L9a49

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L9a48.gif

L9a48

L9a50.gif

L9a50

L9a49.gif Visit L9a49's page at Knotilus!

Visit L9a49's page at the original Knot Atlas!

L9a49 is [math]\displaystyle{ 9^3_{6} }[/math] in the Rolfsen table of links.


L9a49 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{u v^2 w-u v^2+u v w^2-3 u v w+2 u v-u w^2+2 u w-2 v^2 w+v^2-2 v w^2+3 v w-v+w^2-w}{\sqrt{u} v w} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^5+3 q^4+ q^{-4} -4 q^3-2 q^{-3} +7 q^2+5 q^{-2} -7 q-6 q^{-1} +8 }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^2 a^{-4} +a^4+z^4 a^{-2} -2 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} +z^4-z^2-2 z^{-2} -3 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +3 a^2 z^6+4 z^6 a^{-2} +3 z^6 a^{-4} +4 z^6+2 a^3 z^5+2 a z^5-7 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4-4 a^2 z^4-13 z^4 a^{-2} -8 z^4 a^{-4} -10 z^4-2 a^3 z^3-7 a z^3-z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} -2 a^4 z^2+4 a^2 z^2+11 z^2 a^{-2} +5 z^2 a^{-4} +12 z^2+6 a z+6 z a^{-1} +a^4-3 a^2-5 a^{-2} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, 1)
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:L9a49/V 2,1 Data:L9a49/V 3,1 Data:L9a49/V 4,1 Data:L9a49/V 4,2 Data:L9a49/V 4,3 Data:L9a49/V 5,1 Data:L9a49/V 5,2 Data:L9a49/V 5,3 Data:L9a49/V 5,4 Data:L9a49/V 6,1 Data:L9a49/V 6,2 Data:L9a49/V 6,3 Data:L9a49/V 6,4 Data:L9a49/V 6,5 Data:L9a49/V 6,6 Data:L9a49/V 6,7 Data:L9a49/V 6,8 Data:L9a49/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]0 is the signature of L9a49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
11         1-1
9        2 2
7       21 -1
5      52  3
3     44   0
1    43    1
-1   35     2
-3  23      -1
-5  3       3
-712        -1
-91         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 49]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 49]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 10, 17, 9], X[14, 8, 15, 7], 

 X[18, 14, 11, 13], X[10, 16, 5, 15], X[8, 18, 9, 17], X[2, 5, 3, 6], 



  X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 49]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 4, -7, 3, -6}, 
  {9, -2, 5, -4, 6, -3, 7, -5}]
In[5]:=
BR[Link[9, Alternating, 49]]
Out[5]=  
BR[Link[9, Alternating, 49]]
In[6]:=
alex = Alexander[Link[9, Alternating, 49]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 49]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 49]], KnotSignature[Link[9, Alternating, 49]]}
Out[9]=  
{Infinity, 0}
In[10]:=
J=Jones[Link[9, Alternating, 49]][q]
Out[10]=  
     -4   2    5    6            2      3      4    5

8 + q   - -- + -- - - - 7 q + 7 q  - 4 q  + 3 q  - q



          3    2   q


          q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 49]][q]
Out[12]=  
     -14    -12    -10   2    2     -4   5       2      4      6

3 + q    + q    - q    + -- + -- + q   + -- + 5 q  + 3 q  + 2 q  + 



                         8    6          2
                        q    q          q

    8    10    12    14    16


  3 q  - q   + q   + q   - q
In[13]:=
Kauffman[Link[9, Alternating, 49]][a, z]
Out[13]=  
                                    2

    5       2    4   2      1     a     2    2 a   6 z



-8 - -- - 3 a  + a  + -- + ----- + -- - --- - --- + --- + 6 a z + 



     2                2    2  2    2   a z    z     a
    a                z    a  z    z

            2       2                          3      3    3
     2   5 z    11 z       2  2      4  2   2 z    2 z    z
 12 z  + ---- + ----- + 4 a  z  - 2 a  z  - ---- + ---- - -- - 
           4      2                           5      3    a
          a      a                           a      a

                               4       4                      5
      3      3  3       4   8 z    13 z       2  4    4  4   z
 7 a z  - 2 a  z  - 10 z  - ---- - ----- - 4 a  z  + a  z  + -- - 
                              4      2                        5
                             a      a                        a

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

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


   a                          a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 49]], Vassiliev[3][Link[9, Alternating, 49]]}
Out[14]=  
{0, 1}
In[15]:=
Kh[Link[9, Alternating, 49]][q, t]
Out[15]=  
5           1       1       2       3       2      3      3

- + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 
q          9  4    7  4    7  3    5  2    3  2    3     q t



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

    3        3  2      5  2      5  3      7  3    7  4      9  4
 4 q  t + 4 q  t  + 5 q  t  + 2 q  t  + 2 q  t  + q  t  + 2 q  t  + 

  11  5


  q   t
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