L9a33

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L9a32.gif

L9a32

L9a34.gif

L9a34

L9a33.gif Visit L9a33's page at Knotilus!

Visit L9a33's page at the original Knot Atlas!

L9a33 is [math]\displaystyle{ 9^2_{24} }[/math] in the Rolfsen table of links.




Symmetric form
Alternate symmetric version, with three lines touching at center
Alternate symmetric version, with three lines touching at circumference
Form made from 45-degree lines and circular arcs.
Depiction obtained by knotilus.
With an hypotrochoid [1].
Mexican book.

Knot presentations

Planar diagram presentation X8192 X12,3,13,4 X18,10,7,9 X10,14,11,13 X16,5,17,6 X14,18,15,17 X2738 X4,11,5,12 X6,15,1,16
Gauss code {1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{u^2 v^2-3 u^2 v+3 u^2-3 u v^2+7 u v-3 u+3 v^2-3 v+1}{u v} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{10}{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-a^5 z^{-1} +3 a^3 z^3+3 a^3 z-a z^5-2 a z^3+z^3 a^{-1} -3 a z }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^4 z^8-a^2 z^8-3 a^5 z^7-7 a^3 z^7-4 a z^7-2 a^6 z^6-7 a^4 z^6-11 a^2 z^6-6 z^6-a^7 z^5+5 a^5 z^5+9 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^6 z^4+18 a^4 z^4+24 a^2 z^4-z^4 a^{-2} +8 z^4+3 a^7 z^3-3 a^5 z^3-2 a^3 z^3+8 a z^3+4 z^3 a^{-1} -11 a^4 z^2-14 a^2 z^2-3 z^2-3 a^7 z+a^5 z+2 a^3 z-2 a z-a^6+a^7 z^{-1} +a^5 z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{129}{16} }[/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:L9a33/V 2,1 Data:L9a33/V 3,1 Data:L9a33/V 4,1 Data:L9a33/V 4,2 Data:L9a33/V 4,3 Data:L9a33/V 5,1 Data:L9a33/V 5,2 Data:L9a33/V 5,3 Data:L9a33/V 5,4 Data:L9a33/V 6,1 Data:L9a33/V 6,2 Data:L9a33/V 6,3 Data:L9a33/V 6,4 Data:L9a33/V 6,5 Data:L9a33/V 6,6 Data:L9a33/V 6,7 Data:L9a33/V 6,8 Data:L9a33/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 L9a33. 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     64   -2
-4    34    -1
-6   46     2
-8  23      -1
-10  4       4
-1212        -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{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/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}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{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, 33]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 33]]
Out[3]=  
PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 7, 9], X[10, 14, 11, 13], 

 X[16, 5, 17, 6], X[14, 18, 15, 17], X[2, 7, 3, 8], X[4, 11, 5, 12], 



  X[6, 15, 1, 16]]
In[4]:=
GaussCode[Link[9, Alternating, 33]]
Out[4]=  
GaussCode[{1, -7, 2, -8, 5, -9}, 
  {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}]
In[5]:=
BR[Link[9, Alternating, 33]]
Out[5]=  
BR[Link[9, Alternating, 33]]
In[6]:=
alex = Alexander[Link[9, Alternating, 33]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 33]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 33]], KnotSignature[Link[9, Alternating, 33]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 33]][q]
Out[10]=  
  -(13/2)     2      6      7      9      10       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, 33]][q]
Out[12]=  
     -22    2     2     4     2     -8   2    2     2      6    8

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



           20    16    14    10          6    2


           q     q     q     q           q    q
In[13]:=
Kauffman[Link[9, Alternating, 33]][a, z]
Out[13]=  
       5    7

 6   a    a               3      5        7        2       2  2



-a  + -- + -- - 2 a z + 2 a  z + a  z - 3 a  z - 3 z  - 14 a  z  - 



     z    z

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

                                    5
     2  4       4  4      6  4   4 z       5      3  5      5  5
 24 a  z  + 18 a  z  + 3 a  z  - ---- - a z  + 9 a  z  + 5 a  z  - 
                                  a

  7  5      6       2  6      4  6      6  6        7      3  7
 a  z  - 6 z  - 11 a  z  - 7 a  z  - 2 a  z  - 4 a z  - 7 a  z  - 

    5  7    2  8    4  8


  3 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 33]], Vassiliev[3][Link[9, Alternating, 33]]}
Out[14]=  
      129

{0, -(---)}


      16
In[15]:=
Kh[Link[9, Alternating, 33]][q, t]
Out[15]=  
    4      1        1        2        4        2       3       4

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



    2    14  6    12  6    12  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       3      4      6              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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