L8a2

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L8a1.gif

L8a1

L8a3.gif

L8a3

L8a2.gif Visit L8a2's page at Knotilus!

Visit L8a2's page at the original Knot Atlas!

L8a2 is [math]\displaystyle{ 8^2_{10} }[/math] in the Rolfsen table of links.


L8a2 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X16,14,5,13 X14,9,15,10 X8,15,9,16 X2536 X4,12,1,11
Gauss code {1, -7, 2, -8}, {7, -1, 3, -6, 5, -2, 8, -3, 4, -5, 6, -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)^3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{9/2}+3 q^{7/2}-5 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{4}{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+3 z^3 a^{-1} -z^3 a^{-3} +a^3 z-4 a z+4 z a^{-1} -z a^{-3} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^3 a^{-5} +3 z^4 a^{-4} -z^2 a^{-4} +a^3 z^5+5 z^5 a^{-3} -3 a^3 z^3-6 z^3 a^{-3} +3 a^3 z+3 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^6+4 z^6 a^{-2} -5 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +a z^7+z^7 a^{-1} +3 a z^5+7 z^5 a^{-1} -13 a z^3-17 z^3 a^{-1} +10 a z+10 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+5 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, 0)
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:L8a2/V 2,1 Data:L8a2/V 3,1 Data:L8a2/V 4,1 Data:L8a2/V 4,2 Data:L8a2/V 4,3 Data:L8a2/V 5,1 Data:L8a2/V 5,2 Data:L8a2/V 5,3 Data:L8a2/V 5,4 Data:L8a2/V 6,1 Data:L8a2/V 6,2 Data:L8a2/V 6,3 Data:L8a2/V 6,4 Data:L8a2/V 6,5 Data:L8a2/V 6,6 Data:L8a2/V 6,7 Data:L8a2/V 6,8 Data:L8a2/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 L8a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
10        11
8       2 -2
6      31 2
4     22  0
2    43   1
0   34    1
-2  12     -1
-4 13      2
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{4}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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[8, Alternating, 2]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 2]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 14, 5, 13], 
  X[14, 9, 15, 10], X[8, 15, 9, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[4]:=
GaussCode[Link[8, Alternating, 2]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 3, -6, 5, -2, 8, -3, 4, -5, 6, -4}]
In[5]:=
BR[Link[8, Alternating, 2]]
Out[5]=  
BR[Link[8, Alternating, 2]]
In[6]:=
alex = Alexander[Link[8, Alternating, 2]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 2]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 2]], KnotSignature[Link[8, Alternating, 2]]}
Out[9]=  
{Infinity, 1}
In[10]:=
J=Jones[Link[8, Alternating, 2]][q]
Out[10]=  
  -(7/2)    2      4        5                     3/2      5/2

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



           5/2    3/2   Sqrt[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[8, Alternating, 2]][q]
Out[12]=  
     -12    -10   2     2      4      8    10    12    14

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



                  6


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

    1      2    2 a   a    3 z   10 z               3        2   z



1 + ---- + --- + --- + -- - --- - ---- - 10 a z - 3 a  z - 5 z  + -- - 



    3     a z    z    z     3     a                               4
   a  z                    a                                     a

  2              3      3       3                                  4
 z       2  2   z    6 z    17 z          3      3  3       4   3 z
 -- - 3 a  z  - -- + ---- + ----- + 13 a z  + 3 a  z  + 12 z  - ---- + 
  2              5     3      a                                   4
 a              a     a                                          a

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

            7
    2  6   z       7
 2 a  z  - -- - a z


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

4 + 4 q  + ----- + ----- + ----- + ----- + ----- + - + ---- + 3 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    10  4


  2 q  t + 2 q  t  + 3 q  t  + q  t  + 2 q  t  + q   t
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