10 44: Difference between revisions
(Resetting knot page to basic template.) |
No edit summary |
||
Line 1: | Line 1: | ||
<!-- --> |
|||
{{Template:Basic Knot Invariants|name=10_44}} |
|||
<!-- provide an anchor so we can return to the top of the page --> |
|||
<span id="top"></span> |
|||
<!-- this relies on transclusion for next and previous links --> |
|||
{{Knot Navigation Links|ext=gif}} |
|||
{| align=left |
|||
|- valign=top |
|||
|[[Image:{{PAGENAME}}.gif]] |
|||
|{{Rolfsen Knot Site Links|n=10|k=44|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,10,-8,9,-6,3,-4,2,-5,6,-7,8,-9,7,-10,5/goTop.html}} |
|||
|{{:{{PAGENAME}} Quick Notes}} |
|||
|} |
|||
<br style="clear:both" /> |
|||
{{:{{PAGENAME}} Further Notes and Views}} |
|||
{{Knot Presentations}} |
|||
{{3D Invariants}} |
|||
{{4D Invariants}} |
|||
{{Polynomial Invariants}} |
|||
{{Vassiliev Invariants}} |
|||
===[[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>{{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> |
|||
<td width=13.3333%><table cellpadding=0 cellspacing=0> |
|||
<tr><td>\</td><td> </td><td>r</td></tr> |
|||
<tr><td> </td><td> \ </td><td> </td></tr> |
|||
<tr><td>j</td><td> </td><td>\</td></tr> |
|||
</table></td> |
|||
<td width=6.66667%>-6</td ><td width=6.66667%>-5</td ><td width=6.66667%>-4</td ><td width=6.66667%>-3</td ><td width=6.66667%>-2</td ><td width=6.66667%>-1</td ><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=13.3333%>χ</td></tr> |
|||
<tr align=center><td>7</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>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>-2</td></tr> |
|||
<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>4</td><td bgcolor=yellow>1</td><td> </td><td>3</td></tr> |
|||
<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>-3</td></tr> |
|||
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>3</td></tr> |
|||
<tr align=center><td>-3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
|||
<tr align=center><td>-5</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
|||
<tr align=center><td>-7</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> |
|||
<tr align=center><td>-9</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> |
|||
<tr align=center><td>-11</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> |
|||
<tr align=center><td>-13</td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> |
|||
<tr align=center><td>-15</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>1</td></tr> |
|||
</table></center> |
|||
{{Computer Talk Header}} |
|||
<table> |
|||
<tr valign=top> |
|||
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
|||
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</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[Knot[10, 44]]</nowiki></pre></td></tr> |
|||
<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> |
|||
<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, 44]]</nowiki></pre></td></tr> |
|||
<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[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], |
|||
X[13, 20, 14, 1], X[9, 15, 10, 14], X[15, 18, 16, 19], |
|||
X[7, 16, 8, 17], X[17, 8, 18, 9], X[19, 7, 20, 6]]</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[Knot[10, 44]]</nowiki></pre></td></tr> |
|||
<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, 4, -3, 1, -2, 10, -8, 9, -6, 3, -4, 2, -5, 6, -7, 8, -9, |
|||
7, -10, 5]</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[Knot[10, 44]]</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[5, {-1, -1, 2, -1, -3, 2, -3, 4, -3, 4}]</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[Knot[10, 44]][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> -3 7 19 2 3 |
|||
-25 + t - -- + -- + 19 t - 7 t + t |
|||
2 t |
|||
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[Knot[10, 44]][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> 4 6 |
|||
1 - 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>Out[8]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 44], Knot[11, NonAlternating, 154]}</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[Knot[10, 44]], KnotSignature[Knot[10, 44]]}</nowiki></pre></td></tr> |
|||
<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>{79, -2}</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[Knot[10, 44]][q]</nowiki></pre></td></tr> |
|||
<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> -7 4 7 10 13 13 12 2 3 |
|||
-9 + q - -- + -- - -- + -- - -- + -- + 6 q - 3 q + q |
|||
6 5 4 3 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[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]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 44]}</nowiki></pre></td></tr> |
