57143%>5</td ><td width=3.57143%>6</td ><td width=3.57143%>7</td ><td width=3.57143%>8</td ><td width=3.57143%>9</td ><td width=3.57143%>10</td ><td width=3.57143%>11</td ><td width=3.57143%>12</td ><td width=3.57143%>13</td ><td width=3.57143%>14</td ><td width=3.57143%>15</td ><td width=3.57143%>16</td ><td width=3.57143%>17</td ><td width=3.57143%>18</td ><td width=3.57143%>19</td ><td width=3.57143%>20</td ><td width=3.57143%>21</td ><td width=3.57143%>22</td ><td width=3.57143%>23</td ><td width=7.14286%>χ</td></tr> |
<td width=3.57143%>0</td ><td width=3.57143%>1</td ><td width=3.57143%>2</td ><td width=3.57143%>3</td ><td width=3.57143%>4</td ><td width=3.57143%>5</td ><td width=3.57143%>6</td ><td width=3.57143%>7</td ><td width=3.57143%>8</td ><td width=3.57143%>9</td ><td width=3.57143%>10</td ><td width=3.57143%>11</td ><td width=3.57143%>12</td ><td width=3.57143%>13</td ><td width=3.57143%>14</td ><td width=3.57143%>15</td ><td width=3.57143%>16</td ><td width=3.57143%>17</td ><td width=3.57143%>18</td ><td width=3.57143%>19</td ><td width=3.57143%>20</td ><td width=3.57143%>21</td ><td width=3.57143%>22</td ><td width=3.57143%>23</td ><td width=7.14286%>χ</td></tr> |
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<tr align=center><td>69</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </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> |
<tr align=center><td>69</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </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>67</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
<tr align=center><td>67</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
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<tr align=center><td>23</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> </td><td> </td><td> </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>23</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> </td><td> </td><td> </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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<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> </td><td> </td><td> </td><td> </td><td> </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>21</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> </td><td> </td><td> </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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coloured_jones_2 = | |
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coloured_jones_3 = | |
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{{Computer Talk Header}} |
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coloured_jones_4 = | |
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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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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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computer_talk = |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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<table> |
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</tr> |
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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><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[TorusKnot[23, 2]]</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>23</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>TubePlot[TorusKnot[23, 2]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:T(23,2).jpg]]</td></tr><tr valign=top><td><tt><font color=blue>Out[3]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
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X[35, 13, 36, 12], X[13, 37, 14, 36], X[37, 15, 38, 14], |
X[35, 13, 36, 12], X[13, 37, 14, 36], X[37, 15, 38, 14], |
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X[5, 29, 6, 28], X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]</nowiki></pre></td></tr> |
X[5, 29, 6, 28], X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]</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[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[TorusKnot[23, 2]]</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[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, |
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8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, |
8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, |
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1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]</nowiki></pre></td></tr> |
1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]</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[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[TorusKnot[23, 2]]</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[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, |
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1, 1}]</nowiki></pre></td></tr> |
1, 1}]</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[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[TorusKnot[23, 2]][t]</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[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 1 |
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-1 + t - t + t - t + t - t + t - t + t - t + - + |
-1 + t - t + t - t + t - t + t - t + t - t + - + |
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t |
t |
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Line 113: | Line 102: | ||
2 3 4 5 6 7 8 9 10 11 |
2 3 4 5 6 7 8 9 10 11 |
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t - t + t - t + t - t + t - t + t - t + t</nowiki></pre></td></tr> |
t - t + t - t + t - t + t - t + t - t + t</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[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[TorusKnot[23, 2]][z]</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[8]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 8 10 12 |
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1 + 66 z + 715 z + 3003 z + 6435 z + 8008 z + 6188 z + |
1 + 66 z + 715 z + 3003 z + 6435 z + 8008 z + 6188 z + |
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14 16 18 20 22 |
14 16 18 20 22 |
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3060 z + 969 z + 190 z + 21 z + z</nowiki></pre></td></tr> |
3060 z + 969 z + 190 z + 21 z + z</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[9]:=</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[ |
<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>{}</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[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[TorusKnot[23, 2]], KnotSignature[TorusKnot[23, 2]]}</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[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{23, 22}</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[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[TorusKnot[23, 2]][q]</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[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 11 13 14 15 16 17 18 19 20 21 22 23 |
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q + q - q + q - q + q - q + q - q + q - q + q - |
q + q - q + q - q + q - q + q - q + q - q + q - |
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24 25 26 27 28 29 30 31 32 33 34 |
24 25 26 27 28 29 30 31 32 33 34 |
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q + q - q + q - q + q - q + q - q + q - q</nowiki></pre></td></tr> |
q + q - q + 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[ |
<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[], (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[ |
<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>{}</nowiki></pre></td></tr> |
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⚫ | |||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>NotAvailable</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>Kauffman[TorusKnot[23, 2]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>NotAvailable</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>{Vassiliev[2][TorusKnot[23, 2]], Vassiliev[3][TorusKnot[23, 2]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>{66, 506}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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>Kh[TorusKnot[23, 2]][q, t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki> |
<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> 21 23 25 2 29 3 29 4 33 5 33 6 37 7 |
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⚫ | |||
q + q + q t + q t + q t + q t + q t + q t + |
q + q + q t + q t + q t + q t + q t + q t + |
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Line 150: | Line 138: | ||
65 22 69 23 |
65 22 69 23 |
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q t + q t</nowiki></pre></td></tr> |
q t + q t</nowiki></pre></td></tr> |
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</table> |
</table> }} |
Latest revision as of 10:37, 31 August 2005
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See other torus knots | |
Edit T(23,2) Quick Notes
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Edit T(23,2) Further Notes and Views
Knot presentations
Planar diagram presentation | X9,33,10,32 X33,11,34,10 X11,35,12,34 X35,13,36,12 X13,37,14,36 X37,15,38,14 X15,39,16,38 X39,17,40,16 X17,41,18,40 X41,19,42,18 X19,43,20,42 X43,21,44,20 X21,45,22,44 X45,23,46,22 X23,1,24,46 X1,25,2,24 X25,3,26,2 X3,27,4,26 X27,5,28,4 X5,29,6,28 X29,7,30,6 X7,31,8,30 X31,9,32,8 |
Gauss code | -16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15 |
Dowker-Thistlethwaite code | 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22 |
Braid presentation |
Polynomial invariants
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["T(23,2)"];
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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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{ 23, 22 } |
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, ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["T(23,2)"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ , } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
V2 and V3: | (66, 506) |
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 22 is the signature of T(23,2). 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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Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top. |
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