L8n8

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L8n7.gif

L8n7

L9a1.gif

L9a1

L8n8.gif Visit L8n8's page at Knotilus!

Visit L8n8's page at the original Knot Atlas!

L8n8 is [math]\displaystyle{ 8^4_{3} }[/math] in the Rolfsen table of links.



Detail from an 18th century royal decree, Vietnam.

Knot presentations

Planar diagram presentation X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16
Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{(t(2)-t(3)) (t(1)-t(4))}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^{7/2}-q^{3/2}-2 \sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^3 z^{-3} - a^{-3} z^{-3} +a^3 z+2 a^3 z^{-1} -z a^{-3} -2 a^{-3} z^{-1} -a z^3-3 a z^{-3} +z^3 a^{-1} +3 a^{-1} z^{-3} -5 a z-6 a z^{-1} +5 z a^{-1} +6 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^3 z^5-a z^5-z^5 a^{-1} -z^5 a^{-3} -a^2 z^4-z^4 a^{-2} -2 z^4+5 a^3 z^3+7 a z^3+7 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-6 a^3 z-14 a z-14 z a^{-1} -6 z a^{-3} -8 a^2-8 a^{-2} -15+4 a^3 z^{-1} +9 a z^{-1} +9 a^{-1} z^{-1} +4 a^{-3} z^{-1} +3 a^2 z^{-2} +3 a^{-2} z^{-2} +6 z^{-2} -a^3 z^{-3} -3 a z^{-3} -3 a^{-1} z^{-3} - a^{-3} z^{-3} }[/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:L8n8/V 2,1 Data:L8n8/V 3,1 Data:L8n8/V 4,1 Data:L8n8/V 4,2 Data:L8n8/V 4,3 Data:L8n8/V 5,1 Data:L8n8/V 5,2 Data:L8n8/V 5,3 Data:L8n8/V 5,4 Data:L8n8/V 6,1 Data:L8n8/V 6,2 Data:L8n8/V 6,3 Data:L8n8/V 6,4 Data:L8n8/V 6,5 Data:L8n8/V 6,6 Data:L8n8/V 6,7 Data:L8n8/V 6,8 Data:L8n8/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 L8n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
8        11
6        11
4      1  1
2    3    3
0   161   4
-2    3    3
-4  1      1
-61        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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/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, NonAlternating, 8]]
Out[2]=  
8
In[3]:=
PD[Link[8, NonAlternating, 8]]
Out[3]=  
PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[16, 11, 13, 12], X[3, 11, 4, 10], 
  X[9, 1, 10, 4], X[7, 15, 8, 14], X[13, 5, 14, 8], X[12, 15, 9, 16]]
In[4]:=
GaussCode[Link[8, NonAlternating, 8]]
Out[4]=  
GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, 
  {-7, 6, 8, -3}]
In[5]:=
BR[Link[8, NonAlternating, 8]]
Out[5]=  
BR[Link[8, NonAlternating, 8]]
In[6]:=
alex = Alexander[Link[8, NonAlternating, 8]][t]
Out[6]=  
Indeterminate
In[7]:=
Conway[Link[8, NonAlternating, 8]][z]
Out[7]=  
Indeterminate
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, NonAlternating, 8]], KnotSignature[Link[8, NonAlternating, 8]]}
Out[9]=  
{Indeterminate, 0}
In[10]:=
J=Jones[Link[8, NonAlternating, 8]][q]
Out[10]=  
  -(7/2)    -(3/2)      2                   3/2    7/2

-q       - q       - ------- - 2 Sqrt[q] - q    - q


                     Sqrt[q]
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, NonAlternating, 8]][q]
Out[12]=  
      -12    2    4    6    9    12       2      4      6      8

13 + q    + --- + -- + -- + -- + -- + 12 q  + 9 q  + 6 q  + 4 q  + 



            10    8    6    4    2
           q     q    q    q    q

    10    12


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

     8       2     1      3     3 a   a    6      3     3 a     4



-15 - -- - 8 a  - ----- - ---- - --- - -- + -- + ----- + ---- + ---- + 



      2           3  3      3    3     3    2    2  2     2     3
     a           a  z    a z    z     z    z    a  z     z     a  z

                3                                             2
  9    9 a   4 a    6 z   14 z               3         2   6 z
 --- + --- + ---- - --- - ---- - 14 a z - 6 a  z + 12 z  + ---- + 
 a z    z     z      3     a                                 2
                    a                                       a

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

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


  a
In[14]:=
{Vassiliev[2][Link[8, NonAlternating, 8]], Vassiliev[3][Link[8, NonAlternating, 8]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Link[8, NonAlternating, 8]][q, t]
Out[15]=  
    3       2     1       1       1     1        4  2    6  4    8  4

6 + -- + 3 q  + ----- + ----- + ----- + - + t + q  t  + q  t  + q  t



    2           8  4    6  4    4  2   t


    q           q  t    q  t    q  t
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