L9a19

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L9a18.gif

L9a18

L9a20.gif

L9a20

L9a19.gif Visit L9a19's page at Knotilus!

Visit L9a19's page at the original Knot Atlas!

L9a19 is [math]\displaystyle{ 9^2_{38} }[/math] in the Rolfsen table of links.


L9a19 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X18,11,5,12 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code {1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{2 u v^3-5 u v^2+6 u v-2 u-2 v^3+6 v^2-5 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-2 a^3 z^3-3 a^3 z-a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+3 a^6 z^2+4 a^5 z^7-7 a^5 z^5+4 a^5 z^3-2 a^5 z+a^5 z^{-1} +2 a^4 z^8+4 a^4 z^6-13 a^4 z^4+7 a^4 z^2-a^4+10 a^3 z^7-18 a^3 z^5+10 a^3 z^3-3 a^3 z+a^3 z^{-1} +2 a^2 z^8+8 a^2 z^6-17 a^2 z^4+z^4 a^{-2} +6 a^2 z^2+6 a z^7-6 a z^5+4 z^5 a^{-1} +a z^3-3 z^3 a^{-1} +7 z^6-9 z^4+2 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{77}{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:L9a19/V 2,1 Data:L9a19/V 3,1 Data:L9a19/V 4,1 Data:L9a19/V 4,2 Data:L9a19/V 4,3 Data:L9a19/V 5,1 Data:L9a19/V 5,2 Data:L9a19/V 5,3 Data:L9a19/V 5,4 Data:L9a19/V 6,1 Data:L9a19/V 6,2 Data:L9a19/V 6,3 Data:L9a19/V 6,4 Data:L9a19/V 6,5 Data:L9a19/V 6,6 Data:L9a19/V 6,7 Data:L9a19/V 6,8 Data:L9a19/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 L9a19. 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       41 -3
0      53  2
-2     65   -1
-4    54    1
-6   36     3
-8  35      -2
-10 14       3
-12 2        -2
-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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, 19]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 19]]
Out[3]=  
PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], 

 X[18, 11, 5, 12], X[4, 17, 1, 18], X[16, 14, 17, 13], 



  X[14, 8, 15, 7], X[8, 16, 9, 15]]
In[4]:=
GaussCode[Link[9, Alternating, 19]]
Out[4]=  
GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, 
   -5}]
In[5]:=
BR[Link[9, Alternating, 19]]
Out[5]=  
BR[Link[9, Alternating, 19]]
In[6]:=
alex = Alexander[Link[9, Alternating, 19]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 19]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 19]], KnotSignature[Link[9, Alternating, 19]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 19]][q]
Out[10]=  
  -(13/2)     3      6      8      11     10       9

-q        + ----- - ---- + ---- - ---- + ---- - ------- + 7 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, 19]][q]
Out[12]=  
     -20    -18    -16    3     -12    3     -8    -6   2    2

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



                         14           10                4    2
                        q            q                 q    q

    2      6    8


  2 q  + 2 q  - q
In[13]:=
Kauffman[Link[9, Alternating, 19]][a, z]
Out[13]=  
      3    5

4   a    a       3        5      7        2      2  2      4  2



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



    z    z

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

                                    5
     2  4       4  4      6  4   4 z         5       3  5      5  5
 17 a  z  + 13 a  z  + 6 a  z  - ---- + 6 a z  + 18 a  z  + 7 a  z  - 
                                  a

  7  5      6      2  6      4  6      6  6        7       3  7
 a  z  - 7 z  - 8 a  z  - 4 a  z  - 3 a  z  - 6 a z  - 10 a  z  - 

    5  7      2  8      4  8


  4 a  z  - 2 a  z  - 2 a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 19]], Vassiliev[3][Link[9, Alternating, 19]]}
Out[14]=  
      77

{0, -(--)}


      24
In[15]:=
Kh[Link[9, Alternating, 19]][q, t]
Out[15]=  
    5      1        2        1        4        3       5       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

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