L9a45

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L9a44.gif

L9a44

L9a46.gif

L9a46

L9a45.gif Visit L9a45's page at Knotilus!

Visit L9a45's page at the original Knot Atlas!

L9a45 is [math]\displaystyle{ 9^3_{7} }[/math] in the Rolfsen table of links.


L9a45 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X10,3,11,4 X18,12,9,11 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,18,13,17 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{-2 t(2) t(1)+2 t(2) t(3) t(1)-2 t(3) t(1)+3 t(1)+2 t(2)-3 t(2) t(3)+2 t(3)-2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-4 q^3+5 q^2-6 q+7-4 q^{-1} +4 q^{-2} - q^{-3} + q^{-4} }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+z^2+ z^{-2} +2+z^4 a^{-2} +z^2 a^{-2} -z^2 a^{-4} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^5 a^{-5} -2 z^3 a^{-5} +3 z^6 a^{-4} +a^4 z^4-8 z^4 a^{-4} -3 a^4 z^2+3 z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 z^7 a^{-3} +a^3 z^5-8 z^5 a^{-3} +5 z^3 a^{-3} -3 a^3 z+2 a^3 z^{-1} +z^8 a^{-2} +a^2 z^6+z^6 a^{-2} +2 a^2 z^4-5 z^4 a^{-2} -6 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +5 a^2+a z^7+4 z^7 a^{-1} +a z^5-9 z^5 a^{-1} +7 z^3 a^{-1} -3 a z+2 a z^{-1} +z^8-z^6+4 z^4-4 z^2- z^{-2} +3 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{11}{6} }[/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:L9a45/V 2,1 Data:L9a45/V 3,1 Data:L9a45/V 4,1 Data:L9a45/V 4,2 Data:L9a45/V 4,3 Data:L9a45/V 5,1 Data:L9a45/V 5,2 Data:L9a45/V 5,3 Data:L9a45/V 5,4 Data:L9a45/V 6,1 Data:L9a45/V 6,2 Data:L9a45/V 6,3 Data:L9a45/V 6,4 Data:L9a45/V 6,5 Data:L9a45/V 6,6 Data:L9a45/V 6,7 Data:L9a45/V 6,8 Data:L9a45/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]0 is the signature of L9a45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
11         1-1
9        2 2
7       21 -1
5      32  1
3     32   -1
1    43    1
-1   36     3
-3  11      0
-5  3       3
-711        0
-91         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=1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\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 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/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, 45]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 45]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 9, 11], X[16, 14, 17, 13], 

 X[8, 16, 5, 15], X[14, 8, 15, 7], X[12, 18, 13, 17], X[2, 5, 3, 6], 



  X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 45]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 6, -5}, 
  {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}]
In[5]:=
BR[Link[9, Alternating, 45]]
Out[5]=  
BR[Link[9, Alternating, 45]]
In[6]:=
alex = Alexander[Link[9, Alternating, 45]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 45]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 45]], KnotSignature[Link[9, Alternating, 45]]}
Out[9]=  
{Infinity, 0}
In[10]:=
J=Jones[Link[9, Alternating, 45]][q]
Out[10]=  
     -4    -3   4    4            2      3      4    5

7 + q   - q   + -- - - - 6 q + 5 q  - 4 q  + 3 q  - q



                2   q


                q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 45]][q]
Out[12]=  
     -14    2     2    5    5    3    5       2    6    8    10    12

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



           12    10    8    6    4    2
          q     q     q    q    q    q

  14    16


  q   - q
In[13]:=
Kauffman[Link[9, Alternating, 45]][a, z]
Out[13]=  
                           2    4            3

      2      4    -2   2 a    a    2 a   2 a               3



3 + 5 a  + 3 a  - z   - ---- - -- + --- + ---- - 3 a z - 3 a  z - 



                         2     2    z     z
                        z     z

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

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

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


        a     a             a                        a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 45]], Vassiliev[3][Link[9, Alternating, 45]]}
Out[14]=  
    11

{0, --}


    6
In[15]:=
Kh[Link[9, Alternating, 45]][q, t]
Out[15]=  
6           1       1       1       3       1      1      3

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

  11  5


  q   t
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