L8a4

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L8a3.gif

L8a3

L8a5.gif

L8a5

L8a4.gif Visit L8a4's page at Knotilus!

Visit L8a4's page at the original Knot Atlas!

L8a4 is [math]\displaystyle{ 8^2_{12} }[/math] in the Rolfsen table of links.


L8a4 Further Notes and Views

Knot presentations

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

Vassiliev invariants

V2 and V3: (0, 2)
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:L8a4/V 2,1 Data:L8a4/V 3,1 Data:L8a4/V 4,1 Data:L8a4/V 4,2 Data:L8a4/V 4,3 Data:L8a4/V 5,1 Data:L8a4/V 5,2 Data:L8a4/V 5,3 Data:L8a4/V 5,4 Data:L8a4/V 6,1 Data:L8a4/V 6,2 Data:L8a4/V 6,3 Data:L8a4/V 6,4 Data:L8a4/V 6,5 Data:L8a4/V 6,6 Data:L8a4/V 6,7 Data:L8a4/V 6,8 Data:L8a4/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 L8a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123χ
6        1-1
4       2 2
2      21 -1
0     42  2
-2    34   1
-4   32    1
-6  13     2
-8 13      -2
-10 1       1
-121        -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=-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{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[8, Alternating, 4]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 4]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[16, 13, 5, 14], 
  X[14, 7, 15, 8], X[8, 15, 9, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[4]:=
GaussCode[Link[8, Alternating, 4]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}]
In[5]:=
BR[Link[8, Alternating, 4]]
Out[5]=  
BR[Link[8, Alternating, 4]]
In[6]:=
alex = Alexander[Link[8, Alternating, 4]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 4]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 4]], KnotSignature[Link[8, Alternating, 4]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[8, Alternating, 4]][q]
Out[10]=  
 -(11/2)    2      4      6      5        6                     3/2

q        - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q    + 



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

  5/2


  q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 4]][q]
Out[12]=  
     -18    -16    2     -10   2    2    4     -2    6    8

3 - q    - q    - --- + q    + -- + -- + -- + q   + q  - q



                  12           8    6    4


                  q            q    q    q
In[13]:=
Kauffman[Link[8, Alternating, 4]][a, z]
Out[13]=  
                             3    5

   2      4    6   2 a   3 a    a    2 z              3        5



-3 a  - 3 a  - a  + --- + ---- + -- - --- - 7 a z - 7 a  z - 2 a  z + 



                    z     z     z     a

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

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

    5  5      6      2  6      4  6      7    3  7


  2 a  z  - 3 z  - 5 a  z  - 2 a  z  - a z  - a  z
In[14]:=
{Vassiliev[2][Link[8, Alternating, 4]], Vassiliev[3][Link[8, Alternating, 4]]}
Out[14]=  
{0, 2}
In[15]:=
Kh[Link[8, Alternating, 4]][q, t]
Out[15]=  
    4      1        1        1       3       1       3       3

4 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 



    2    12  5    10  4    8  4    8  3    6  3    6  2    4  2
   q    q   t    q   t    q  t    q  t    q  t    q  t    q  t

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


  q  t   q  t
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