L9a51

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L9a50.gif

L9a50

L9a52.gif

L9a52

L9a51.gif Visit L9a51's page at Knotilus!

Visit L9a51's page at the original Knot Atlas!

L9a51 is [math]\displaystyle{ 9^3_{11} }[/math] in the Rolfsen table of links.


L9a51 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code {1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{u v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^7-3 q^6+7 q^5-8 q^4+11 q^3-10 q^2- q^{-2} +9 q+4 q^{-1} -6 }[/math] (db)
Signature 2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} -2 z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -3 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} + a^{-2} -z^4-z^2+1 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -10 z^4 a^{-6} +10 z^2 a^{-6} + a^{-6} z^{-2} -5 a^{-6} +5 z^7 a^{-5} -3 z^5 a^{-5} -4 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} +8 z^6 a^{-4} -21 z^4 a^{-4} +17 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +10 z^7 a^{-3} -16 z^5 a^{-3} +2 z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} +6 z^6 a^{-2} -18 z^4 a^{-2} +9 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +5 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-8 z^4+3 z^2+1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{23}{3} }[/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:L9a51/V 2,1 Data:L9a51/V 3,1 Data:L9a51/V 4,1 Data:L9a51/V 4,2 Data:L9a51/V 4,3 Data:L9a51/V 5,1 Data:L9a51/V 5,2 Data:L9a51/V 5,3 Data:L9a51/V 5,4 Data:L9a51/V 6,1 Data:L9a51/V 6,2 Data:L9a51/V 6,3 Data:L9a51/V 6,4 Data:L9a51/V 6,5 Data:L9a51/V 6,6 Data:L9a51/V 6,7 Data:L9a51/V 6,8 Data:L9a51/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]2 is the signature of L9a51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
15         11
13        31-2
11       4  4
9      43  -1
7     74   3
5    45    1
3   56     -1
1  36      3
-1 13       -2
-3 3        3
-51         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 51]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 51]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 12, 9, 11], X[18, 8, 11, 7], 

 X[16, 13, 17, 14], X[14, 6, 15, 5], X[10, 16, 5, 15], X[2, 9, 3, 10], 



  X[4, 18, 1, 17]]
In[4]:=
GaussCode[Link[9, Alternating, 51]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, 
  {3, -2, 5, -6, 7, -5, 9, -4}]
In[5]:=
BR[Link[9, Alternating, 51]]
Out[5]=  
BR[Link[9, Alternating, 51]]
In[6]:=
alex = Alexander[Link[9, Alternating, 51]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 51]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 51]], KnotSignature[Link[9, Alternating, 51]]}
Out[9]=  
{Infinity, 2}
In[10]:=
J=Jones[Link[9, Alternating, 51]][q]
Out[10]=  
      -2   4             2       3      4      5      6    7

-6 - q   + - + 9 q - 10 q  + 11 q  - 8 q  + 7 q  - 3 q  + q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 51]][q]
Out[12]=  
     -6   2       2      4      6    8      10      12      14

1 - q   + -- + 3 q  - 2 q  + 3 q  + q  + 5 q   + 6 q   + 3 q   + 



          4
         q

    16    22


  5 q   + q
In[13]:=
Kauffman[Link[9, Alternating, 51]][a, z]
Out[13]=  
    5    8    3      1       2       1      2      2     6 z   6 z

1 - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 



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

         2       2       2      2      3      3      3      3
    2   z    10 z    17 z    9 z    2 z    4 z    2 z    3 z       3
 3 z  - -- + ----- + ----- + ---- - ---- - ---- + ---- + ---- - a z  - 
         8     6       4       2      7      5      3     a
        a     a       a       a      a      a      a

         4       4       4       4      5      5       5      5
    4   z    10 z    21 z    18 z    3 z    3 z    16 z    9 z
 8 z  + -- - ----- - ----- - ----- + ---- - ---- - ----- - ---- + 
         8     6       4       2       7      5      3      a
        a     a       a       a       a      a      a

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


                 a      a      a      a      a              a      a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 51]], Vassiliev[3][Link[9, Alternating, 51]]}
Out[14]=  
      23

{0, -(--)}


      3
In[15]:=
Kh[Link[9, Alternating, 51]][q, t]
Out[15]=  
         3     1       3      1      3    3 q      3        5

6 q + 5 q  + ----- + ----- + ---- + --- + --- + 6 q  t + 4 q  t + 



             5  3    3  2      2   q t    t
            q  t    q  t    q t

    5  2      7  2      7  3      9  3      9  4      11  4
 5 q  t  + 7 q  t  + 4 q  t  + 4 q  t  + 3 q  t  + 4 q   t  + 

    13  5    13  6    15  6


  3 q   t  + q   t  + q   t
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