L9a34

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L9a33.gif

L9a33

L9a35.gif

L9a35

L9a34.gif Visit L9a34's page at Knotilus!

Visit L9a34's page at the original Knot Atlas!

L9a34 is [math]\displaystyle{ 9^2_{6} }[/math] in the Rolfsen table of links.


L9a34 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{29}{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:L9a34/V 2,1 Data:L9a34/V 3,1 Data:L9a34/V 4,1 Data:L9a34/V 4,2 Data:L9a34/V 4,3 Data:L9a34/V 5,1 Data:L9a34/V 5,2 Data:L9a34/V 5,3 Data:L9a34/V 5,4 Data:L9a34/V 6,1 Data:L9a34/V 6,2 Data:L9a34/V 6,3 Data:L9a34/V 6,4 Data:L9a34/V 6,5 Data:L9a34/V 6,6 Data:L9a34/V 6,7 Data:L9a34/V 6,8 Data:L9a34/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 L9a34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-1012χ
2         11
0        1 -1
-2       31 2
-4      32  -1
-6     32   1
-8    33    0
-10   33     0
-12  14      3
-14 12       -1
-16 1        1
-181         -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=-7 }[/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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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 }[/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, 34]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 34]]
Out[3]=  
PD[X[10, 1, 11, 2], X[14, 5, 15, 6], X[12, 3, 13, 4], X[16, 8, 17, 7], 

 X[18, 15, 9, 16], X[4, 13, 5, 14], X[6, 18, 7, 17], X[2, 9, 3, 10], 



  X[8, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 34]]
Out[4]=  
GaussCode[{1, -8, 3, -6, 2, -7, 4, -9}, 
  {8, -1, 9, -3, 6, -2, 5, -4, 7, -5}]
In[5]:=
BR[Link[9, Alternating, 34]]
Out[5]=  
BR[Link[9, Alternating, 34]]
In[6]:=
alex = Alexander[Link[9, Alternating, 34]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 34]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 34]], KnotSignature[Link[9, Alternating, 34]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 34]][q]
Out[10]=  
 -(17/2)     2       3       6      6      6      5      4

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



           15/2    13/2    11/2    9/2    7/2    5/2    3/2
          q       q       q       q      q      q      q

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


  Sqrt[q]
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 34]][q]
Out[12]=  
  -26    3     -16    2     -12   2     -6    -4    2

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



        18           14           8


        q            q            q
In[13]:=
Kauffman[Link[9, Alternating, 34]][a, z]
Out[13]=  
      3    5

4   a    a               3        5      7        9        2  2



a  - -- - -- - 2 a z + 3 a  z + 6 a  z - a  z - 2 a  z - 2 a  z  - 



    z    z

  4  2    6  2      10  2        3      3  3       5  3      7  3
 a  z  - a  z  + 2 a   z  + 3 a z  - 2 a  z  - 12 a  z  - 3 a  z  + 

    9  3      2  4      4  4      8  4    10  4      5      3  5
 4 a  z  + 5 a  z  + 2 a  z  + 2 a  z  - a   z  - a z  + 4 a  z  + 

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

    5  7      7  7    4  8    6  8


  4 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 34]], Vassiliev[3][Link[9, Alternating, 34]]}
Out[14]=  
      29

{0, -(--)}


      24
In[15]:=
Kh[Link[9, Alternating, 34]][q, t]
Out[15]=  
2    3      1        1        1        2        1        4

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



4    2    18  7    16  6    14  6    14  5    12  5    12  4



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



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


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