L9a11

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L9a10.gif

L9a10

L9a12.gif

L9a12

L9a11.gif Visit L9a11's page at Knotilus!

Visit L9a11's page at the original Knot Atlas!

L9a11 is [math]\displaystyle{ 9^2_{26} }[/math] in the Rolfsen table of links.


L9a11 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{u v^3-4 u v^2+6 u v-2 u-2 v^3+6 v^2-4 v+1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{7}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^7 z^{-1} -3 a^5 z-2 a^5 z^{-1} +3 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^4 z^8-a^2 z^8-2 a^5 z^7-6 a^3 z^7-4 a z^7-2 a^6 z^6-5 a^4 z^6-9 a^2 z^6-6 z^6-a^7 z^5+5 a^3 z^5-4 z^5 a^{-1} +4 a^6 z^4+12 a^4 z^4+17 a^2 z^4-z^4 a^{-2} +8 z^4+3 a^7 z^3+7 a^5 z^3+7 a^3 z^3+7 a z^3+4 z^3 a^{-1} -2 a^6 z^2-7 a^4 z^2-7 a^2 z^2-2 z^2-3 a^7 z-7 a^5 z-8 a^3 z-4 a z+a^4+a^7 z^{-1} +2 a^5 z^{-1} +2 a^3 z^{-1} +a z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{137}{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:L9a11/V 2,1 Data:L9a11/V 3,1 Data:L9a11/V 4,1 Data:L9a11/V 4,2 Data:L9a11/V 4,3 Data:L9a11/V 5,1 Data:L9a11/V 5,2 Data:L9a11/V 5,3 Data:L9a11/V 5,4 Data:L9a11/V 6,1 Data:L9a11/V 6,2 Data:L9a11/V 6,3 Data:L9a11/V 6,4 Data:L9a11/V 6,5 Data:L9a11/V 6,6 Data:L9a11/V 6,7 Data:L9a11/V 6,8 Data:L9a11/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 L9a11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-10123χ
6         1-1
4        3 3
2       31 -2
0      53  2
-2     54   -1
-4    44    0
-6   35     2
-8  24      -2
-10 14       3
-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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/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, 11]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 11]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[18, 16, 5, 15], 

 X[16, 9, 17, 10], X[8, 17, 9, 18], X[10, 14, 11, 13], X[2, 5, 3, 6], 



  X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 11]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 3, -6, 5, -7, 9, -2, 7, -3, 4, -5, 6, 
   -4}]
In[5]:=
BR[Link[9, Alternating, 11]]
Out[5]=  
BR[Link[9, Alternating, 11]]
In[6]:=
alex = Alexander[Link[9, Alternating, 11]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 11]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 11]], KnotSignature[Link[9, Alternating, 11]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 11]][q]
Out[10]=  
  -(13/2)     2      5      7      9      9        8

-q        + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 



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

    3/2    5/2


  4 q    + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 11]][q]
Out[12]=  
     -22    2     -16    2     2     -10   2     -2    2      6    8

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



           20           14    12           4


           q            q     q            q
In[13]:=
Kauffman[Link[9, Alternating, 11]][a, z]
Out[13]=  
            3      5    7

4   a   2 a    2 a    a               3        5        7        2



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



    z    z      z     z

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

                   4                                      5
    7  3      4   z        2  4       4  4      6  4   4 z
 3 a  z  + 8 z  - -- + 17 a  z  + 12 a  z  + 4 a  z  - ---- + 
                   2                                    a
                  a

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

    3  7      5  7    2  8    4  8


  6 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 11]], Vassiliev[3][Link[9, Alternating, 11]]}
Out[14]=  
      137

{0, -(---)}


      24
In[15]:=
Kh[Link[9, Alternating, 11]][q, t]
Out[15]=  
    4      1        1        1        4        2       4       3

5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



    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

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


  q  t    q  t    q  t   q  t
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