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{{Knot Navigation Links|prev=T(3,2)|next=T(7,2)|imageext=jpg}}
{{Knot Navigation Links|ext=jpg}}


{| align=left
{| align=left
|- valign=top
|- valign=top
|[[Image:T(5,2).jpg]]
|[[Image:{{PAGENAME}}.jpg]]
|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-1,2,-3,4,-5,1,-2,3/goTop.html T(5,2)'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-1,2,-3,4,-5,1,-2,3/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!


Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/5.2.html T(5,2)'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/5.2.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!


{{:T(5,2) Quick Notes}}
{{:{{PAGENAME}} Quick Notes}}
|}
|}


<br style="clear:both" />
<br style="clear:both" />


{{:T(5,2) Further Notes and Views}}
{{:{{PAGENAME}} Further Notes and Views}}

{{Knot Presentations}}


===Knot presentations===
===Knot presentations===
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|-
|-
|'''[[Gauss Codes|Gauss code]]'''
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {-4, 5, -1, 2, -3, 4, -5, 1, -2, 3}
|style="padding-left: 1em;" | <math>\{-4,5,-1,2,-3,4,-5,1,-2,3\}</math>
|-
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
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{{Polynomial Invariants|name=T(5,2)}}
{{Polynomial Invariants}}
{{Vassiliev Invariants}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V<sub>2</sub> and V<sub>3</sub>'''
|style="padding-left: 1em;" | {0, 5}
|}


===[[Khovanov Homology]]===
===[[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>4 is the signature of T(5,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
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>.


<center><table border=1>
<center><table border=1>
<tr align=center>
<tr align=center>
<td width=20.%><table cellpadding=0 cellspacing=0>
<td width=20.%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
</table></td>
<td width=10.%>0</td ><td width=10.%>1</td ><td width=10.%>2</td ><td width=10.%>3</td ><td width=10.%>4</td ><td width=10.%>5</td ><td width=20.%>&chi;</td></tr>
<td width=10.%>0</td ><td width=10.%>1</td ><td width=10.%>2</td ><td width=10.%>3</td ><td width=10.%>4</td ><td width=10.%>5</td ><td width=20.%>&chi;</td></tr>
<tr align=center><td>15</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>15</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>0</td></tr>
<tr align=center><td>13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>0</td></tr>
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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 19, 2005, 13:11:25)...</pre></td></tr>
<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>
<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[5, 2]]</nowiki></pre></td></tr>
<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[5, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[3, 9, 4, 8], X[9, 5, 10, 4], X[5, 1, 6, 10], X[1, 7, 2, 6],
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[3, 9, 4, 8], X[9, 5, 10, 4], X[5, 1, 6, 10], X[1, 7, 2, 6],
X[7, 3, 8, 2]]</nowiki></pre></td></tr>
X[7, 3, 8, 2]]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-4, 5, -1, 2, -3, 4, -5, 1, -2, 3]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-4, 5, -1, 2, -3, 4, -5, 1, -2, 3]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1}]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1}]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][t]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 1 2
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 1 2
1 + t - - - t + t
1 + t - - - t + t
t</nowiki></pre></td></tr>
t</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][z]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4
1 + 3 z + z</nowiki></pre></td></tr>
1 + 3 z + z</nowiki></pre></td></tr>
<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>
<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>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]], KnotSignature[TorusKnot[5, 2]]}</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]], KnotSignature[TorusKnot[5, 2]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{5, 4}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{5, 4}</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][q]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 5 6 7
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 5 6 7
q + q - q + q - q</nowiki></pre></td></tr>
q + q - q + q - q</nowiki></pre></td></tr>
<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>
<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>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki></pre></td></tr>
Include[ColouredJonesM.mhtml]
<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
<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[TorusKnot[5, 2]][q]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6 8 10 12 14 18 20 22
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6 8 10 12 14 18 20 22
q + q + 2 q + q + q - q - q - q</nowiki></pre></td></tr>
q + q + 2 q + q + q - q - q - q</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][a, z]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 3 3 4 4
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 3 3 4 4
2 3 z z 2 z z 3 z 4 z z z z z
2 3 z z 2 z z 3 z 4 z z z z z
-- + -- + -- - -- - --- + -- - ---- - ---- + -- + -- + -- + --
-- + -- + -- - -- - --- + -- - ---- - ---- + -- + -- + -- + --
6 4 9 7 5 8 6 4 7 5 6 4
6 4 9 7 5 8 6 4 7 5 6 4
a a a a a a a a a a a a</nowiki></pre></td></tr>
a a a a a a a a a a a a</nowiki></pre></td></tr>
<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][TorusKnot[5, 2]], Vassiliev[3][TorusKnot[5, 2]]}</nowiki></pre></td></tr>
<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][TorusKnot[5, 2]], Vassiliev[3][TorusKnot[5, 2]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 5}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 5}</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][q, t]</nowiki></pre></td></tr>
<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[TorusKnot[5, 2]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 5 7 2 11 3 11 4 15 5
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 5 7 2 11 3 11 4 15 5
q + q + q t + q t + q t + q t</nowiki></pre></td></tr>
q + q + q t + q t + q t + q t</nowiki></pre></td></tr>
</table>
</table>

Revision as of 20:24, 27 August 2005


T(3,2).jpg

T(3,2)

T(7,2).jpg

T(7,2)

T(5,2).jpg Visit T(5,2)'s page at Knotilus!

