L9a23

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L9a22.gif

L9a22

L9a24.gif

L9a24

L9a23.gif Visit L9a23's page at Knotilus!

Visit L9a23's page at the original Knot Atlas!

L9a23 is [math]\displaystyle{ 9^2_{22} }[/math] in the Rolfsen table of links.


L9a23 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X10,3,11,4 X18,16,7,15 X14,5,15,6 X4,13,5,14 X12,18,13,17 X16,12,17,11 X2738 X6,9,1,10
Gauss code {1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{t(1) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^5 z^3+2 a^5 z+2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-5 a^3 z-3 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -a z+a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^4 z^8-a^2 z^8-3 a^5 z^7-6 a^3 z^7-3 a z^7-2 a^6 z^6-4 a^4 z^6-6 a^2 z^6-4 z^6-a^7 z^5+8 a^5 z^5+13 a^3 z^5+a z^5-3 z^5 a^{-1} +4 a^6 z^4+13 a^4 z^4+15 a^2 z^4-z^4 a^{-2} +5 z^4+3 a^7 z^3-10 a^5 z^3-15 a^3 z^3+2 a z^3+4 z^3 a^{-1} -a^6 z^2-11 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -2 z^2-2 a^7 z+8 a^5 z+11 a^3 z-z a^{-1} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{105}{16} }[/math])
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:L9a23/V 2,1 Data:L9a23/V 3,1 Data:L9a23/V 4,1 Data:L9a23/V 4,2 Data:L9a23/V 4,3 Data:L9a23/V 5,1 Data:L9a23/V 5,2 Data:L9a23/V 5,3 Data:L9a23/V 5,4 Data:L9a23/V 6,1 Data:L9a23/V 6,2 Data:L9a23/V 6,3 Data:L9a23/V 6,4 Data:L9a23/V 6,5 Data:L9a23/V 6,6 Data:L9a23/V 6,7 Data:L9a23/V 6,8 Data:L9a23/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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-1 is the signature of L9a23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-10123χ
6         1-1
4        2 2
2       31 -2
0      42  2
-2     54   -1
-4    33    0
-6   35     2
-8  23      -1
-10  3       3
-1212        -1
-141         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Link[9, Alternating, 23]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 23]]
Out[3]=  
PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[18, 16, 7, 15], X[14, 5, 15, 6], 

 X[4, 13, 5, 14], X[12, 18, 13, 17], X[16, 12, 17, 11], X[2, 7, 3, 8], 



  X[6, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 23]]
Out[4]=  
GaussCode[{1, -8, 2, -5, 4, -9}, 
  {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}]
In[5]:=
BR[Link[9, Alternating, 23]]
Out[5]=  
BR[Link[9, Alternating, 23]]
In[6]:=
alex = Alexander[Link[9, Alternating, 23]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 23]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 23]], KnotSignature[Link[9, Alternating, 23]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 23]][q]
Out[10]=  
  -(13/2)     2      5      6      8      8        7

-q        + ----- - ---- + ---- - ---- + ---- - ------- + 5 Sqrt[q] - 



            11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q      q      q      q

    3/2    5/2


  3 q    + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 23]][q]
Out[12]=  
     -20    2     4     -12    3     -6   2     2    6    8

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



           16    14           10          2


           q     q            q           q
In[13]:=
Kauffman[Link[9, Alternating, 23]][a, z]
Out[13]=  
                         3      5

      2      4   a   3 a    2 a    z       3        5        7



1 + 3 a  + 3 a  - - - ---- - ---- - - + 11 a  z + 8 a  z - 2 a  z - 



                 z    z      z     a

         2                                    3
    2   z        2  2       4  2    6  2   4 z         3       3  3
 2 z  + -- - 13 a  z  - 11 a  z  - a  z  + ---- + 2 a z  - 15 a  z  - 
         2                                  a
        a

                              4
     5  3      7  3      4   z        2  4       4  4      6  4
 10 a  z  + 3 a  z  + 5 z  - -- + 15 a  z  + 13 a  z  + 4 a  z  - 
                              2
                             a

    5
 3 z       5       3  5      5  5    7  5      6      2  6      4  6
 ---- + a z  + 13 a  z  + 8 a  z  - a  z  - 4 z  - 6 a  z  - 4 a  z  - 
  a

    6  6        7      3  7      5  7    2  8    4  8


  2 a  z  - 3 a z  - 6 a  z  - 3 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 23]], Vassiliev[3][Link[9, Alternating, 23]]}
Out[14]=  
      105

{0, -(---)}


      16
In[15]:=
Kh[Link[9, Alternating, 23]][q, t]
Out[15]=  
    4      1        1        2        3        2       3       3

4 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



    2    14  6    12  6    12  5    10  4    8  4    8  3    6  3
   q    q   t    q   t    q   t    q   t    q  t    q  t    q  t

   5       3      3      5              2      2  2      4  2    6  3
 ----- + ----- + ---- + ---- + 2 t + 3 q  t + q  t  + 2 q  t  + q  t
  6  2    4  2    4      2


  q  t    q  t    q  t   q  t
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