L9a30

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L9a29.gif

L9a29

L9a31.gif

L9a31

L9a30.gif Visit L9a30's page at Knotilus!

Visit L9a30's page at the original Knot Atlas!

L9a30 is [math]\displaystyle{ 9^2_{3} }[/math] in the Rolfsen table of links.


L9a30 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X18,11,7,12 X12,17,13,18
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{2 u^2 v^2-2 u^2 v-2 u v^2+3 u v-2 u-2 v+2}{u v} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^7 \left(-z^3\right)-2 a^7 z+a^5 z^5+3 a^5 z^3+2 a^5 z+a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} -a z^3-3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-4 a^9 z^3+a^9 z+2 a^8 z^6-3 a^8 z^4+a^8 z^2+2 a^7 z^7-5 a^7 z^5+7 a^7 z^3-2 a^7 z+a^6 z^8-2 a^6 z^6+4 a^6 z^4-a^6 z^2+3 a^5 z^7-9 a^5 z^5+12 a^5 z^3-4 a^5 z+a^4 z^8-3 a^4 z^6+6 a^4 z^4-5 a^4 z^2+a^3 z^7-a^3 z^5-3 a^3 z^3+3 a^3 z-a^3 z^{-1} +a^2 z^6-2 a^2 z^4-a^2 z^2+a^2+a z^5-4 a z^3+4 a z-a z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{79}{48} }[/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:L9a30/V 2,1 Data:L9a30/V 3,1 Data:L9a30/V 4,1 Data:L9a30/V 4,2 Data:L9a30/V 4,3 Data:L9a30/V 5,1 Data:L9a30/V 5,2 Data:L9a30/V 5,3 Data:L9a30/V 5,4 Data:L9a30/V 6,1 Data:L9a30/V 6,2 Data:L9a30/V 6,3 Data:L9a30/V 6,4 Data:L9a30/V 6,5 Data:L9a30/V 6,6 Data:L9a30/V 6,7 Data:L9a30/V 6,8 Data:L9a30/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 L9a30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-1012χ
2         11
0          0
-2       31 2
-4      21  -1
-6     32   1
-8    33    0
-10   22     0
-12  13      2
-14 12       -1
-16 1        1
-181         -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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/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, 30]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 30]]
Out[3]=  
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], 

 X[16, 13, 17, 14], X[14, 6, 15, 5], X[4, 16, 5, 15], 



  X[18, 11, 7, 12], X[12, 17, 13, 18]]
In[4]:=
GaussCode[Link[9, Alternating, 30]]
Out[4]=  
GaussCode[{1, -2, 3, -7, 6, -4}, 
  {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}]
In[5]:=
BR[Link[9, Alternating, 30]]
Out[5]=  
BR[Link[9, Alternating, 30]]
In[6]:=
alex = Alexander[Link[9, Alternating, 30]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 30]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 30]], KnotSignature[Link[9, Alternating, 30]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[9, Alternating, 30]][q]
Out[10]=  
 -(17/2)     2       3       5      5      5      4      3

q        - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 



           15/2    13/2    11/2    9/2    7/2    5/2    3/2
          q       q       q       q      q      q      q

    1
 ------- - Sqrt[q]


  Sqrt[q]
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 30]][q]
Out[12]=  
     -26    2     -14   2    2     -2    2

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



           18           8    4


           q            q    q
In[13]:=
Kauffman[Link[9, Alternating, 30]][a, z]
Out[13]=  
           3

 2   a   a               3        5        7      9      2  2



-a  + - + -- - 4 a z - 3 a  z + 4 a  z + 2 a  z - a  z + a  z  + 



     z   z

    4  2    6  2    8  2      10  2        3      3  3       5  3
 5 a  z  + a  z  - a  z  + 2 a   z  + 4 a z  + 3 a  z  - 12 a  z  - 

    7  3      9  3      2  4      4  4      6  4      8  4    10  4
 7 a  z  + 4 a  z  + 2 a  z  - 6 a  z  - 4 a  z  + 3 a  z  - a   z  - 

    5    3  5      5  5      7  5      9  5    2  6      4  6
 a z  + a  z  + 9 a  z  + 5 a  z  - 2 a  z  - a  z  + 3 a  z  + 

    6  6      8  6    3  7      5  7      7  7    4  8    6  8


  2 a  z  - 2 a  z  - a  z  - 3 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 30]], Vassiliev[3][Link[9, Alternating, 30]]}
Out[14]=  
    79

{0, --}


    48
In[15]:=
Kh[Link[9, Alternating, 30]][q, t]
Out[15]=  
 -4   3      1        1        1        2        1        3

q   + -- + ------ + ------ + ------ + ------ + ------ + ------ + 



      2    18  7    16  6    14  6    14  5    12  5    12  4
     q    q   t    q   t    q   t    q   t    q   t    q   t

   2        2        3       3       3      2      2     t     2  2
 ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + q  t
  10  4    10  3    8  3    8  2    6  2    6      4      2


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