L9a2

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L9a1.gif

L9a1

L9a3.gif

L9a3

L9a2.gif Visit L9a2's page at Knotilus!

Visit L9a2's page at the original Knot Atlas!

L9a2 is [math]\displaystyle{ 9^2_{31} }[/math] in the Rolfsen table of links.


L9a2 Further Notes and Views

Knot presentations

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

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)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+3 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}} }[/math] (db)
Signature 3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^7 a^{-3} -z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +z a^{-3} -z a^{-5} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +4 z^6 a^{-6} -3 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} +4 z^7 a^{-5} -8 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -2 z^6 a^{-4} -3 z^4 a^{-4} +3 a^{-4} +7 z^7 a^{-3} -24 z^5 a^{-3} +22 z^3 a^{-3} -3 z a^{-3} -3 a^{-3} z^{-1} +2 z^8 a^{-2} -5 z^6 a^{-2} +z^2 a^{-2} +3 a^{-2} +3 z^7 a^{-1} -12 z^5 a^{-1} +12 z^3 a^{-1} -z a^{-1} -2 a^{-1} z^{-1} +z^6-3 z^4+z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{3}{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:L9a2/V 2,1 Data:L9a2/V 3,1 Data:L9a2/V 4,1 Data:L9a2/V 4,2 Data:L9a2/V 4,3 Data:L9a2/V 5,1 Data:L9a2/V 5,2 Data:L9a2/V 5,3 Data:L9a2/V 5,4 Data:L9a2/V 6,1 Data:L9a2/V 6,2 Data:L9a2/V 6,3 Data:L9a2/V 6,4 Data:L9a2/V 6,5 Data:L9a2/V 6,6 Data:L9a2/V 6,7 Data:L9a2/V 6,8 Data:L9a2/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]3 is the signature of L9a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
16         11
14        2 -2
12       21 1
10      42  -2
8     32   1
6    34    1
4   33     0
2  25      3
0 11       0
-2 2        2
-41         -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=4 }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 2]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 2]]
Out[3]=  
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], 

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



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

q       - ------- + 3 Sqrt[q] - 6 q    + 6 q    - 7 q    + 6 q    - 



         Sqrt[q]

    11/2      13/2    15/2


  4 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, 2]][q]
Out[12]=  
     -4    -2      2      4      6      8    10      14      16    20

1 - q   + q   + 3 q  + 4 q  + 2 q  + 4 q  - q   - 2 q   - 2 q   - q   + 



  22


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

 -6   3    3     1      3      2    2 z   3 z   z    2   2 z    2 z



-a   - -- - -- + ---- + ---- + --- + --- + --- + - - z  + ---- + ---- - 



       4    2    5      3     a z    5     3    a          8      6
      a    a    a  z   a  z         a     a               a      a

  2    3      3      3       3       3             4      4      4
 z    z    3 z    6 z    22 z    12 z       4   3 z    3 z    3 z
 -- - -- + ---- - ---- - ----- - ----- + 3 z  - ---- + ---- + ---- - 
  2    9     7      5      3       a              8      6      4
 a    a     a      a      a                      a      a      a

    5      5       5       5           6      6      6      7      7
 4 z    8 z    24 z    12 z     6   4 z    2 z    5 z    4 z    7 z
 ---- + ---- + ----- + ----- - z  - ---- + ---- + ---- - ---- - ---- - 
   7      5      3       a            6      4      2      5      3
  a      a      a                    a      a      a      a      a

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


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

{0, -(-)}


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

  2      4     1      -2     2     1   2 q       4        6



5 q  + 3 q  + ----- + t   + ----- + - + ---- + 3 q  t + 3 q  t + 



              4  3          2  2   t    t
             q  t          q  t

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

  12  5      14  5    16  6


  q   t  + 2 q   t  + q   t
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