L9a35

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L9a34.gif

L9a34

L9a36.gif

L9a36

L9a35.gif Visit L9a35's page at Knotilus!

Visit L9a35's page at the original Knot Atlas!

L9a35 is [math]\displaystyle{ 9^2_{9} }[/math] in the Rolfsen table of links.



More symmetric depiction

Knot presentations

Planar diagram presentation X10,1,11,2 X12,3,13,4 X16,8,17,7 X14,6,15,5 X18,13,9,14 X6,16,7,15 X4,18,5,17 X2,9,3,10 X8,11,1,12
Gauss code {1, -8, 2, -7, 4, -6, 3, -9}, {8, -1, 9, -2, 5, -4, 6, -3, 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)-1) (t(2)-1) \left(t(2) t(1)^2+t(2)^2 t(1)-t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{9/2}+\frac{1}{q^{9/2}}+3 q^{7/2}-\frac{2}{q^{7/2}}-5 q^{5/2}+\frac{3}{q^{5/2}}+6 q^{3/2}-\frac{6}{q^{3/2}}-7 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature 1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a z^5+z^5 a^{-1} -a^3 z^3+3 a z^3+2 z^3 a^{-1} -z^3 a^{-3} -2 a^3 z+3 a z-z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^2 z^8-z^8-2 a^3 z^7-5 a z^7-3 z^7 a^{-1} -a^4 z^6-5 z^6 a^{-2} -4 z^6+8 a^3 z^5+14 a z^5+z^5 a^{-1} -5 z^5 a^{-3} +4 a^4 z^4+7 a^2 z^4+6 z^4 a^{-2} -3 z^4 a^{-4} +12 z^4-10 a^3 z^3-12 a z^3+4 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -4 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -5 z^2+4 a^3 z+6 a z-2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1} }[/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:L9a35/V 2,1 Data:L9a35/V 3,1 Data:L9a35/V 4,1 Data:L9a35/V 4,2 Data:L9a35/V 4,3 Data:L9a35/V 5,1 Data:L9a35/V 5,2 Data:L9a35/V 5,3 Data:L9a35/V 5,4 Data:L9a35/V 6,1 Data:L9a35/V 6,2 Data:L9a35/V 6,3 Data:L9a35/V 6,4 Data:L9a35/V 6,5 Data:L9a35/V 6,6 Data:L9a35/V 6,7 Data:L9a35/V 6,8 Data:L9a35/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]1 is the signature of L9a35. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
10         11
8        2 -2
6       31 2
4      32  -1
2     43   1
0    45    1
-2   22     0
-4  14      3
-6 12       -1
-8 1        1
-101         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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 }[/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, 35]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 35]]
Out[3]=  
PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[14, 6, 15, 5], 

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



  X[8, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 35]]
Out[4]=  
GaussCode[{1, -8, 2, -7, 4, -6, 3, -9}, 
  {8, -1, 9, -2, 5, -4, 6, -3, 7, -5}]
In[5]:=
BR[Link[9, Alternating, 35]]
Out[5]=  
BR[Link[9, Alternating, 35]]
In[6]:=
alex = Alexander[Link[9, Alternating, 35]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 35]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 35]], KnotSignature[Link[9, Alternating, 35]]}
Out[9]=  
{Infinity, 1}
In[10]:=
J=Jones[Link[9, Alternating, 35]][q]
Out[10]=  
 -(9/2)    2      3      6        6                     3/2      5/2

q       - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 6 q    - 5 q    + 



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

    7/2    9/2


  3 q    - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 35]][q]
Out[12]=  
     -14   3     -4   2       4    6    8    12    14

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



           6          2


           q          q
In[13]:=
Kauffman[Link[9, Alternating, 35]][a, z]
Out[13]=  
                                             2      2

    1    a   2 z              3        2   z    2 z       2  2



1 - --- - - - --- + 6 a z + 4 a  z - 5 z  + -- - ---- - 6 a  z  - 



   a z   z    3                             4     2
             a                             a     a

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

    4                          5    5
 6 z       2  4      4  4   5 z    z          5      3  5      6
 ---- + 7 a  z  + 4 a  z  - ---- + -- + 14 a z  + 8 a  z  - 4 z  - 
   2                          3    a
  a                          a

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


   a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 35]], Vassiliev[3][Link[9, Alternating, 35]]}
Out[14]=  
    3

{0, -}


    2
In[15]:=
Kh[Link[9, Alternating, 35]][q, t]
Out[15]=  
       2     1        1       1       2       1       4       2     4

5 + 4 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 



           10  5    8  4    6  4    6  3    4  3    4  2    2  2   t
          q   t    q  t    q  t    q  t    q  t    q  t    q  t

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


  q  t
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