L9a1

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L8n8.gif

L8n8

L9a2.gif

L9a2

L9a1.gif Visit L9a1's page at Knotilus!

Visit L9a1's page at the original Knot Atlas!

L9a1 is [math]\displaystyle{ 9^2_{32} }[/math] in the Rolfsen table of links.


L9a1 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,12,5,11 X12,18,13,17 X16,10,17,9 X2,14,3,13
Gauss code {1, -9, 5, -3}, {4, -1, 2, -5, 8, -4, 6, -7, 9, -2, 3, -8, 7, -6}

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(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-10 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} }[/math] (db)
Signature 1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^5 a^{-1} +z^5 a^{-3} -a z^3+z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} -z a^{-1} +2 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -5 z^6 a^{-2} -2 z^6 a^{-4} -3 z^6 a^{-6} -6 z^6-4 a z^5+4 z^5 a^{-1} +18 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+11 z^4 a^{-2} +10 z^4 a^{-4} +7 z^4 a^{-6} +7 z^4+4 a z^3-13 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -4 z^2 a^{-2} -7 z^2 a^{-4} -4 z^2 a^{-6} -z^2+2 z a^{-1} +4 z a^{-3} +2 z a^{-5} +1-a z^{-1} - a^{-1} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{1}{2} }[/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:L9a1/V 2,1 Data:L9a1/V 3,1 Data:L9a1/V 4,1 Data:L9a1/V 4,2 Data:L9a1/V 4,3 Data:L9a1/V 5,1 Data:L9a1/V 5,2 Data:L9a1/V 5,3 Data:L9a1/V 5,4 Data:L9a1/V 6,1 Data:L9a1/V 6,2 Data:L9a1/V 6,3 Data:L9a1/V 6,4 Data:L9a1/V 6,5 Data:L9a1/V 6,6 Data:L9a1/V 6,7 Data:L9a1/V 6,8 Data:L9a1/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 L9a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
14         11
12        2 -2
10       31 2
8      52  -3
6     53   2
4    45    1
2   55     0
0  36      3
-2 13       -2
-4 3        3
-61         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/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, 1]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 1]]
Out[3]=  
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], 

 X[8, 4, 9, 3], X[18, 12, 5, 11], X[12, 18, 13, 17], X[16, 10, 17, 9], 



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

q       - ---- + ------- - 9 Sqrt[q] + 9 q    - 10 q    + 8 q    - 



          3/2   Sqrt[q]
         q

    9/2      11/2    13/2


  5 q    + 3 q     - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 1]][q]
Out[12]=  
     -8   2     -4      4    6    10      12    14    18    20

4 - q   + -- + q   + 3 q  + q  + q   - 3 q   + q   - q   + q



          6


          q
In[13]:=
Kauffman[Link[9, Alternating, 1]][a, z]
Out[13]=  
                                        2      2      2      3      3

    1    a   2 z   4 z   2 z    2   4 z    7 z    4 z    2 z    7 z



1 - --- - - + --- + --- + --- - z  - ---- - ---- - ---- + ---- - ---- - 



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

     3                      4       4       4            5      5
 13 z         3      4   7 z    10 z    11 z     2  4   z    9 z
 ----- + 4 a z  + 7 z  + ---- + ----- + ----- - a  z  - -- + ---- + 
   3                       6      4       2              7     5
  a                       a      a       a              a     a

     5      5                      6      6      6      7      7
 18 z    4 z         5      6   3 z    2 z    5 z    4 z    9 z
 ----- + ---- - 4 a z  - 6 z  - ---- - ---- - ---- - ---- - ---- - 
   3      a                       6      4      2      5      3
  a                              a      a      a      a      a

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


          a      a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 1]], Vassiliev[3][Link[9, Alternating, 1]]}
Out[14]=  
    1

{0, -}


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

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



           6  3    4  2    2  2   t    2
          q  t    q  t    q  t        q  t

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

    12  5    14  6


  2 q   t  + q   t
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