L9a44

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L9a43.gif

L9a43

L9a45.gif

L9a45

L9a44.gif Visit L9a44's page at Knotilus!

Visit L9a44's page at the original Knot Atlas!

L9a44 is [math]\displaystyle{ 9^3_{3} }[/math] in the Rolfsen table of links.


L9a44 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+2 t(1) t(3)^2-t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-2 t(1) t(3)+t(1) t(2) t(3)-2 t(2) t(3)+t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -q^6+3 q^5-6 q^4+7 q^3-7 q^2+8 q-5+5 q^{-1} - q^{-2} + q^{-3} }[/math] (db)
Signature 2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} -7 z^2+3 a^2+8 a^{-2} -2 a^{-4} -9+2 a^2 z^{-2} +4 a^{-2} z^{-2} - a^{-4} z^{-2} -5 z^{-2} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^3 a^{-7} +3 z^4 a^{-6} +6 z^5 a^{-5} -6 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} +7 z^6 a^{-4} -11 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} -2 a^{-4} +4 z^7 a^{-3} -z^5 a^{-3} -12 z^3 a^{-3} +13 z a^{-3} -5 a^{-3} z^{-1} +z^8 a^{-2} +a^2 z^6+7 z^6 a^{-2} -5 a^2 z^4-21 z^4 a^{-2} +9 a^2 z^2+16 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} -7 a^2-10 a^{-2} +a z^7+5 z^7 a^{-1} -a z^5-8 z^5 a^{-1} -6 a z^3-11 z^3 a^{-1} +11 a z+21 z a^{-1} -5 a z^{-1} -9 a^{-1} z^{-1} +z^8+z^6-12 z^4+19 z^2+5 z^{-2} -14 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{17}{6} }[/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:L9a44/V 2,1 Data:L9a44/V 3,1 Data:L9a44/V 4,1 Data:L9a44/V 4,2 Data:L9a44/V 4,3 Data:L9a44/V 5,1 Data:L9a44/V 5,2 Data:L9a44/V 5,3 Data:L9a44/V 5,4 Data:L9a44/V 6,1 Data:L9a44/V 6,2 Data:L9a44/V 6,3 Data:L9a44/V 6,4 Data:L9a44/V 6,5 Data:L9a44/V 6,6 Data:L9a44/V 6,7 Data:L9a44/V 6,8 Data:L9a44/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]2 is the signature of L9a44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
13         1-1
11        2 2
9       41 -3
7      32  1
5     44   0
3    43    1
1   47     3
-1  11      0
-3  4       4
-511        0
-71         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=3 }[/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} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2 }[/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}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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, 44]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 44]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 14, 9, 13], X[16, 12, 17, 11], 

 X[12, 18, 13, 17], X[8, 16, 5, 15], X[14, 8, 15, 7], X[2, 5, 3, 6], 



  X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 44]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 7, -6}, 
  {9, -2, 4, -5, 3, -7, 6, -4, 5, -3}]
In[5]:=
BR[Link[9, Alternating, 44]]
Out[5]=  
BR[Link[9, Alternating, 44]]
In[6]:=
alex = Alexander[Link[9, Alternating, 44]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 44]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 44]], KnotSignature[Link[9, Alternating, 44]]}
Out[9]=  
{Infinity, 2}
In[10]:=
J=Jones[Link[9, Alternating, 44]][q]
Out[10]=  
      -3    -2   5            2      3      4      5    6

-5 + q   - q   + - + 8 q - 7 q  + 7 q  - 6 q  + 3 q  - q


                 q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 44]][q]
Out[12]=  
     -10   2    3    7    6       2    4      6      8    10      12

7 + q    + -- + -- + -- + -- + 6 q  + q  + 2 q  - 3 q  - q   - 2 q   - 



           8    6    4    2
          q    q    q    q

    14    16    18


  2 q   + q   - q
In[13]:=
Kauffman[Link[9, Alternating, 44]][a, z]
Out[13]=  
                                               2

     2    10      2   5      1       4     2 a     1      5      9



-14 - -- - -- - 7 a  + -- + ----- + ----- + ---- - ---- - ---- - --- - 



      4    2           2    4  2    2  2     2     5      3     a z
     a    a           z    a  z    a  z     z     a  z   a  z

                                               2       2
 5 a   3 z   13 z   21 z                2   6 z    16 z       2  2
 --- + --- + ---- + ---- + 11 a z + 19 z  + ---- + ----- + 9 a  z  + 
  z     5      3     a                        4      2
       a      a                              a      a

  3      3       3       3                       4       4       4
 z    6 z    12 z    11 z         3       4   3 z    11 z    21 z
 -- - ---- - ----- - ----- - 6 a z  - 12 z  + ---- - ----- - ----- - 
  7     5      3       a                        6      4       2
 a     a      a                                a      a       a

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

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


                     a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 44]], Vassiliev[3][Link[9, Alternating, 44]]}
Out[14]=  
    17

{0, --}


    6
In[15]:=
Kh[Link[9, Alternating, 44]][q, t]
Out[15]=  
         3     1       1       1       4      1      1    4 q

7 q + 4 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 



             7  4    5  4    5  3    3  2      2   q t    t
            q  t    q  t    q  t    q  t    q t

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

    11  4    13  5


  2 q   t  + q   t
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