L9a31

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L9a30.gif

L9a30

L9a32.gif

L9a32

L9a31.gif Visit L9a31's page at Knotilus!

Visit L9a31's page at the original Knot Atlas!

L9a31 is [math]\displaystyle{ 9^2_{39} }[/math] in the Rolfsen table of links.


L9a31 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X16,9,17,10 X6718 X18,13,7,14 X10,4,11,3 X14,6,15,5 X4,12,5,11 X12,17,13,18 X2,16,3,15
Gauss code {1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{\left(v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^2} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{7/2}+3 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +9 a z^3-3 z^3 a^{-1} +7 a z-3 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^6 z^4-a^6 z^2+3 a^5 z^5-3 a^5 z^3+5 a^4 z^6-7 a^4 z^4+4 a^4 z^2-a^4+5 a^3 z^7-8 a^3 z^5+z^5 a^{-3} +8 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +a^3 z^{-1} +2 a^2 z^8+4 a^2 z^6+3 z^6 a^{-2} -12 a^2 z^4-6 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+9 a z^7+4 z^7 a^{-1} -20 a z^5-8 z^5 a^{-1} +18 a z^3+5 z^3 a^{-1} -9 a z-4 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^8+2 z^6-10 z^4+7 z^2-3 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{89}{48} }[/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:L9a31/V 2,1 Data:L9a31/V 3,1 Data:L9a31/V 4,1 Data:L9a31/V 4,2 Data:L9a31/V 4,3 Data:L9a31/V 5,1 Data:L9a31/V 5,2 Data:L9a31/V 5,3 Data:L9a31/V 5,4 Data:L9a31/V 6,1 Data:L9a31/V 6,2 Data:L9a31/V 6,3 Data:L9a31/V 6,4 Data:L9a31/V 6,5 Data:L9a31/V 6,6 Data:L9a31/V 6,7 Data:L9a31/V 6,8 Data:L9a31/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 L9a31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
8         11
6        2 -2
4       41 3
2      43  -1
0     63   3
-2    45    1
-4   45     -1
-6  24      2
-8 14       -3
-10 2        2
-121         -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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/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, 31]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 31]]
Out[3]=  
PD[X[8, 1, 9, 2], X[16, 9, 17, 10], X[6, 7, 1, 8], X[18, 13, 7, 14], 

 X[10, 4, 11, 3], X[14, 6, 15, 5], X[4, 12, 5, 11], X[12, 17, 13, 18], 



  X[2, 16, 3, 15]]
In[4]:=
GaussCode[Link[9, Alternating, 31]]
Out[4]=  
GaussCode[{1, -9, 5, -7, 6, -3}, 
  {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}]
In[5]:=
BR[Link[9, Alternating, 31]]
Out[5]=  
BR[Link[9, Alternating, 31]]
In[6]:=
alex = Alexander[Link[9, Alternating, 31]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 31]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 31]], KnotSignature[Link[9, Alternating, 31]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 31]][q]
Out[10]=  
 -(11/2)    3      6      8      9       10                     3/2

q        - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q    + 



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

    5/2    7/2


  3 q    - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 31]][q]
Out[12]=  
     -16    -14    2    2    3       2    4      6    8    10

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



                  12    6    4


                  q     q    q
In[13]:=
Kauffman[Link[9, Alternating, 31]][a, z]
Out[13]=  
                             3

      2    4    2    3 a   a    z    4 z              3        2



3 + 3 a  + a  - --- - --- - -- - -- + --- + 9 a z + 4 a  z - 7 z  - 



               a z    z    z     3    a
                                a

    2                                   3      3
 2 z        2  2      4  2    6  2   2 z    5 z          3      3  3
 ---- - 10 a  z  - 4 a  z  + a  z  + ---- - ---- - 18 a z  - 8 a  z  + 
   2                                   3     a
  a                                   a

                      4                                 5      5
    5  3       4   6 z        2  4      4  4    6  4   z    8 z
 3 a  z  + 10 z  + ---- + 12 a  z  + 7 a  z  - a  z  - -- + ---- + 
                     2                                  3    a
                    a                                  a

                                         6
       5      3  5      5  5      6   3 z       2  6      4  6
 20 a z  + 8 a  z  - 3 a  z  - 2 z  - ---- - 4 a  z  - 5 a  z  - 
                                        2
                                       a

    7
 4 z         7      3  7      8      2  8
 ---- - 9 a z  - 5 a  z  - 2 z  - 2 a  z


   a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 31]], Vassiliev[3][Link[9, Alternating, 31]]}
Out[14]=  
    89

{0, --}


    48
In[15]:=
Kh[Link[9, Alternating, 31]][q, t]
Out[15]=  
    5      1        2        1       4       2       4       4

6 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 



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

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

  8  4


  q  t
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