Visit T(5,2)'s page at the original Knot Atlas!

An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).

When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1




A kolam of a 2x3 dot array
The VISA Interlink Logo [1]
Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]
The Utah State Parks logo
As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).
Non-pentagonal shape.
Pentagram of circles.
Alternate pentagram of intersecting circles.
3D-looking rendition.
Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.
Knotted epitrochoid
Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation X3948 X9,5,10,4 X5,1,6,10 X1726 X7382
Gauss code -4, 5, -1, 2, -3, 4, -5, 1, -2, 3
Dowker-Thistlethwaite code 6 8 10 2 4
Conway Notation Data:T(5,2)/Conway Notation

Knot presentations

Planar diagram presentation X3948 X9,5,10,4 X5,1,6,10 X1726 X7382
Gauss code
Dowker-Thistlethwaite code 6 8 10 2 4

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 5, 4 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:T(5,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(5,2)/QuantumInvariant/G2/1,0

Vassiliev invariants

V2 and V3: (3, 5)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:T(5,2)/V 2,1 Data:T(5,2)/V 3,1 Data:T(5,2)/V 4,1 Data:T(5,2)/V 4,2 Data:T(5,2)/V 4,3 Data:T(5,2)/V 5,1 Data:T(5,2)/V 5,2 Data:T(5,2)/V 5,3 Data:T(5,2)/V 5,4 Data:T(5,2)/V 6,1 Data:T(5,2)/V 6,2 Data:T(5,2)/V 6,3 Data:T(5,2)/V 6,4 Data:T(5,2)/V 6,5 Data:T(5,2)/V 6,6 Data:T(5,2)/V 6,7 Data:T(5,2)/V 6,8 Data:T(5,2)/V 6,9

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 T(5,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
012345χ
15     1-1
13      0
11   11 0
9      0
7  1   1
51     1
31     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[TorusKnot[5, 2]]
Out[2]=  
5
In[3]:=
PD[TorusKnot[5, 2]]
Out[3]=  
PD[X[3, 9, 4, 8], X[9, 5, 10, 4], X[5, 1, 6, 10], X[1, 7, 2, 6], 
  X[7, 3, 8, 2]]
In[4]:=
GaussCode[TorusKnot[5, 2]]
Out[4]=  
GaussCode[-4, 5, -1, 2, -3, 4, -5, 1, -2, 3]
In[5]:=
BR[TorusKnot[5, 2]]
Out[5]=  
BR[2, {1, 1, 1, 1, 1}]
In[6]:=
alex = Alexander[TorusKnot[5, 2]][t]
Out[6]=  
     -2   1        2

1 + t - - - t + t

t
In[7]:=
Conway[TorusKnot[5, 2]][z]
Out[7]=  
       2    4
1 + 3 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[5, 1], Knot[10, 132]}
In[9]:=
{KnotDet[TorusKnot[5, 2]], KnotSignature[TorusKnot[5, 2]]}
Out[9]=  
{5, 4}
In[10]:=
J=Jones[TorusKnot[5, 2]][q]
Out[10]=  
 2    4    5    6    7
q  + q  - q  + q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[5, 1], Knot[10, 132]}
In[12]:=
A2Invariant[TorusKnot[5, 2]][q]
Out[12]=  
 6    8      10    12    14    18    20    22
q  + q  + 2 q   + q   + q   - q   - q   - q
In[13]:=
Kauffman[TorusKnot[5, 2]][a, z]
Out[13]=  
                           2      2      2    3    3    4    4

2 3 z z 2 z z 3 z 4 z z z z z -- + -- + -- - -- - --- + -- - ---- - ---- + -- + -- + -- + --

6    4    9    7    5     8     6      4     7    5    6    4
a a a a a a a a a a a a
In[14]:=
{Vassiliev[2][TorusKnot[5, 2]], Vassiliev[3][TorusKnot[5, 2]]}
Out[14]=  
{0, 5}
In[15]:=
Kh[TorusKnot[5, 2]][q, t]
Out[15]=  
 3    5    7  2    11  3    11  4    15  5
q  + q  + q  t  + q   t  + q   t  + q   t