L9a46

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L9a45.gif

L9a45

L9a47.gif

L9a47

L9a46.gif Visit L9a46's page at Knotilus!

Visit L9a46's page at the original Knot Atlas!

L9a46 is [math]\displaystyle{ 9^3_{10} }[/math] in the Rolfsen table of links.


L9a46 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,12,17,11 X18,9,11,10 X10,17,5,18 X4,15,1,16
Gauss code {1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{(u-1) (v-1)^2 (w-1)^2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial [math]\displaystyle{ - q^{-5} +q^4+4 q^{-4} -4 q^3-6 q^{-3} +7 q^2+10 q^{-2} -9 q-10 q^{-1} +12 }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^4 z^2+a^4 z^{-2} +2 a^2 z^4+z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} -a^2-z^6-3 z^4-3 z^2+ z^{-2} +1 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^5 z^5-a^5 z^3+4 a^4 z^6-8 a^4 z^4+z^4 a^{-4} +4 a^4 z^2-a^4 z^{-2} +a^4+5 a^3 z^7-8 a^3 z^5+4 z^5 a^{-3} +2 a^3 z^3-3 z^3 a^{-3} -a^3 z+2 a^3 z^{-1} +2 a^2 z^8+8 a^2 z^6+7 z^6 a^{-2} -23 a^2 z^4-9 z^4 a^{-2} +12 a^2 z^2+4 z^2 a^{-2} -2 a^2 z^{-2} +a^2+11 a z^7+6 z^7 a^{-1} -17 a z^5-4 z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} -a z+2 a z^{-1} +2 z^8+11 z^6-25 z^4+12 z^2- z^{-2} +1 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{5}{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:L9a46/V 2,1 Data:L9a46/V 3,1 Data:L9a46/V 4,1 Data:L9a46/V 4,2 Data:L9a46/V 4,3 Data:L9a46/V 5,1 Data:L9a46/V 5,2 Data:L9a46/V 5,3 Data:L9a46/V 5,4 Data:L9a46/V 6,1 Data:L9a46/V 6,2 Data:L9a46/V 6,3 Data:L9a46/V 6,4 Data:L9a46/V 6,5 Data:L9a46/V 6,6 Data:L9a46/V 6,7 Data:L9a46/V 6,8 Data:L9a46/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 L9a46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
9         11
7        3 -3
5       41 3
3      53  -2
1     74   3
-1    57    2
-3   55     0
-5  37      4
-7 13       -2
-9 3        3
-111         -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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/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}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/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, 46]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 46]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], 

 X[14, 7, 15, 8], X[16, 12, 17, 11], X[18, 9, 11, 10], 



  X[10, 17, 5, 18], X[4, 15, 1, 16]]
In[4]:=
GaussCode[Link[9, Alternating, 46]]
Out[4]=  
GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, 
  {6, -2, 4, -5, 9, -6, 8, -7}]
In[5]:=
BR[Link[9, Alternating, 46]]
Out[5]=  
BR[Link[9, Alternating, 46]]
In[6]:=
alex = Alexander[Link[9, Alternating, 46]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 46]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 46]], KnotSignature[Link[9, Alternating, 46]]}
Out[9]=  
{Infinity, 0}
In[10]:=
J=Jones[Link[9, Alternating, 46]][q]
Out[10]=  
      -5   4    6    10   10            2      3    4

12 - q   + -- - -- + -- - -- - 9 q + 7 q  - 4 q  + q



           4    3    2   q


           q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 46]][q]
Out[12]=  
     -16    -14    3     -10   6    4    4    6       2      4    8

1 - q    + q    + --- + q    + -- + -- + -- + -- + 4 q  - 2 q  + q  - 



                  12           8    6    4    2
                 q            q    q    q    q

    10    12


  2 q   + q
In[13]:=
Kauffman[Link[9, Alternating, 46]][a, z]
Out[13]=  
                       2    4            3

    2    4    -2   2 a    a    2 a   2 a           3         2



1 + a  + a  - z   - ---- - -- + --- + ---- - a z - a  z + 12 z  + 



                     2     2    z     z
                    z     z

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

          4      4                           5      5
     4   z    9 z        2  4      4  4   4 z    4 z          5
 25 z  + -- - ---- - 23 a  z  - 8 a  z  + ---- - ---- - 17 a z  - 
          4     2                           3     a
         a     a                           a

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

    3  7      8      2  8


  5 a  z  + 2 z  + 2 a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 46]], Vassiliev[3][Link[9, Alternating, 46]]}
Out[14]=  
    5

{0, -}


    6
In[15]:=
Kh[Link[9, Alternating, 46]][q, t]
Out[15]=  
7           1        3       1       3       3       7       5

- + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          11  5    9  4    7  4    7  3    5  3    5  2    3  2



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

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

  9  4


  q  t
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