|||
<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[Knot[10, 44]][q]</nowiki></pre></td></tr> |
|||
<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> -22 -20 2 2 2 -12 2 -8 3 2 2 |
|||
q - q - --- + --- - --- + q + --- - q + -- - -- + -- - |
|||
18 16 14 10 6 4 2 |
|||
q q q q q q q |
|||
2 4 6 10 |
|||
2 q + 2 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[Knot[10, 44]][a, z]</nowiki></pre></td></tr> |
|||
<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 4 2 z 3 2 3 z 2 2 |
|||
-2 - a - 3 a - a - --- - 4 a z - 2 a z + 9 z + ---- + 13 a z + |
|||
a 2 |
|||
a |
|||
3 |
|||
4 2 6 2 8 z 3 3 3 7 3 4 |
|||
10 a z + 3 a z + ---- + 20 a z + 15 a z - 3 a z - 6 z - |
|||
a |
|||
4 5 |
|||
3 z 2 4 4 4 6 4 8 4 9 z 5 |
|||
---- - 12 a z - 18 a z - 8 a z + a z - ---- - 26 a z - |
|||
2 a |
|||
a |
|||
6 |
|||
3 5 5 5 7 5 6 z 2 6 4 6 |
|||
27 a z - 6 a z + 4 a z - 4 z + -- - 7 a z + 5 a z + |
|||
2 |
|||
a |
|||
7 |
|||
6 6 3 z 7 3 7 5 7 8 2 8 |
|||
7 a z + ---- + 8 a z + 12 a z + 7 a z + 3 z + 7 a z + |
|||
a |
|||
4 8 9 3 9 |
|||
4 a z + a z + a z</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][Knot[10, 44]], Vassiliev[3][Knot[10, 44]]}</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, -1}</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[Knot[10, 44]][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>6 7 1 3 1 4 3 6 4 |
|||
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + |
|||
3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 |
|||
q q t q t q t q t q t q t q t |
|||
7 6 6 7 4 t 2 3 2 |
|||
----- + ----- + ---- + ---- + --- + 5 q t + 2 q t + 4 q t + |
|||
7 2 5 2 5 3 q |
|||
q t q t q t q t |
|||
3 3 5 3 7 4 |
|||
q t + 2 q t + q t</nowiki></pre></td></tr> |
|||
</table> |
Revision as of 21:48, 27 August 2005
|
|
Visit 10 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 44's page at Knotilus! Visit 10 44's page at the original Knot Atlas! |
10 44 Quick Notes |
Knot presentations
Planar diagram presentation | X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,20,14,1 X9,15,10,14 X15,18,16,19 X7,16,8,17 X17,8,18,9 X19,7,20,6 |
Gauss code | -1, 4, -3, 1, -2, 10, -8, 9, -6, 3, -4, 2, -5, 6, -7, 8, -9, 7, -10, 5 |
Dowker-Thistlethwaite code | 4 10 12 16 14 2 20 18 8 6 |
Conway Notation | [2121112] |
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | |
1,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
G2 Invariants.
Weight | Invariant |
---|---|
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["10 44"];
|
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]=
|
{ 79, -2 } |
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: | (0, -1) |
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 -2 is the signature of 10 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
|
-6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | χ | |||||||||
7 | 1 | 1 | |||||||||||||||||||
5 | 2 | -2 | |||||||||||||||||||
3 | 4 | 1 | 3 | ||||||||||||||||||
1 | 5 | 2 | -3 | ||||||||||||||||||
-1 | 7 | 4 | 3 | ||||||||||||||||||
-3 | 7 | 6 | -1 | ||||||||||||||||||
-5 | 6 | 6 | 0 | ||||||||||||||||||
-7 | 4 | 7 | 3 | ||||||||||||||||||
-9 | 3 | 6 | -3 | ||||||||||||||||||
-11 | 1 | 4 | 3 | ||||||||||||||||||
-13 | 3 | -3 | |||||||||||||||||||
-15 | 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, 44]] |
Out[2]= | 10 |
In[3]:= | PD[Knot[10, 44]] |
Out[3]= | PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2],X[13, 20, 14, 1], X[9, 15, 10, 14], X[15, 18, 16, 19],X[7, 16, 8, 17], X[17, 8, 18, 9], X[19, 7, 20, 6]] |
In[4]:= | GaussCode[Knot[10, 44]] |
Out[4]= | GaussCode[-1, 4, -3, 1, -2, 10, -8, 9, -6, 3, -4, 2, -5, 6, -7, 8, -9, 7, -10, 5] |
In[5]:= | BR[Knot[10, 44]] |
Out[5]= | BR[5, {-1, -1, 2, -1, -3, 2, -3, 4, -3, 4}] |
In[6]:= | alex = Alexander[Knot[10, 44]][t] |
Out[6]= | -3 7 19 2 3 |
In[7]:= | Conway[Knot[10, 44]][z] |
Out[7]= | 4 6 1 - z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 44], Knot[11, NonAlternating, 154]} |
In[9]:= | {KnotDet[Knot[10, 44]], KnotSignature[Knot[10, 44]]} |
Out[9]= | {79, -2} |
In[10]:= | J=Jones[Knot[10, 44]][q] |
Out[10]= | -7 4 7 10 13 13 12 2 3 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[10, 44]} |
In[12]:= | A2Invariant[Knot[10, 44]][q] |
Out[12]= | -22 -20 2 2 2 -12 2 -8 3 2 2 |
In[13]:= | Kauffman[Knot[10, 44]][a, z] |
Out[13]= | 2-2 2 4 2 z 3 2 3 z 2 2 |
In[14]:= | {Vassiliev[2][Knot[10, 44]], Vassiliev[3][Knot[10, 44]]} |
Out[14]= | {0, -1} |
In[15]:= | Kh[Knot[10, 44]][q, t] |
Out[15]= | 6 7 1 3 1 4 3 6 4 |