L8a6

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L8a5.gif

L8a5

L8a7.gif

L8a7

L8a6.gif Visit L8a6's page at Knotilus!

Visit L8a6's page at the original Knot Atlas!

L8a6 is [math]\displaystyle{ 8^2_{6} }[/math] in the Rolfsen table of links.


L8a6 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{41}{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:L8a6/V 2,1 Data:L8a6/V 3,1 Data:L8a6/V 4,1 Data:L8a6/V 4,2 Data:L8a6/V 4,3 Data:L8a6/V 5,1 Data:L8a6/V 5,2 Data:L8a6/V 5,3 Data:L8a6/V 5,4 Data:L8a6/V 6,1 Data:L8a6/V 6,2 Data:L8a6/V 6,3 Data:L8a6/V 6,4 Data:L8a6/V 6,5 Data:L8a6/V 6,6 Data:L8a6/V 6,7 Data:L8a6/V 6,8 Data:L8a6/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 L8a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
8        11
6       1 -1
4      11 0
2     21  -1
0    21   1
-2   23    1
-4  11     0
-6  2      2
-811       0
-101        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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=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[8, Alternating, 6]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 6]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 5, 7], X[14, 10, 15, 9], 
  X[10, 14, 11, 13], X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[8, Alternating, 6]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}]
In[5]:=
BR[Link[8, Alternating, 6]]
Out[5]=  
BR[Link[8, Alternating, 6]]
In[6]:=
alex = Alexander[Link[8, Alternating, 6]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 6]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 6]], KnotSignature[Link[8, Alternating, 6]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[8, Alternating, 6]][q]
Out[10]=  
  -(9/2)    -(7/2)    3      3        4                     3/2

-q       + q       - ---- + ---- - ------- + 3 Sqrt[q] - 2 q    + 



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

    5/2    7/2


  2 q    - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 6]][q]
Out[12]=  
     -16    2     -12    2    2     -4    4    8    12

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



           14           10    8


           q            q     q
In[13]:=
Kauffman[Link[8, Alternating, 6]][a, z]
Out[13]=  
      3    5                                          2      3

4   a    a    z             3        5        2   5 z    3 z



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



    z    z     3                                    2      3
              a                                    a      a

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

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


                  a
In[14]:=
{Vassiliev[2][Link[8, Alternating, 6]], Vassiliev[3][Link[8, Alternating, 6]]}
Out[14]=  
      41

{0, -(--)}


      24
In[15]:=
Kh[Link[8, Alternating, 6]][q, t]
Out[15]=  
    3      1        1       1       2       1      1      2

2 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + t + 



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

    2      2  2    4  2    4  3    6  3    8  4


  2 q  t + q  t  + q  t  + q  t  + q  t  + q  t
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