L9a40

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L9a39.gif

L9a39

L9a41.gif

L9a41

L9a40.gif Visit L9a40's page at Knotilus!

Visit L9a40's page at the original Knot Atlas!

L9a40 is [math]\displaystyle{ 9^2_{4} }[/math] in the Rolfsen table of links.



National swiss museum
Povray depiction

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{(u-1) (v-1) \left(u^2 v+u v^2+u v+u+v\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^3 a^5-2 z a^5+z^5 a^3+3 z^3 a^3+z a^3+z^5 a+4 z^3 a+4 z a+a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^8 z^2+2 a^7 z^3+3 a^6 z^4-2 a^6 z^2+4 a^5 z^5-8 a^5 z^3+4 a^5 z+3 a^4 z^6-6 a^4 z^4+a^4 z^2+2 a^3 z^7-4 a^3 z^5-2 a^3 z^3+2 a^3 z+a^2 z^8-2 a^2 z^6-2 a^2 z^4+2 a^2 z^2+3 a z^7+z^7 a^{-1} -14 a z^5-6 z^5 a^{-1} +19 a z^3+11 z^3 a^{-1} -8 a z+a z^{-1} -6 z a^{-1} + a^{-1} z^{-1} +z^8-5 z^6+7 z^4-2 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{5}{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:L9a40/V 2,1 Data:L9a40/V 3,1 Data:L9a40/V 4,1 Data:L9a40/V 4,2 Data:L9a40/V 4,3 Data:L9a40/V 5,1 Data:L9a40/V 5,2 Data:L9a40/V 5,3 Data:L9a40/V 5,4 Data:L9a40/V 6,1 Data:L9a40/V 6,2 Data:L9a40/V 6,3 Data:L9a40/V 6,4 Data:L9a40/V 6,5 Data:L9a40/V 6,6 Data:L9a40/V 6,7 Data:L9a40/V 6,8 Data:L9a40/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 L9a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
6         11
4          0
2       21 1
0      1   -1
-2     32   1
-4    23    1
-6   21     1
-8  12      1
-10 12       -1
-12 1        1
-141         -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=-2 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 40]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 40]]
Out[3]=  
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], 

 X[18, 13, 9, 14], X[14, 8, 15, 7], X[16, 6, 17, 5], X[6, 16, 7, 15], 



  X[4, 18, 5, 17]]
In[4]:=
GaussCode[Link[9, Alternating, 40]]
Out[4]=  
GaussCode[{1, -2, 3, -9, 7, -8, 6, -4}, 
  {4, -1, 2, -3, 5, -6, 8, -7, 9, -5}]
In[5]:=
BR[Link[9, Alternating, 40]]
Out[5]=  
BR[Link[9, Alternating, 40]]
In[6]:=
alex = Alexander[Link[9, Alternating, 40]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 40]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 40]], KnotSignature[Link[9, Alternating, 40]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 40]][q]
Out[10]=  
 -(13/2)     2      3      4      3      4        3

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



           11/2    9/2    7/2    5/2    3/2   Sqrt[q]
          q       q      q      q      q

  3/2    5/2


  q    - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 40]][q]
Out[12]=  
  -20    -12   2     -6    -4    -2    2    4    6    8

-q    + q    + -- + q   + q   + q   + q  + q  + q  + q



               8


               q
In[13]:=
Kauffman[Link[9, Alternating, 40]][a, z]
Out[13]=  
     1    a   6 z              3        5        2      2  2    4  2

1 - --- - - + --- + 8 a z - 2 a  z - 4 a  z + 2 z  - 2 a  z  - a  z  + 



   a z   z    a

                       3
    6  2    8  2   11 z          3      3  3      5  3      7  3
 2 a  z  - a  z  - ----- - 19 a z  + 2 a  z  + 8 a  z  - 2 a  z  - 
                     a

                                         5
    4      2  4      4  4      6  4   6 z          5      3  5
 7 z  + 2 a  z  + 6 a  z  - 3 a  z  + ---- + 14 a z  + 4 a  z  - 
                                       a

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

  2  8


  a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 40]], Vassiliev[3][Link[9, Alternating, 40]]}
Out[14]=  
    5

{0, -}


    2
In[15]:=
Kh[Link[9, Alternating, 40]][q, t]
Out[15]=  
3    3      1        1        1        2        1       2       2

-- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



4    2    14  5    12  4    10  4    10  3    8  3    8  2    6  2



q    q    q   t    q   t    q   t    q   t    q  t    q  t    q  t



  1      2         2 t      2  2    2  3    6  4
 ---- + ---- + t + --- + 2 q  t  + q  t  + q  t
  6      4          2


  q  t   q  t       q
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