L9a41

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L9a40.gif

L9a40

L9a42.gif

L9a42

L9a41.gif Visit L9a41's page at Knotilus!

Visit L9a41's page at the original Knot Atlas!

L9a41 is [math]\displaystyle{ 9^2_{23} }[/math] in the Rolfsen table of links.


L9a41 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{(t(1) t(2)+1) (t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^5 z^5+4 a^5 z^3+5 a^5 z+a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-11 a^3 z-a^3 z^{-1} +a z^5+4 a z^3+4 a z }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^3 a^9+z a^9-2 z^4 a^8+z^2 a^8-3 z^5 a^7+2 z^3 a^7-z a^7-4 z^6 a^6+7 z^4 a^6-6 z^2 a^6-3 z^7 a^5+5 z^5 a^5-4 z^3 a^5+4 z a^5-a^5 z^{-1} -z^8 a^4-3 z^6 a^4+13 z^4 a^4-10 z^2 a^4+a^4-5 z^7 a^3+16 z^5 a^3-17 z^3 a^3+11 z a^3-a^3 z^{-1} -z^8 a^2+8 z^4 a^2-7 z^2 a^2-2 z^7 a+8 z^5 a-10 z^3 a+5 z a-z^6+4 z^4-4 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{125}{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:L9a41/V 2,1 Data:L9a41/V 3,1 Data:L9a41/V 4,1 Data:L9a41/V 4,2 Data:L9a41/V 4,3 Data:L9a41/V 5,1 Data:L9a41/V 5,2 Data:L9a41/V 5,3 Data:L9a41/V 5,4 Data:L9a41/V 6,1 Data:L9a41/V 6,2 Data:L9a41/V 6,3 Data:L9a41/V 6,4 Data:L9a41/V 6,5 Data:L9a41/V 6,6 Data:L9a41/V 6,7 Data:L9a41/V 6,8 Data:L9a41/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 L9a41. 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       21 -1
-2      41  3
-4     23   1
-6    43    1
-8   22     0
-10  24      -2
-12 13       2
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 41]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 41]]
Out[3]=  
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 5, 9, 6], 

 X[14, 8, 15, 7], X[16, 14, 17, 13], X[8, 16, 1, 15], X[6, 9, 7, 10], 



  X[4, 17, 5, 18]]
In[4]:=
GaussCode[Link[9, Alternating, 41]]
Out[4]=  
GaussCode[{1, -2, 3, -9, 4, -8, 5, -7}, 
  {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}]
In[5]:=
BR[Link[9, Alternating, 41]]
Out[5]=  
BR[Link[9, Alternating, 41]]
In[6]:=
alex = Alexander[Link[9, Alternating, 41]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 41]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 41]], KnotSignature[Link[9, Alternating, 41]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 41]][q]
Out[10]=  
  -(15/2)     2       4      6      6      5      6        3

-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, 41]][q]
Out[12]=  
      -22    2     -14    -10   4    2    3     -2    4

-1 + q    + --- - q    - q    + -- + -- + -- + q   - q



            18                  8    6    4


            q                   q    q    q
In[13]:=
Kauffman[Link[9, Alternating, 41]][a, z]
Out[13]=  
      3    5

4   a    a                3        5      7      9        2



a  - -- - -- + 5 a z + 11 a  z + 4 a  z - a  z + a  z - 4 z  - 



    z    z

    2  2       4  2      6  2    8  2         3       3  3      5  3
 7 a  z  - 10 a  z  - 6 a  z  + a  z  - 10 a z  - 17 a  z  - 4 a  z  + 

    7  3    9  3      4      2  4       4  4      6  4      8  4
 2 a  z  - a  z  + 4 z  + 8 a  z  + 13 a  z  + 7 a  z  - 2 a  z  + 

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

      7      3  7      5  7    2  8    4  8


  2 a z  - 5 a  z  - 3 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 41]], Vassiliev[3][Link[9, Alternating, 41]]}
Out[14]=  
      125

{0, -(---)}


      24
In[15]:=
Kh[Link[9, Alternating, 41]][q, t]
Out[15]=  
3    4      1        1        1        3        2        4        2

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



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       4      3      2           t     2    2  2    4  3
 ----- + ----- + ---- + ---- + 2 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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