L9a3

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L9a2.gif

L9a2

L9a4.gif

L9a4

L9a3.gif Visit L9a3's page at Knotilus!

Visit L9a3's page at the original Knot Atlas!

L9a3 is [math]\displaystyle{ 9^2_{33} }[/math] in the Rolfsen table of links.


L9a3 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X14,10,15,9 X10,14,11,13 X18,12,5,11 X2,16,3,15
Gauss code {1, -9, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -4, 7, -6, 9, -2, 3, -8}

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) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -6 q^{9/2}+8 q^{7/2}-10 q^{5/2}+\frac{1}{q^{5/2}}+9 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature 1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^5 a^{-1} +z^5 a^{-3} -a z^3+2 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a z+3 z a^{-1} -2 z a^{-3} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -2 z^8 a^{-2} -2 z^8 a^{-4} -4 z^7 a^{-1} -9 z^7 a^{-3} -5 z^7 a^{-5} -4 z^6 a^{-2} -4 z^6 a^{-4} -4 z^6 a^{-6} -4 z^6-3 a z^5+2 z^5 a^{-1} +16 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+9 z^4 a^{-2} +13 z^4 a^{-4} +8 z^4 a^{-6} +3 z^4+4 a z^3+3 z^3 a^{-1} -6 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-2 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2 a^{-6} -2 a z-5 z a^{-1} -3 z a^{-3} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{3}{2} }[/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:L9a3/V 2,1 Data:L9a3/V 3,1 Data:L9a3/V 4,1 Data:L9a3/V 4,2 Data:L9a3/V 4,3 Data:L9a3/V 5,1 Data:L9a3/V 5,2 Data:L9a3/V 5,3 Data:L9a3/V 5,4 Data:L9a3/V 6,1 Data:L9a3/V 6,2 Data:L9a3/V 6,3 Data:L9a3/V 6,4 Data:L9a3/V 6,5 Data:L9a3/V 6,6 Data:L9a3/V 6,7 Data:L9a3/V 6,8 Data:L9a3/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 L9a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
14         11
12        3 -3
10       31 2
8      53  -2
6     53   2
4    45    1
2   55     0
0  26      4
-2 13       -2
-4 2        2
-61         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/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, 3]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 3]]
Out[3]=  
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], 

 X[8, 4, 9, 3], X[14, 10, 15, 9], X[10, 14, 11, 13], X[18, 12, 5, 11], 



  X[2, 16, 3, 15]]
In[4]:=
GaussCode[Link[9, Alternating, 3]]
Out[4]=  
GaussCode[{1, -9, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -4, 7, -6, 9, -2, 3, 
   -8}]
In[5]:=
BR[Link[9, Alternating, 3]]
Out[5]=  
BR[Link[9, Alternating, 3]]
In[6]:=
alex = Alexander[Link[9, Alternating, 3]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 3]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 3]], KnotSignature[Link[9, Alternating, 3]]}
Out[9]=  
{Infinity, 1}
In[10]:=
J=Jones[Link[9, Alternating, 3]][q]
Out[10]=  
 -(5/2)    3        5                     3/2       5/2      7/2

q       - ---- + ------- - 9 Sqrt[q] + 9 q    - 10 q    + 8 q    - 



          3/2   Sqrt[q]
         q

    9/2      11/2    13/2


  6 q    + 4 q     - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 3]][q]
Out[12]=  
     -8    -6    -2    2      4      6    8      10      12    16

4 - q   + q   - q   + q  + 4 q  + 3 q  + q  + 2 q   - 3 q   - q   - 



    18    20


  2 q   + q
In[13]:=
Kauffman[Link[9, Alternating, 3]][a, z]
Out[13]=  
                                                            2      2

 -6   3    3     1      3      2    3 z   5 z           2 z    3 z



-a   - -- - -- + ---- + ---- + --- - --- - --- - 2 a z - ---- - ---- - 



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

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

     4      4            5       5       5      5
 13 z    9 z     2  4   z    10 z    16 z    2 z         5      6
 ----- + ---- - a  z  - -- + ----- + ----- + ---- - 3 a z  - 4 z  - 
   4       2             7     5       3      a
  a       a             a     a       a

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


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

{0, -(-)}


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

6 + 5 q  + ----- + ----- + ----- + - + ---- + 5 q  t + 4 q  t + 



           6  3    4  2    2  2   t    2
          q  t    q  t    q  t        q  t

    4  2      6  2      6  3      8  3      8  4      10  4    10  5
 5 q  t  + 5 q  t  + 3 q  t  + 5 q  t  + 3 q  t  + 3 q   t  + q   t  + 

    12  5    14  6


  3 q   t  + q   t
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