L9a9

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L9a8.gif

L9a8

L9a10.gif

L9a10

L9a9.gif Visit L9a9's page at Knotilus!

Visit L9a9's page at the original Knot Atlas!

L9a9 is [math]\displaystyle{ 9^2_{37} }[/math] in the Rolfsen table of links.


L9a9 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{3}{q^{9/2}}-q^{7/2}+\frac{5}{q^{7/2}}+3 q^{5/2}-\frac{7}{q^{5/2}}-5 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^3 z^5-3 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +8 a z^3-3 z^3 a^{-1} +4 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -2 a^2 z^8-2 z^8-4 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 a^4 z^6-3 z^6 a^{-2} +z^6-3 a^5 z^5+7 a^3 z^5+22 a z^5+11 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+5 a^4 z^4+3 a^2 z^4+7 z^4 a^{-2} +4 z^4+4 a^5 z^3-8 a^3 z^3-24 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -3 z^2+4 a^3 z+8 a z+4 z a^{-1} +1-a z^{-1} - a^{-1} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, 1)
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:L9a9/V 2,1 Data:L9a9/V 3,1 Data:L9a9/V 4,1 Data:L9a9/V 4,2 Data:L9a9/V 4,3 Data:L9a9/V 5,1 Data:L9a9/V 5,2 Data:L9a9/V 5,3 Data:L9a9/V 5,4 Data:L9a9/V 6,1 Data:L9a9/V 6,2 Data:L9a9/V 6,3 Data:L9a9/V 6,4 Data:L9a9/V 6,5 Data:L9a9/V 6,6 Data:L9a9/V 6,7 Data:L9a9/V 6,8 Data:L9a9/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 L9a9. 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       31 2
2      32  -1
0     63   3
-2    45    1
-4   34     -1
-6  24      2
-8 13       -2
-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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}\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, 9]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 9]]
Out[3]=  
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[16, 9, 17, 10], X[8, 15, 9, 16], 

 X[4, 17, 1, 18], X[12, 6, 13, 5], X[10, 4, 11, 3], X[18, 12, 5, 11], 



  X[2, 14, 3, 13]]
In[4]:=
GaussCode[Link[9, Alternating, 9]]
Out[4]=  
GaussCode[{1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, 
   -8}]
In[5]:=
BR[Link[9, Alternating, 9]]
Out[5]=  
BR[Link[9, Alternating, 9]]
In[6]:=
alex = Alexander[Link[9, Alternating, 9]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 9]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 9]], KnotSignature[Link[9, Alternating, 9]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 9]][q]
Out[10]=  
 -(11/2)    3      5      7      8        9                     3/2

q        - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 5 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, 9]][q]
Out[12]=  
     -16    -14    -12    -10    -8    -6   3     2    6    8    10

4 - q    + q    - q    + q    + q   - q   + -- + q  + q  - q  + q



                                            4


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

    1    a   4 z              3        2   2 z       2  2    4  2



1 - --- - - + --- + 8 a z + 4 a  z - 3 z  - ---- - 3 a  z  - a  z  + 



   a z   z    a                              2
                                            a

            3       3                                           4
  6  2   2 z    10 z          3      3  3      5  3      4   7 z
 a  z  + ---- - ----- - 24 a z  - 8 a  z  + 4 a  z  + 4 z  + ---- + 
           3      a                                            2
          a                                                   a

                              5       5
    2  4      4  4    6  4   z    11 z          5      3  5
 3 a  z  + 5 a  z  - a  z  - -- + ----- + 22 a z  + 7 a  z  - 
                              3     a
                             a

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

    2  8


  2 a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 9]], Vassiliev[3][Link[9, Alternating, 9]]}
Out[14]=  
{0, 1}
In[15]:=
Kh[Link[9, Alternating, 9]][q, t]
Out[15]=  
    5      1        2        1       3       2       4       3

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

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

  8  4


  q  t
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