L8a3

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L8a2.gif

L8a2

L8a4.gif

L8a4

L8a3.gif Visit L8a3's page at Knotilus!

Visit L8a3's page at the original Knot Atlas!

L8a3 is [math]\displaystyle{ 8^2_{9} }[/math] in the Rolfsen table of links.


L8a3 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{2 u v^2-4 u v+u+v^3-4 v^2+2 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^7 z^{-1} -3 z a^5-2 a^5 z^{-1} +2 z^3 a^3+3 z a^3+2 a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^5 a^7+3 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-2 z^2 a^6-z^7 a^5-2 z^5 a^5+11 z^3 a^5-8 z a^5+2 a^5 z^{-1} -5 z^6 a^4+11 z^4 a^4-6 z^2 a^4+a^4-z^7 a^3-4 z^5 a^3+12 z^3 a^3-9 z a^3+2 a^3 z^{-1} -3 z^6 a^2+4 z^4 a^2-3 z^2 a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{125}{24} }[/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:L8a3/V 2,1 Data:L8a3/V 3,1 Data:L8a3/V 4,1 Data:L8a3/V 4,2 Data:L8a3/V 4,3 Data:L8a3/V 5,1 Data:L8a3/V 5,2 Data:L8a3/V 5,3 Data:L8a3/V 5,4 Data:L8a3/V 6,1 Data:L8a3/V 6,2 Data:L8a3/V 6,3 Data:L8a3/V 6,4 Data:L8a3/V 6,5 Data:L8a3/V 6,6 Data:L8a3/V 6,7 Data:L8a3/V 6,8 Data:L8a3/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 L8a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012χ
4        11
2       1 -1
0      31 2
-2     32  -1
-4    22   0
-6   23    1
-8  22     0
-10 13      2
-12 1       -1
-141        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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}_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, 3]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 3]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[16, 11, 5, 12], 
  X[12, 15, 13, 16], X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[8, Alternating, 3]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 3, -6, 8, -2, 4, -5, 6, -3, 5, -4}]
In[5]:=
BR[Link[8, Alternating, 3]]
Out[5]=  
BR[Link[8, Alternating, 3]]
In[6]:=
alex = Alexander[Link[8, Alternating, 3]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 3]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 3]], KnotSignature[Link[8, Alternating, 3]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[8, Alternating, 3]][q]
Out[10]=  
  -(13/2)     2      4      4      5      5        4

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



            11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q      q      q      q

  3/2


  q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 3]][q]
Out[12]=  
     -22    2     -16    -14    -12    -10    -6    -4    2    6

2 + q    + --- + q    + q    - q    + q    + q   + q   - q  + q



           20


           q
In[13]:=
Kauffman[Link[8, Alternating, 3]][a, z]
Out[13]=  
            3      5    7

4   a   2 a    2 a    a    z              3        5        7      2



a  + - + ---- + ---- + -- + - - 3 a z - 9 a  z - 8 a  z - 3 a  z + z  - 



    z    z      z     z    a

                                3
    2  2      4  2      6  2   z         3       3  3       5  3
 3 a  z  - 6 a  z  - 2 a  z  - -- + 3 a z  + 12 a  z  + 11 a  z  + 
                               a

    7  3      4      2  4       4  4      6  4        5      3  5
 3 a  z  - 2 z  + 4 a  z  + 11 a  z  + 5 a  z  - 3 a z  - 4 a  z  - 

    5  5    7  5      2  6      4  6      6  6    3  7    5  7


  2 a  z  - a  z  - 3 a  z  - 5 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[8, Alternating, 3]], Vassiliev[3][Link[8, Alternating, 3]]}
Out[14]=  
      125

{0, -(---)}


      24
In[15]:=
Kh[Link[8, Alternating, 3]][q, t]
Out[15]=  
    2      1        1        1        3        2       2       2

3 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



    2    14  6    12  5    10  5    10  4    8  4    8  3    6  3
   q    q   t    q   t    q   t    q   t    q  t    q  t    q  t

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


  q  t    q  t    q  t   q  t
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