L9a36

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L9a35.gif

L9a35

L9a37.gif

L9a37

L9a36.gif Visit L9a36's page at Knotilus!

Visit L9a36's page at the original Knot Atlas!

L9a36 is [math]\displaystyle{ 9^2_{1} }[/math] in the Rolfsen table of links.


L9a36 Further Notes and Views

Knot presentations

Planar diagram presentation X10,1,11,2 X12,4,13,3 X18,12,9,11 X14,6,15,5 X16,8,17,7 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,18,1,17
Gauss code {1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{u^3 v^3-u^3 v^2-u^2 v^3+u^2 v^2-u^2 v-u v^2+u v-u-v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -3 q^{9/2}+2 q^{7/2}-2 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+2 q^{15/2}-3 q^{13/2}+3 q^{11/2}-\sqrt{q} }[/math] (db)
Signature 5 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^5 a^{-7} +4 z^3 a^{-7} +3 z a^{-7} -z^7 a^{-5} -6 z^5 a^{-5} -11 z^3 a^{-5} -7 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} +5 z^6 a^{-4} +3 z^6 a^{-6} -2 z^6 a^{-8} +6 z^5 a^{-3} +15 z^5 a^{-5} +7 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-4} +4 z^4 a^{-8} -2 z^4 a^{-10} -11 z^3 a^{-3} -21 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -2 z^3 a^{-11} -2 z^2 a^{-6} +z^2 a^{-10} -z^2 a^{-12} +7 z a^{-3} +9 z a^{-5} +2 z a^{-7} +z a^{-9} +z a^{-11} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{41}{24} }[/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:L9a36/V 2,1 Data:L9a36/V 3,1 Data:L9a36/V 4,1 Data:L9a36/V 4,2 Data:L9a36/V 4,3 Data:L9a36/V 5,1 Data:L9a36/V 5,2 Data:L9a36/V 5,3 Data:L9a36/V 5,4 Data:L9a36/V 6,1 Data:L9a36/V 6,2 Data:L9a36/V 6,3 Data:L9a36/V 6,4 Data:L9a36/V 6,5 Data:L9a36/V 6,6 Data:L9a36/V 6,7 Data:L9a36/V 6,8 Data:L9a36/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]5 is the signature of L9a36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
20         1-1
18        1 1
16       11 0
14      21  1
12     22   0
10    11    0
8   12     1
6  11      0
4 12       1
2          0
01         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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=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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/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, 36]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 36]]
Out[3]=  
PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[14, 6, 15, 5], 

 X[16, 8, 17, 7], X[2, 9, 3, 10], X[4, 14, 5, 13], X[6, 16, 7, 15], 



  X[8, 18, 1, 17]]
In[4]:=
GaussCode[Link[9, Alternating, 36]]
Out[4]=  
GaussCode[{1, -6, 2, -7, 4, -8, 5, -9}, 
  {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}]
In[5]:=
BR[Link[9, Alternating, 36]]
Out[5]=  
BR[Link[9, Alternating, 36]]
In[6]:=
alex = Alexander[Link[9, Alternating, 36]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 36]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 36]], KnotSignature[Link[9, Alternating, 36]]}
Out[9]=  
{Infinity, 5}
In[10]:=
J=Jones[Link[9, Alternating, 36]][q]
Out[10]=  
            3/2      5/2      7/2      9/2      11/2      13/2

-Sqrt[q] + q    - 2 q    + 2 q    - 3 q    + 3 q     - 3 q     + 



    15/2      17/2    19/2


  2 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, 36]][q]
Out[12]=  
 2    4    6    8    12      16    18    20    22    28
q  + q  + q  + q  + q   + 2 q   + q   + q   + q   - q
In[13]:=
Kauffman[Link[9, Alternating, 36]][a, z]
Out[13]=  
                                                  2     2       2

-4    1      1      z    z    2 z   9 z   7 z   z     z     2 z



a   - ---- - ---- + --- + -- + --- + --- + --- - --- + --- - ---- - 



      5      3      11    9    7     5     3     12    10     6
     a  z   a  z   a     a    a     a     a     a     a      a

    3      3      3       3       3      4      4      4      5
 2 z    2 z    6 z    21 z    11 z    2 z    4 z    6 z    2 z
 ---- + ---- - ---- - ----- - ----- - ---- + ---- - ---- - ---- + 
  11      9      7      5       3      10      8      4      9
 a       a      a      a       a      a       a      a      a

    5       5      5      6      6      6      7      7    7    8    8
 7 z    15 z    6 z    2 z    3 z    5 z    2 z    3 z    z    z    z
 ---- + ----- + ---- - ---- + ---- + ---- - ---- - ---- - -- - -- - --
   7      5       3      8      6      4      7      5     3    6    4


   a      a       a      a      a      a      a      a     a    a    a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 36]], Vassiliev[3][Link[9, Alternating, 36]]}
Out[14]=  
    41

{0, --}


    24
In[15]:=
Kh[Link[9, Alternating, 36]][q, t]
Out[15]=  
                   4

  4    6    -2   q     6      8        8  2    10  2    10  3



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



                 t

    12  3      12  4      14  4    14  5    16  5    16  6    18  6
 2 q   t  + 2 q   t  + 2 q   t  + q   t  + q   t  + q   t  + q   t  + 

  20  7


  q   t
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