T(15,2): Difference between revisions
No edit summary |
No edit summary |
||
Line 1: | Line 1: | ||
<!-- --> |
<!-- --> |
||
<!-- This knot page was produced from [[Torus Knots Splice Template]] --> |
|||
<!-- --> |
<!-- --> |
||
<!-- --> |
|||
<span id="top"></span> |
<span id="top"></span> |
||
<!-- --> |
|||
{{Knot Navigation Links|ext=jpg}} |
{{Knot Navigation Links|ext=jpg}} |
||
⚫ | |||
{| align=left |
|||
|- valign=top |
|||
|[[Image:{{PAGENAME}}.jpg]] |
|||
⚫ | |||
{{:{{PAGENAME}} Quick Notes}} |
|||
|} |
|||
<br style="clear:both" /> |
<br style="clear:both" /> |
||
Line 23: | Line 17: | ||
{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
||
{{Khovanov Homology|table=<table border=1> |
|||
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> |
|||
<tr align=center> |
<tr align=center> |
||
<td width=10.%><table cellpadding=0 cellspacing=0> |
<td width=10.%><table cellpadding=0 cellspacing=0> |
||
Line 52: | Line 42: | ||
<tr align=center><td>15</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>1</td></tr> |
<tr align=center><td>15</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>1</td></tr> |
||
<tr align=center><td>13</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>1</td></tr> |
<tr align=center><td>13</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>1</td></tr> |
||
</table> |
</table>}} |
||
{{Computer Talk Header}} |
{{Computer Talk Header}} |
||
Line 81: | Line 71: | ||
<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[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]</nowiki></pre></td></tr> |
<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[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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[15, 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[15, 2]][t]</nowiki></pre></td></tr> |
||
<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> |
<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> -7 -6 -5 -4 |
||
-1 + |
-1 + Alternating - Alternating + Alternating - Alternating + |
||
⚫ | |||
⚫ | |||
6 7 |
|||
Alternating - Alternating + ----------- + Alternating - |
|||
⚫ | |||
Alternating |
|||
2 3 4 5 |
|||
Alternating + Alternating - Alternating + Alternating - |
|||
6 7 |
|||
⚫ | |||
<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[15, 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[15, 2]][z]</nowiki></pre></td></tr> |
||
<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 8 10 12 14 |
<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 8 10 12 14 |
||
Line 157: | Line 153: | ||
<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, 140}</nowiki></pre></td></tr> |
<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, 140}</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[15, 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[15, 2]][q, t]</nowiki></pre></td></tr> |
||
<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> 13 15 |
<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> 13 15 2 17 3 21 4 21 |
||
q + q + |
q + q + Alternating q + Alternating q + Alternating q + |
||
5 25 6 25 7 29 |
|||
Alternating q + Alternating q + Alternating q + |
|||
8 29 9 33 10 33 |
|||
Alternating q + Alternating q + Alternating q + |
|||
11 37 12 37 13 41 |
|||
Alternating q + Alternating q + Alternating q + |
|||
14 41 15 45 |
|||
45 15 |
|||
q |
Alternating q + Alternating q</nowiki></pre></td></tr> |
||
</table> |
</table> |
||
[[Category:Knot Page]] |
Revision as of 20:44, 28 August 2005
|
|
![]() |
Visit [[[:Template:KnotilusURL]] T(15,2)'s page] at Knotilus!
Visit T(15,2)'s page at the original Knot Atlas! |
T(15,2) Quick Notes |
T(15,2) Further Notes and Views
Knot presentations
Planar diagram presentation | X9,25,10,24 X25,11,26,10 X11,27,12,26 X27,13,28,12 X13,29,14,28 X29,15,30,14 X15,1,16,30 X1,17,2,16 X17,3,18,2 X3,19,4,18 X19,5,20,4 X5,21,6,20 X21,7,22,6 X7,23,8,22 X23,9,24,8 |
Gauss code | -8, 9, -10, 11, -12, 13, -14, 15, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 1, -2, 3, -4, 5, -6, 7 |
Dowker-Thistlethwaite code | 16 18 20 22 24 26 28 30 2 4 6 8 10 12 14 |
Conway Notation | Data:T(15,2)/Conway Notation |
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["T(15,2)"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
In[6]:=
|
Alexander[K, 2][t]
|
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.
|
Out[6]=
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 15, 14 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
Vassiliev invariants
V2 and V3: | (28, 140) |
V2,1 through V6,9: |
|
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 14 is the signature of T(15,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
Integral Khovanov Homology
(db, data source) |
|
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[15, 2]] |
Out[2]= | 15 |
In[3]:= | PD[TorusKnot[15, 2]] |
Out[3]= | PD[X[9, 25, 10, 24], X[25, 11, 26, 10], X[11, 27, 12, 26],X[27, 13, 28, 12], X[13, 29, 14, 28], X[29, 15, 30, 14], X[15, 1, 16, 30], X[1, 17, 2, 16], X[17, 3, 18, 2], X[3, 19, 4, 18], X[19, 5, 20, 4], X[5, 21, 6, 20], X[21, 7, 22, 6], X[7, 23, 8, 22],X[23, 9, 24, 8]] |
In[4]:= | GaussCode[TorusKnot[15, 2]] |
Out[4]= | GaussCode[-8, 9, -10, 11, -12, 13, -14, 15, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 1, -2, 3, -4, 5, -6, 7] |
In[5]:= | BR[TorusKnot[15, 2]] |
Out[5]= | BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}] |
In[6]:= | alex = Alexander[TorusKnot[15, 2]][t] |
Out[6]= | -7 -6 -5 -4 |
In[7]:= | Conway[TorusKnot[15, 2]][z] |
Out[7]= | 2 4 6 8 10 12 14 1 + 28 z + 126 z + 210 z + 165 z + 66 z + 13 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {} |
In[9]:= | {KnotDet[TorusKnot[15, 2]], KnotSignature[TorusKnot[15, 2]]} |
Out[9]= | {15, 14} |
In[10]:= | J=Jones[TorusKnot[15, 2]][q] |
Out[10]= | 7 9 10 11 12 13 14 15 16 17 18 19 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {} |
In[12]:= | A2Invariant[TorusKnot[15, 2]][q] |
Out[12]= | 26 28 30 32 34 58 60 62 q + q + 2 q + q + q - q - q - q |
In[13]:= | Kauffman[TorusKnot[15, 2]][a, z] |
Out[13]= | 2 |
In[14]:= | {Vassiliev[2][TorusKnot[15, 2]], Vassiliev[3][TorusKnot[15, 2]]} |
Out[14]= | {0, 140} |
In[15]:= | Kh[TorusKnot[15, 2]][q, t] |
Out[15]= | 13 15 2 17 3 21 4 21 |