L8a17

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L8a16.gif

L8a16

L8a18.gif

L8a18

L8a17.gif Visit L8a17's page at Knotilus!

Visit L8a17's page at the original Knot Atlas!

L8a17 is [math]\displaystyle{ 8^3_{2} }[/math] in the Rolfsen table of links.


L8a17 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{t(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1)+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{-10} -2 q^{-9} +4 q^{-8} -4 q^{-7} +6 q^{-6} -4 q^{-5} +4 q^{-4} -2 q^{-3} + q^{-2} }[/math] (db)
Signature -4 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^{10} z^{-2} +a^{10}-3 a^8 z^2-2 a^8 z^{-2} -6 a^8+2 a^6 z^4+6 a^6 z^2+a^6 z^{-2} +5 a^6+a^4 z^4+2 a^4 z^2 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+a^{10} z^{-2} -3 a^{10}+a^9 z^7+2 a^9 z^5-6 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +5 a^8 z^6-12 a^8 z^4+15 a^8 z^2+2 a^8 z^{-2} -8 a^8+a^7 z^7+2 a^7 z^5-6 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +3 a^6 z^6-8 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -5 a^6+2 a^5 z^5-3 a^5 z^3+a^4 z^4-2 a^4 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{100}{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:L8a17/V 2,1 Data:L8a17/V 3,1 Data:L8a17/V 4,1 Data:L8a17/V 4,2 Data:L8a17/V 4,3 Data:L8a17/V 5,1 Data:L8a17/V 5,2 Data:L8a17/V 5,3 Data:L8a17/V 5,4 Data:L8a17/V 6,1 Data:L8a17/V 6,2 Data:L8a17/V 6,3 Data:L8a17/V 6,4 Data:L8a17/V 6,5 Data:L8a17/V 6,6 Data:L8a17/V 6,7 Data:L8a17/V 6,8 Data:L8a17/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 L8a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-10χ
-3        11
-5       21-1
-7      2  2
-9     22  0
-11    42   2
-13   13    2
-15  33     0
-17 13      2
-19 1       -1
-211        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/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=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/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, 17]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 17]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[8, 15, 9, 16], 
  X[14, 7, 15, 8], X[16, 9, 11, 10], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[8, Alternating, 17]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 5, -4, 6, -3}, {8, -2, 3, -5, 4, -6}]
In[5]:=
BR[Link[8, Alternating, 17]]
Out[5]=  
BR[Link[8, Alternating, 17]]
In[6]:=
alex = Alexander[Link[8, Alternating, 17]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 17]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 17]], KnotSignature[Link[8, Alternating, 17]]}
Out[9]=  
{Infinity, -4}
In[10]:=
J=Jones[Link[8, Alternating, 17]][q]
Out[10]=  
 -10   2    4    4    6    4    4    2     -2

q    - -- + -- - -- + -- - -- + -- - -- + q



       9    8    7    6    5    4    3


       q    q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 17]][q]
Out[12]=  
 -32    2     -28    3     3     3     5     3     4     -14    -10

q    + --- + q    + --- + --- + --- + --- + --- + --- + q    + q    - 



       30           26    24    22    20    18    16
      q            q     q     q     q     q     q

  -8    -6


  q   + q
In[13]:=
Kauffman[Link[8, Alternating, 17]][a, z]
Out[13]=  
                              6      8    10      7      9

   6      8      10    12   a    2 a    a     2 a    2 a       7



-5 a  - 8 a  - 3 a   + a   + -- + ---- + --- - ---- - ---- + 6 a  z + 



                             2     2     2     z      z
                            z     z     z

    9        4  2       6  2       8  2    10  2      12  2
 6 a  z - 2 a  z  + 10 a  z  + 15 a  z  + a   z  - 2 a   z  - 

    5  3      7  3      9  3      11  3    4  4      6  4       8  4
 3 a  z  - 6 a  z  - 6 a  z  - 3 a   z  + a  z  - 8 a  z  - 12 a  z  - 

    10  4    12  4      5  5      7  5      9  5      11  5
 2 a   z  + a   z  + 2 a  z  + 2 a  z  + 2 a  z  + 2 a   z  + 

    6  6      8  6      10  6    7  7    9  7


  3 a  z  + 5 a  z  + 2 a   z  + a  z  + a  z
In[14]:=
{Vassiliev[2][Link[8, Alternating, 17]], Vassiliev[3][Link[8, Alternating, 17]]}
Out[14]=  
    100

{0, ---}


     3
In[15]:=
Kh[Link[8, Alternating, 17]][q, t]
Out[15]=  
 -5    -3     1        1        1        3        3        3

q   + q   + ------ + ------ + ------ + ------ + ------ + ------ + 



            21  8    19  7    17  7    17  6    15  6    15  5
           q   t    q   t    q   t    q   t    q   t    q   t

   1        3        4        2        2       2       2      2
 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----
  13  5    13  4    11  4    11  3    9  3    9  2    7  2    5


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