L8a18

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L8a17.gif

L8a17

L8a19.gif

L8a19

L8a18.gif Visit L8a18's page at Knotilus!

Visit L8a18's page at the original Knot Atlas!

L8a18 is [math]\displaystyle{ 8^3_{1} }[/math] in the Rolfsen table of links.


L8a18 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{u v^2 w^2-u v w^2+u v w-u w+u-v^2 w^2+v^2 w-v w+v-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^8-2 q^7+3 q^6-3 q^5+4 q^4-2 q^3+3 q^2-q+1 }[/math] (db)
Signature 4 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^4 a^{-6} +3 z^2 a^{-6} + a^{-6} z^{-2} +2 a^{-6} -z^6 a^{-4} -5 z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} -6 a^{-4} +z^4 a^{-2} +4 z^2 a^{-2} + a^{-2} z^{-2} +4 a^{-2} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^2 a^{-10} +2 z^3 a^{-9} +3 z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +3 z^5 a^{-7} -4 z^3 a^{-7} +3 z^6 a^{-6} -8 z^4 a^{-6} +6 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +z^7 a^{-5} -7 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +4 z^6 a^{-4} -16 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -5 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{26}{3} }[/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:L8a18/V 2,1 Data:L8a18/V 3,1 Data:L8a18/V 4,1 Data:L8a18/V 4,2 Data:L8a18/V 4,3 Data:L8a18/V 5,1 Data:L8a18/V 5,2 Data:L8a18/V 5,3 Data:L8a18/V 5,4 Data:L8a18/V 6,1 Data:L8a18/V 6,2 Data:L8a18/V 6,3 Data:L8a18/V 6,4 Data:L8a18/V 6,5 Data:L8a18/V 6,6 Data:L8a18/V 6,7 Data:L8a18/V 6,8 Data:L8a18/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]4 is the signature of L8a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-10123456χ
17        11
15       21-1
13      1  1
11     22  0
9    21   1
7   13    2
5  21     1
3 13      2
1         0
-11        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/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}^{2} }[/math]
[math]\displaystyle{ r=1 }[/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}^{2} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/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, 18]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 18]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 16, 9, 15], X[14, 8, 15, 7], 
  X[16, 10, 11, 9], X[10, 12, 5, 11], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=
GaussCode[Link[8, Alternating, 18]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 4, -3, 5, -6}, {6, -2, 8, -4, 3, -5}]
In[5]:=
BR[Link[8, Alternating, 18]]
Out[5]=  
BR[Link[8, Alternating, 18]]
In[6]:=
alex = Alexander[Link[8, Alternating, 18]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 18]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 18]], KnotSignature[Link[8, Alternating, 18]]}
Out[9]=  
{Infinity, 4}
In[10]:=
J=Jones[Link[8, Alternating, 18]][q]
Out[10]=  
           2      3      4      5      6      7    8
1 - q + 3 q  - 2 q  + 4 q  - 3 q  + 3 q  - 2 q  + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 18]][q]
Out[12]=  
     2      4      6      8      10      12      14      16    18

1 + q  + 2 q  + 3 q  + 3 q  + 5 q   + 3 q   + 4 q   + 2 q   + q   + 



  20    24


  q   + q
In[13]:=
Kauffman[Link[8, Alternating, 18]][a, z]
Out[13]=  
 -8   3    8    5      1       2       1      2      2     6 z   6 z

a   - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 



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

  2       2      2       2      2      3      3      3    3      4
 z     3 z    6 z    18 z    8 z    2 z    4 z    7 z    z    3 z
 --- - ---- + ---- + ----- + ---- + ---- - ---- - ---- - -- + ---- - 
  10     8      6      4       2      9      7      5     3     8
 a      a      a      a       a      a      a      a     a     a

    4       4      4      5      5      6      6    6    7    7
 8 z    16 z    5 z    3 z    3 z    3 z    4 z    z    z    z
 ---- - ----- - ---- + ---- - ---- + ---- + ---- + -- + -- + --
   6      4       2      7      3      6      4     2    5    3


   a      a       a      a      a      a      a     a    a    a
In[14]:=
{Vassiliev[2][Link[8, Alternating, 18]], Vassiliev[3][Link[8, Alternating, 18]]}
Out[14]=  
      26

{0, -(--)}


      3
In[15]:=
Kh[Link[8, Alternating, 18]][q, t]
Out[15]=  
                      3

  3      5    1     q     5      7        7  2      9  2    9  3



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



                2   t
             q t

    11  3      11  4    13  4      15  5    15  6    17  6


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