L8a16

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L8a15.gif

L8a15

L8a17.gif

L8a17

L8a16.gif Visit L8a16's page at Knotilus!

Visit L8a16's page at the original Knot Atlas!

L8a16 is [math]\displaystyle{ 8^3_{5} }[/math] in the Rolfsen table of links.



Depiction obtained by knotilus

Knot presentations

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,10,11,9 X10,12,5,11 X4,15,1,16
Gauss code {1, -4, 3, -8}, {2, -1, 5, -3, 6, -7}, {7, -2, 4, -5, 8, -6}

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) (t(3)-1) (t(2) t(3)+1)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^6+3 q^5-4 q^4+6 q^3-5 q^2- q^{-2} +6 q+3 q^{-1} -3 }[/math] (db)
Signature 2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -2 z^2 a^{-4} -2 z^2- a^{-2} +1-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +7 z^6 a^{-2} +4 z^6 a^{-4} +3 z^6+a z^5-3 z^5 a^{-1} +4 z^5 a^{-5} -17 z^4 a^{-2} -5 z^4 a^{-4} +3 z^4 a^{-6} -9 z^4-2 a z^3-z^3 a^{-1} -3 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +10 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2-z a^{-1} -z a^{-3} + a^{-2} + a^{-4} +1+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{5}{6} }[/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:L8a16/V 2,1 Data:L8a16/V 3,1 Data:L8a16/V 4,1 Data:L8a16/V 4,2 Data:L8a16/V 4,3 Data:L8a16/V 5,1 Data:L8a16/V 5,2 Data:L8a16/V 5,3 Data:L8a16/V 5,4 Data:L8a16/V 6,1 Data:L8a16/V 6,2 Data:L8a16/V 6,3 Data:L8a16/V 6,4 Data:L8a16/V 6,5 Data:L8a16/V 6,6 Data:L8a16/V 6,7 Data:L8a16/V 6,8 Data:L8a16/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]2 is the signature of L8a16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-1012345χ
13        1-1
11       2 2
9      21 -1
7     42  2
5    34   1
3   32    1
1  25     3
-1 11      0
-3 2       2
-51        -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=3 }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/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, 16]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 16]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], 
  X[14, 7, 15, 8], X[16, 10, 11, 9], X[10, 12, 5, 11], X[4, 15, 1, 16]]
In[4]:=
GaussCode[Link[8, Alternating, 16]]
Out[4]=  
GaussCode[{1, -4, 3, -8}, {2, -1, 5, -3, 6, -7}, {7, -2, 4, -5, 8, -6}]
In[5]:=
BR[Link[8, Alternating, 16]]
Out[5]=  
BR[Link[8, Alternating, 16]]
In[6]:=
alex = Alexander[Link[8, Alternating, 16]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 16]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 16]], KnotSignature[Link[8, Alternating, 16]]}
Out[9]=  
{Infinity, 2}
In[10]:=
J=Jones[Link[8, Alternating, 16]][q]
Out[10]=  
      -2   3            2      3      4      5    6

-3 - q   + - + 6 q - 5 q  + 6 q  - 4 q  + 3 q  - q


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

3 - q   + q   + q   + 5 q  + 3 q  + 6 q  + 3 q  + 4 q   + 2 q   + q   - 



  18


  q
In[13]:=
Kauffman[Link[8, Alternating, 16]][a, z]
Out[13]=  
     -4    -2    -2     1       2      2      2    z    z      2

1 + a   + a   - z   - ----- - ----- + ---- + --- - -- - - + 6 z  - 



                      4  2    2  2    3     a z    3   a
                     a  z    a  z    a  z         a

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

    4       4      5      5                    6      6      7      7
 5 z    17 z    4 z    3 z       5      6   4 z    7 z    2 z    2 z
 ---- - ----- + ---- - ---- + a z  + 3 z  + ---- + ---- + ---- + ----
   4      2       5     a                     4      2      3     a


   a      a       a                           a      a      a
In[14]:=
{Vassiliev[2][Link[8, Alternating, 16]], Vassiliev[3][Link[8, Alternating, 16]]}
Out[14]=  
      5

{0, -(-)}


      6
In[15]:=
Kh[Link[8, Alternating, 16]][q, t]
Out[15]=  
         3     1       2      1      1    2 q      3        5

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



             5  3    3  2      2   q t    t
            q  t    q  t    q t

    5  2      7  2      7  3      9  3    9  4      11  4    13  5


  4 q  t  + 4 q  t  + 2 q  t  + 2 q  t  + q  t  + 2 q   t  + q   t
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