10 22: Difference between revisions

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{{Rolfsen Knot Page Header|n=10|k=22|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-9,5,-1,6,-8,7,-10,2,-3,9,-5,4,-2,10,-6,8,-7/goTop.html}}
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|{{Rolfsen Knot Site Links|n=10|k=22|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-9,5,-1,6,-8,7,-10,2,-3,9,-5,4,-2,10,-6,8,-7/goTop.html}}
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{{Vassiliev Invariants}}
{{Vassiliev Invariants}}


===[[Khovanov Homology]]===
{{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=13.3333%><table cellpadding=0 cellspacing=0>
<td width=13.3333%><table cellpadding=0 cellspacing=0>
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<tr align=center><td>-7</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-7</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-9</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-9</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table></center>
</table>}}

{{Computer Talk Header}}
{{Computer Talk Header}}


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q t + q t + q t</nowiki></pre></td></tr>
q t + q t + q t</nowiki></pre></td></tr>
</table>
</table>

[[Category:Knot Page]]

Revision as of 19:07, 28 August 2005

10 21.gif

10_21

10 23.gif

10_23

10 22.gif Visit 10 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 22's page at Knotilus!

Visit 10 22's page at the original Knot Atlas!

10 22 Quick Notes


10 22 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17
Gauss code 1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7
Dowker-Thistlethwaite code 6 12 14 18 20 16 4 2 10 8
Conway Notation [3313]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-7]
Hyperbolic Volume 9.98187
A-Polynomial See Data:10 22/A-polynomial

[edit Notes for 10 22's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 0

[edit Notes for 10 22's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 49, 0 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

Vassiliev invariants

V2 and V3: (-4, -2)
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

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 0 is the signature of 10 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
13          11
11         1 -1
9        31 2
7       31  -2
5      43   1
3     43    -1
1    44     0
-1   35      2
-3  13       -2
-5 13        2
-7 1         -1
-91          1
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[Knot[10, 22]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 22]]
Out[3]=  
PD[X[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16], 
 X[14, 5, 15, 6], X[18, 8, 19, 7], X[20, 10, 1, 9], X[8, 20, 9, 19], 

X[4, 13, 5, 14], X[10, 18, 11, 17]]
In[4]:=
GaussCode[Knot[10, 22]]
Out[4]=  
GaussCode[1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, 
  -6, 8, -7]
In[5]:=
BR[Knot[10, 22]]
Out[5]=  
BR[4, {1, 1, 1, 1, 2, -1, -3, 2, -3, -3, -3}]
In[6]:=
alex = Alexander[Knot[10, 22]][t]
Out[6]=  
     2    6    10             2      3

13 - -- + -- - -- - 10 t + 6 t - 2 t

     3    2   t
t t
In[7]:=
Conway[Knot[10, 22]][z]
Out[7]=  
       2      4      6
1 - 4 z  - 6 z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 22]}
In[9]:=
{KnotDet[Knot[10, 22]], KnotSignature[Knot[10, 22]]}
Out[9]=  
{49, 0}
In[10]:=
J=Jones[Knot[10, 22]][q]
Out[10]=  
     -4   2    4    6            2      3      4      5    6

8 + q - -- + -- - - - 8 q + 7 q - 6 q + 4 q - 2 q + q

          3    2   q
q q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 22], Knot[10, 35]}
In[12]:=
A2Invariant[Knot[10, 22]][q]
Out[12]=  
      -12    -8    -6    -4   2     4      6    8    10    12    14

-1 + q + q + q - q + -- - q - 2 q + q - q + q + q +

                              2
                             q

  18
q
In[13]:=
Kauffman[Knot[10, 22]][a, z]
Out[13]=  
                                                    2      2       2
    2    2       2   z    z    z            2   4 z    6 z    12 z

-1 + -- + -- - 2 a - -- + -- + - - a z + 6 z + ---- - ---- - ----- +

     4    2           5    3   a                  6      4      2
    a    a           a    a                      a      a      a

                        3      3                              4
    2  2      4  2   6 z    4 z         3      3  3    4   4 z
 6 a  z  - 2 a  z  + ---- - ---- + 7 a z  - 3 a  z  - z  - ---- + 
                       5      3                              6
                      a      a                              a

    4       4                        5    5
 6 z    16 z       2  4    4  4   7 z    z         5      3  5
 ---- + ----- - 6 a  z  + a  z  - ---- - -- - 6 a z  + 2 a  z  - 
   4      2                         5    a
  a      a                         a

         6      6       6                7    7
    6   z    6 z    12 z       2  6   2 z    z         7      8
 2 z  + -- - ---- - ----- + 3 a  z  + ---- - -- + 3 a z  + 2 z  + 
         6     4      2                 5     3
        a     a      a                 a     a

    8      8    9    9
 2 z    4 z    z    z
 ---- + ---- + -- + --
   4      2     3   a
a a a
In[14]:=
{Vassiliev[2][Knot[10, 22]], Vassiliev[3][Knot[10, 22]]}
Out[14]=  
{0, -2}
In[15]:=
Kh[Knot[10, 22]][q, t]
Out[15]=  
5           1       1       1       3       1      3      3

- + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 4 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t

         q  t    q  t    q  t    q  t    q  t    q  t

    3        3  2      5  2      5  3      7  3    7  4      9  4
 4 q  t + 3 q  t  + 4 q  t  + 3 q  t  + 3 q  t  + q  t  + 3 q  t  + 

  9  5    11  5    13  6
q t + q t + q t