L9a39

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L9a38.gif

L9a38

L9a40.gif

L9a40

L9a39.gif Visit L9a39's page at Knotilus!

Visit L9a39's page at the original Knot Atlas!

L9a39 is [math]\displaystyle{ 9^2_{2} }[/math] in the Rolfsen table of links.


L9a39 Further Notes and Views

Knot presentations

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X16,8,17,7 X18,16,9,15 X8,18,1,17 X4,13,5,14
Gauss code {1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}+q^{3/2}-\frac{4}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^3 z^7+a^5 z^5-6 a^3 z^5+a z^5+4 a^5 z^3-12 a^3 z^3+4 a z^3+4 a^5 z-9 a^3 z+3 a z+a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^3 a^9+z a^9-2 z^4 a^8+2 z^2 a^8-2 z^5 a^7+z^3 a^7-2 z^6 a^6+2 z^4 a^6-z^2 a^6-2 z^7 a^5+5 z^5 a^5-7 z^3 a^5+5 z a^5-a^5 z^{-1} -z^8 a^4+z^6 a^4+3 z^4 a^4-4 z^2 a^4+a^4-4 z^7 a^3+16 z^5 a^3-20 z^3 a^3+10 z a^3-a^3 z^{-1} -z^8 a^2+2 z^6 a^2+3 z^4 a^2-4 z^2 a^2-2 z^7 a+9 z^5 a-11 z^3 a+4 z a-z^6+4 z^4-3 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{113}{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:L9a39/V 2,1 Data:L9a39/V 3,1 Data:L9a39/V 4,1 Data:L9a39/V 4,2 Data:L9a39/V 4,3 Data:L9a39/V 5,1 Data:L9a39/V 5,2 Data:L9a39/V 5,3 Data:L9a39/V 5,4 Data:L9a39/V 6,1 Data:L9a39/V 6,2 Data:L9a39/V 6,3 Data:L9a39/V 6,4 Data:L9a39/V 6,5 Data:L9a39/V 6,6 Data:L9a39/V 6,7 Data:L9a39/V 6,8 Data:L9a39/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]-3 is the signature of L9a39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-10123χ
4         1-1
2        1 1
0       11 0
-2      31  2
-4     22   0
-6    32    1
-8   12     1
-10  23      -1
-12 12       1
-14 1        -1
-161         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 39]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 39]]
Out[3]=  
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], 

 X[6, 9, 7, 10], X[16, 8, 17, 7], X[18, 16, 9, 15], X[8, 18, 1, 17], 



  X[4, 13, 5, 14]]
In[4]:=
GaussCode[Link[9, Alternating, 39]]
Out[4]=  
GaussCode[{1, -2, 3, -9, 4, -5, 6, -8}, 
  {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}]
In[5]:=
BR[Link[9, Alternating, 39]]
Out[5]=  
BR[Link[9, Alternating, 39]]
In[6]:=
alex = Alexander[Link[9, Alternating, 39]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 39]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 39]], KnotSignature[Link[9, Alternating, 39]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 39]][q]
Out[10]=  
  -(15/2)     2       3      4      5      4      4        2

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



            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q

              3/2


  2 Sqrt[q] + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 39]][q]
Out[12]=  
 -22    -18    -12   3     -6   2     -2    4

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



                     8          4


                     q          q
In[13]:=
Kauffman[Link[9, Alternating, 39]][a, z]
Out[13]=  
      3    5

4   a    a                3        5      9        2      2  2



a  - -- - -- + 4 a z + 10 a  z + 5 a  z + a  z - 3 z  - 4 a  z  - 



    z    z

    4  2    6  2      8  2         3       3  3      5  3    7  3
 4 a  z  - a  z  + 2 a  z  - 11 a z  - 20 a  z  - 7 a  z  + a  z  - 

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

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

      7      3  7      5  7    2  8    4  8


  2 a z  - 4 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 39]], Vassiliev[3][Link[9, Alternating, 39]]}
Out[14]=  
      113

{0, -(---)}


      24
In[15]:=
Kh[Link[9, Alternating, 39]][q, t]
Out[15]=  
2    3      1        1        1        2        2        3        1

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



4    2    16  6    14  5    12  5    12  4    10  4    10  3    8  3



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



   2       3      2      2         t     2    2  2    4  3
 ----- + ----- + ---- + ---- + t + -- + t  + q  t  + q  t
  8  2    6  2    6      4          2


  q  t    q  t    q  t   q  t       q
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