L9a8

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L9a7.gif

L9a7

L9a9.gif

L9a9

L9a8.gif Visit L9a8's page at Knotilus!

Visit L9a8's page at the original Knot Atlas!

L9a8 is [math]\displaystyle{ 9^2_{25} }[/math] in the Rolfsen table of links.


L9a8 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X10,4,11,3 X16,8,17,7 X18,13,5,14 X14,17,15,18 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -8, 2, -9}, {8, -1, 3, -7, 6, -2, 9, -6, 4, -5, 7, -3, 5, -4}

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)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-8 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature 1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^5 a^{-1} -2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-3 a z+3 z a^{-1} -2 z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -4 z^3 a^{-5} +2 z a^{-5} +4 z^6 a^{-4} -4 z^4 a^{-4} +z^2 a^{-4} +3 z^7 a^{-3} +a^3 z^5+z^5 a^{-3} -3 a^3 z^3-7 z^3 a^{-3} +3 a^3 z+5 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +z^8 a^{-2} +2 a^2 z^6+6 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +2 a^2 z^2+3 z^2 a^{-2} +2 a z^7+5 z^7 a^{-1} -3 z^5 a^{-1} -8 a z^3-8 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8+4 z^6-9 z^4+3 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, -1)
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:L9a8/V 2,1 Data:L9a8/V 3,1 Data:L9a8/V 4,1 Data:L9a8/V 4,2 Data:L9a8/V 4,3 Data:L9a8/V 5,1 Data:L9a8/V 5,2 Data:L9a8/V 5,3 Data:L9a8/V 5,4 Data:L9a8/V 6,1 Data:L9a8/V 6,2 Data:L9a8/V 6,3 Data:L9a8/V 6,4 Data:L9a8/V 6,5 Data:L9a8/V 6,6 Data:L9a8/V 6,7 Data:L9a8/V 6,8 Data:L9a8/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 L9a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
12         1-1
10        2 2
8       31 -2
6      52  3
4     33   0
2    55    0
0   45     1
-2  13      -2
-4 14       3
-6 1        -1
-81         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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/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, 8]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 8]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 17, 7], X[18, 13, 5, 14], 

 X[14, 17, 15, 18], X[12, 10, 13, 9], X[8, 16, 9, 15], X[2, 5, 3, 6], 



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

-q       + ---- - ---- + ------- - 8 Sqrt[q] + 8 q    - 8 q    + 



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

    7/2      9/2    11/2


  5 q    - 3 q    + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 8]][q]
Out[12]=  
     -12    -10   3     -2      2      4      8      10    12      14

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



                  6
                 q

  18


  q
In[13]:=
Kauffman[Link[9, Alternating, 8]][a, z]
Out[13]=  
                        3

    1      2    2 a   a    2 z   5 z   8 z              3        2



1 + ---- + --- + --- + -- - --- - --- - --- - 8 a z - 3 a  z - 3 z  + 



    3     a z    z    z     5     3     a
   a  z                    a     a

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

         4      4       4                5    5      5
    4   z    4 z    10 z       2  4   3 z    z    3 z     3  5
 9 z  - -- + ---- + ----- + 4 a  z  - ---- - -- + ---- - a  z  - 
         6     4      2                 5     3    a
        a     a      a                 a     a

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


          a      a                a                          a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 8]], Vassiliev[3][Link[9, Alternating, 8]]}
Out[14]=  
{0, -1}
In[15]:=
Kh[Link[9, Alternating, 8]][q, t]
Out[15]=  
       2     1       1       1       4       1     4    3        2

5 + 5 q  + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q  t + 



           8  4    6  3    4  3    4  2    2  2   t    2
          q  t    q  t    q  t    q  t    q  t        q  t

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

  12  5


  q   t
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