L8a20

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L8a19.gif

L8a19

L8a21.gif

L8a21

L8a20.gif Visit L8a20's page at Knotilus!

Visit L8a20's page at the original Knot Atlas!

L8a20 is [math]\displaystyle{ 8^3_{4} }[/math] in the Rolfsen table of links.



Depiction obtained with knotilus

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{(w-1) \left(u v w-u v-2 u w-2 v w-w^2+w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^4-2 q^3+5 q^2-5 q+6-5 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4} }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^4+ a^{-4} -2 a^2 z^2+a^2 z^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +z^4-2 z^{-2} -2 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^4 z^4+z^4 a^{-4} -2 a^4 z^2-2 z^2 a^{-4} +a^4+ a^{-4} +2 a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3-2 z^3 a^{-3} +3 a^2 z^6+3 z^6 a^{-2} -5 a^2 z^4-5 z^4 a^{-2} +5 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-4 a^{-2} +a z^7+z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} -12 a z^3-12 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+14 z^2+2 z^{-2} -9 }[/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:L8a20/V 2,1 Data:L8a20/V 3,1 Data:L8a20/V 4,1 Data:L8a20/V 4,2 Data:L8a20/V 4,3 Data:L8a20/V 5,1 Data:L8a20/V 5,2 Data:L8a20/V 5,3 Data:L8a20/V 5,4 Data:L8a20/V 6,1 Data:L8a20/V 6,2 Data:L8a20/V 6,3 Data:L8a20/V 6,4 Data:L8a20/V 6,5 Data:L8a20/V 6,6 Data:L8a20/V 6,7 Data:L8a20/V 6,8 Data:L8a20/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]0 is the signature of L8a20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
9        11
7       21-1
5      3  3
3     22  0
1    43   1
-1   34    1
-3  22     0
-5  3      3
-712       -1
-91        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/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}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{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}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 20]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 20]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 12, 13, 11], X[14, 8, 15, 7], 
  X[8, 14, 9, 13], X[12, 16, 5, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[8, Alternating, 20]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {5, -4, 6, -3}, {7, -1, 4, -5, 8, -2, 3, -6}]
In[5]:=
BR[Link[8, Alternating, 20]]
Out[5]=  
BR[Link[8, Alternating, 20]]
In[6]:=
alex = Alexander[Link[8, Alternating, 20]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 20]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 20]], KnotSignature[Link[8, Alternating, 20]]}
Out[9]=  
{Infinity, 0}
In[10]:=
J=Jones[Link[8, Alternating, 20]][q]
Out[10]=  
     -4   2    5    5            2      3    4

6 + q   - -- + -- - - - 5 q + 5 q  - 2 q  + q



          3    2   q


          q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 20]][q]
Out[12]=  
     -14    -12    -10   2    2    2    5       2      4      6

3 + q    + q    - q    + -- + -- + -- + -- + 5 q  + 2 q  + 2 q  + 



                         8    6    4    2
                        q    q    q    q

    8    10    12    14


  2 q  - q   + q   + q
In[13]:=
Kauffman[Link[8, Alternating, 20]][a, z]
Out[13]=  
                                          2

     -4   4       2    4   2      1     a     2    2 a   8 z



-9 + a   - -- - 4 a  + a  + -- + ----- + -- - --- - --- + --- + 8 a z + 



           2                2    2  2    2   a z    z     a
          a                z    a  z    z

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

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

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


                             a
In[14]:=
{Vassiliev[2][Link[8, Alternating, 20]], Vassiliev[3][Link[8, Alternating, 20]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Link[8, Alternating, 20]][q, t]
Out[15]=  
4           1       1       2       3       2      2      3

- + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 
q          9  4    7  4    7  3    5  2    3  2    3     q t



         q  t    q  t    q  t    q  t    q  t    q  t

    3        3  2      5  2      7  3    7  4    9  4


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