L9a22

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L9a21.gif

L9a21

L9a23.gif

L9a23

L9a22.gif Visit L9a22's page at Knotilus!

Visit L9a22's page at the original Knot Atlas!

L9a22 is [math]\displaystyle{ 9^2_{35} }[/math] in the Rolfsen table of links.


L9a22 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,12,17,11 X12,6,13,5 X4,17,5,18 X14,7,15,8 X18,13,7,14 X6,16,1,15
Gauss code {1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+1}{u v^2} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^3 z^7+a^5 z^5-5 a^3 z^5+a z^5+3 a^5 z^3-8 a^3 z^3+3 a z^3+2 a^5 z-4 a^3 z+a z+a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^3 a^9-3 z^4 a^8+z^2 a^8-5 z^5 a^7+4 z^3 a^7-z a^7-6 z^6 a^6+8 z^4 a^6-3 z^2 a^6-5 z^7 a^5+8 z^5 a^5-3 z^3 a^5+z a^5-2 z^8 a^4-2 z^6 a^4+13 z^4 a^4-7 z^2 a^4-8 z^7 a^3+24 z^5 a^3-19 z^3 a^3+4 z a^3+a^3 z^{-1} -2 z^8 a^2+3 z^6 a^2+5 z^4 a^2-5 z^2 a^2-a^2-3 z^7 a+11 z^5 a-11 z^3 a+2 z a+a z^{-1} -z^6+3 z^4-2 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{65}{48} }[/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:L9a22/V 2,1 Data:L9a22/V 3,1 Data:L9a22/V 4,1 Data:L9a22/V 4,2 Data:L9a22/V 4,3 Data:L9a22/V 5,1 Data:L9a22/V 5,2 Data:L9a22/V 5,3 Data:L9a22/V 5,4 Data:L9a22/V 6,1 Data:L9a22/V 6,2 Data:L9a22/V 6,3 Data:L9a22/V 6,4 Data:L9a22/V 6,5 Data:L9a22/V 6,6 Data:L9a22/V 6,7 Data:L9a22/V 6,8 Data:L9a22/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 L9a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-10123χ
4         1-1
2        2 2
0       21 -1
-2      52  3
-4     33   0
-6    54    1
-8   34     1
-10  24      -2
-12 13       2
-14 2        -2
-161         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=-6 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/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, 22]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 22]]
Out[3]=  
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 12, 17, 11], 

 X[12, 6, 13, 5], X[4, 17, 5, 18], X[14, 7, 15, 8], X[18, 13, 7, 14], 



  X[6, 16, 1, 15]]
In[4]:=
GaussCode[Link[9, Alternating, 22]]
Out[4]=  
GaussCode[{1, -2, 3, -6, 5, -9}, 
  {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}]
In[5]:=
BR[Link[9, Alternating, 22]]
Out[5]=  
BR[Link[9, Alternating, 22]]
In[6]:=
alex = Alexander[Link[9, Alternating, 22]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 22]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 22]], KnotSignature[Link[9, Alternating, 22]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 22]][q]
Out[10]=  
  -(15/2)     3       5      7      8      7      7        4

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



            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q

              3/2


  3 Sqrt[q] + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 22]][q]
Out[12]=  
 -22    -20    -18    -16    -14    -12    -10   4     -6   3    2

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



                                                 8          4    2
                                                q          q    q

  2    4


  q  - q
In[13]:=
Kauffman[Link[9, Alternating, 22]][a, z]
Out[13]=  
           3

 2   a   a               3      5      7        2      2  2



-a  + - + -- + 2 a z + 4 a  z + a  z - a  z - 2 z  - 5 a  z  - 



     z   z

    4  2      6  2    8  2         3       3  3      5  3      7  3
 7 a  z  - 3 a  z  + a  z  - 11 a z  - 19 a  z  - 3 a  z  + 4 a  z  - 

  9  3      4      2  4       4  4      6  4      8  4         5
 a  z  + 3 z  + 5 a  z  + 13 a  z  + 8 a  z  - 3 a  z  + 11 a z  + 

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

      7      3  7      5  7      2  8      4  8


  3 a z  - 8 a  z  - 5 a  z  - 2 a  z  - 2 a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 22]], Vassiliev[3][Link[9, Alternating, 22]]}
Out[14]=  
      65

{0, -(--)}


      48
In[15]:=
Kh[Link[9, Alternating, 22]][q, t]
Out[15]=  
3    5      1        2        1        3        2        4        3

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



4    2    16  6    14  5    12  5    12  4    10  4    10  3    8  3



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



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


  q  t    q  t    q  t   q  t         q
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