L9a52

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L9a51.gif

L9a51

L9a53.gif

L9a53

L9a52.gif Visit L9a52's page at Knotilus!

Visit L9a52's page at the original Knot Atlas!

L9a52 is [math]\displaystyle{ 9^3_{8} }[/math] in the Rolfsen table of links.


L9a52 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X10,3,11,4 X16,11,17,12 X14,8,15,7 X8,14,9,13 X18,15,13,16 X12,17,5,18 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{(v-1) (w-1) \left(2 u w-u+w^2-2 w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ - q^{-8} +3 q^{-7} -5 q^{-6} +8 q^{-5} -7 q^{-4} +9 q^{-3} -7 q^{-2} +q+5 q^{-1} -2 }[/math] (db)
Signature -2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^8+3 a^6 z^2+a^6 z^{-2} +3 a^6-2 a^4 z^4-4 a^4 z^2-2 a^4 z^{-2} -4 a^4-a^2 z^4+a^2 z^{-2} +a^2+z^2+1 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-7 a^8 z^4+5 a^8 z^2-2 a^8+3 a^7 z^7-3 a^7 z^5-5 a^7 z^3+3 a^7 z+a^6 z^8+7 a^6 z^6-22 a^6 z^4+20 a^6 z^2+a^6 z^{-2} -9 a^6+6 a^5 z^7-8 a^5 z^5-2 a^5 z^3+7 a^5 z-2 a^5 z^{-1} +a^4 z^8+7 a^4 z^6-19 a^4 z^4+20 a^4 z^2+2 a^4 z^{-2} -10 a^4+3 a^3 z^7-2 a^3 z^5-a^3 z^3+5 a^3 z-2 a^3 z^{-1} +3 a^2 z^6-3 a^2 z^4+3 a^2 z^2+a^2 z^{-2} -3 a^2+2 a z^5-2 a z^3+z^4-2 z^2+1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{14}{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:L9a52/V 2,1 Data:L9a52/V 3,1 Data:L9a52/V 4,1 Data:L9a52/V 4,2 Data:L9a52/V 4,3 Data:L9a52/V 5,1 Data:L9a52/V 5,2 Data:L9a52/V 5,3 Data:L9a52/V 5,4 Data:L9a52/V 6,1 Data:L9a52/V 6,2 Data:L9a52/V 6,3 Data:L9a52/V 6,4 Data:L9a52/V 6,5 Data:L9a52/V 6,6 Data:L9a52/V 6,7 Data:L9a52/V 6,8 Data:L9a52/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 L9a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-1012χ
3         11
1        1 -1
-1       41 3
-3      53  -2
-5     42   2
-7    35    2
-9   54     1
-11  25      3
-13 13       -2
-15 2        2
-171         -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=-1 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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[9, Alternating, 52]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 52]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[14, 8, 15, 7], 

 X[8, 14, 9, 13], X[18, 15, 13, 16], X[12, 17, 5, 18], X[2, 5, 3, 6], 



  X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 52]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, 
  {8, -1, 4, -5, 9, -2, 3, -7}]
In[5]:=
BR[Link[9, Alternating, 52]]
Out[5]=  
BR[Link[9, Alternating, 52]]
In[6]:=
alex = Alexander[Link[9, Alternating, 52]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 52]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 52]], KnotSignature[Link[9, Alternating, 52]]}
Out[9]=  
{Infinity, -2}
In[10]:=
J=Jones[Link[9, Alternating, 52]][q]
Out[10]=  
      -8   3    5    8    7    9    7    5

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



           7    6    5    4    3    2   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[9, Alternating, 52]][q]
Out[12]=  
  -26    -24    2     2     5     3     6     3    3    2     -4   3

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



               22    18    16    14    12    10    8    6          2
              q     q     q     q     q     q     q    q          q

  4


  q
In[13]:=
Kauffman[Link[9, Alternating, 52]][a, z]
Out[13]=  
                                  2      4    6      3      5

      2       4      6      8   a    2 a    a    2 a    2 a



1 - 3 a  - 10 a  - 9 a  - 2 a  + -- + ---- + -- - ---- - ---- + 



                                 2     2     2    z      z
                                z     z     z

    3        5        7      9        2      2  2       4  2
 5 a  z + 7 a  z + 3 a  z + a  z - 2 z  + 3 a  z  + 20 a  z  + 

     6  2      8  2        3    3  3      5  3      7  3      9  3
 20 a  z  + 5 a  z  - 2 a z  - a  z  - 2 a  z  - 5 a  z  - 2 a  z  + 

  4      2  4       4  4       6  4      8  4        5      3  5
 z  - 3 a  z  - 19 a  z  - 22 a  z  - 7 a  z  + 2 a z  - 2 a  z  - 

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

    3  7      5  7      7  7    4  8    6  8


  3 a  z  + 6 a  z  + 3 a  z  + a  z  + a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 52]], Vassiliev[3][Link[9, Alternating, 52]]}
Out[14]=  
    14

{0, --}


    3
In[15]:=
Kh[Link[9, Alternating, 52]][q, t]
Out[15]=  
3    4     1        2        1        3        2        5        5

-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 



3   q    17  7    15  6    13  6    13  5    11  5    11  4    9  4



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



   4       3       5       4      2      5     t          3  2
 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q  t
  9  3    7  3    7  2    5  2    5      3     q


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