L9a38

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L9a37.gif

L9a37

L9a39.gif

L9a39

L9a38.gif Visit L9a38's page at Knotilus!

Visit L9a38's page at the original Knot Atlas!

L9a38 is [math]\displaystyle{ 9^2_{5} }[/math] in the Rolfsen table of links.


L9a38 Further Notes and Views

Knot presentations

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

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) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -2 q^{9/2}+3 q^{7/2}-\frac{1}{q^{7/2}}-5 q^{5/2}+\frac{2}{q^{5/2}}+5 q^{3/2}-\frac{3}{q^{3/2}}+q^{11/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature 1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^7 a^{-1} +a z^5-6 z^5 a^{-1} +z^5 a^{-3} +4 a z^3-12 z^3 a^{-1} +4 z^3 a^{-3} +4 a z-8 z a^{-1} +4 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^4 a^{-6} -2 z^2 a^{-6} +2 z^5 a^{-5} -4 z^3 a^{-5} +z a^{-5} +2 z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +a^3 z-4 z a^{-3} +z^8 a^{-2} +2 a^2 z^6-z^6 a^{-2} -6 a^2 z^4+3 a^2 z^2+3 z^2 a^{-2} +2 a z^7+4 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +6 a z^3+21 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-z^6-2 z^4+3 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, -2)
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:L9a38/V 2,1 Data:L9a38/V 3,1 Data:L9a38/V 4,1 Data:L9a38/V 4,2 Data:L9a38/V 4,3 Data:L9a38/V 5,1 Data:L9a38/V 5,2 Data:L9a38/V 5,3 Data:L9a38/V 5,4 Data:L9a38/V 6,1 Data:L9a38/V 6,2 Data:L9a38/V 6,3 Data:L9a38/V 6,4 Data:L9a38/V 6,5 Data:L9a38/V 6,6 Data:L9a38/V 6,7 Data:L9a38/V 6,8 Data:L9a38/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 L9a38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
12         1-1
10        1 1
8       21 -1
6      31  2
4     22   0
2    43    1
0   24     2
-2  12      -1
-4 12       1
-6 1        -1
-81         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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/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, 38]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 38]]
Out[3]=  
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], 

 X[14, 6, 15, 5], X[18, 14, 9, 13], X[16, 8, 17, 7], X[6, 16, 7, 15], 



  X[8, 18, 1, 17]]
In[4]:=
GaussCode[Link[9, Alternating, 38]]
Out[4]=  
GaussCode[{1, -2, 3, -4, 5, -8, 7, -9}, 
  {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}]
In[5]:=
BR[Link[9, Alternating, 38]]
Out[5]=  
BR[Link[9, Alternating, 38]]
In[6]:=
alex = Alexander[Link[9, Alternating, 38]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 38]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 38]], KnotSignature[Link[9, Alternating, 38]]}
Out[9]=  
{Infinity, 1}
In[10]:=
J=Jones[Link[9, Alternating, 38]][q]
Out[10]=  
  -(7/2)    2      3        4                     3/2      5/2

-q       + ---- - ---- + ------- - 6 Sqrt[q] + 5 q    - 5 q    + 



           5/2    3/2   Sqrt[q]
          q      q

    7/2      9/2    11/2


  3 q    - 2 q    + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 38]][q]
Out[12]=  
     -10    -6      4    6      8    10    12    16
2 + q    + q   + 3 q  + q  + 2 q  + q   - q   - q
In[13]:=
Kauffman[Link[9, Alternating, 38]][a, z]
Out[13]=  
                                                         2    2

    1    a   z    4 z   10 z            3        2   2 z    z



1 - --- - - - -- + --- + ---- + 4 a z - a  z - 3 z  + ---- - -- - 



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

    2                3      3       3                              4
 3 z       2  2   4 z    8 z    21 z         3      3  3      4   z
 ---- - 3 a  z  + ---- - ---- - ----- - 6 a z  + 3 a  z  + 2 z  - -- + 
   2                5      3      a                                6
  a                a      a                                       a

    4                5      5       5                            6
 3 z       2  4   2 z    5 z    14 z         5    3  5    6   2 z
 ---- + 6 a  z  - ---- + ---- + ----- + 6 a z  - a  z  + z  - ---- + 
   4                5      3      a                             4
  a                a      a                                    a

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


  a               a                          a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 38]], Vassiliev[3][Link[9, Alternating, 38]]}
Out[14]=  
{0, -2}
In[15]:=
Kh[Link[9, Alternating, 38]][q, t]
Out[15]=  
       2     1       1       1       2       1     2    2        2

4 + 4 q  + ----- + ----- + ----- + ----- + ----- + - + ---- + 3 q  t + 



           8  4    6  3    4  3    4  2    2  2   t    2
          q  t    q  t    q  t    q  t    q  t        q  t

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


  2 q  t + 2 q  t  + 3 q  t  + q  t  + 2 q  t  + q  t  + q   t  + q   t
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