L9a15

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L9a14.gif

L9a14

L9a16.gif

L9a16

L9a15.gif Visit L9a15's page at Knotilus!

Visit L9a15's page at the original Knot Atlas!

L9a15 is [math]\displaystyle{ 9^2_{15} }[/math] in the Rolfsen table of links.


L9a15 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{15}{2} }[/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:L9a15/V 2,1 Data:L9a15/V 3,1 Data:L9a15/V 4,1 Data:L9a15/V 4,2 Data:L9a15/V 4,3 Data:L9a15/V 5,1 Data:L9a15/V 5,2 Data:L9a15/V 5,3 Data:L9a15/V 5,4 Data:L9a15/V 6,1 Data:L9a15/V 6,2 Data:L9a15/V 6,3 Data:L9a15/V 6,4 Data:L9a15/V 6,5 Data:L9a15/V 6,6 Data:L9a15/V 6,7 Data:L9a15/V 6,8 Data:L9a15/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 L9a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
18         1-1
16        1 1
14       31 -2
12      31  2
10     43   -1
8    33    0
6   24     2
4  33      0
2 14       3
0 1        -1
-21         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/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, 15]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 15]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[18, 12, 5, 11], 

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



  X[4, 14, 1, 13]]
In[4]:=
GaussCode[Link[9, Alternating, 15]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 3, -5, 6, -7, 4, -2, 9, -3, 7, -6, 5, 
   -4}]
In[5]:=
BR[Link[9, Alternating, 15]]
Out[5]=  
BR[Link[9, Alternating, 15]]
In[6]:=
alex = Alexander[Link[9, Alternating, 15]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 15]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 15]], KnotSignature[Link[9, Alternating, 15]]}
Out[9]=  
{Infinity, 3}
In[10]:=
J=Jones[Link[9, Alternating, 15]][q]
Out[10]=  
     1                      3/2      5/2      7/2      9/2      11/2

-(-------) + 2 Sqrt[q] - 5 q    + 5 q    - 7 q    + 7 q    - 6 q     + 



 Sqrt[q]

    13/2      15/2    17/2


  4 q     - 2 q     + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 15]][q]
Out[12]=  
 -2    2      4    6      8    10    12    14    16    18      20

q   + q  + 3 q  + q  + 4 q  + q   + q   + q   - q   + q   - 2 q   - 



  22    26


  q   - q
In[13]:=
Kauffman[Link[9, Alternating, 15]][a, z]
Out[13]=  
                                                              2

2    5    3     -2    1      2      1    4 z   7 z   3 z   2 z
-- + -- + -- - a   - ---- - ---- + --- + --- + --- - --- + ---- - 



8    6    4          7      5     a z    7     5     a     10



a    a    a          a  z   a  z         a     a           a



    2       2      2      3       3       3      3      3    4
 6 z    15 z    7 z    3 z    10 z    13 z    3 z    3 z    z
 ---- - ----- - ---- + ---- - ----- - ----- + ---- + ---- - --- + 
   8      6       4      9      7       5       3     a      10
  a      a       a      a      a       a       a            a

    4       4      4      4      5      5       5      5    5      6
 6 z    11 z    8 z    4 z    2 z    7 z    12 z    2 z    z    3 z
 ---- + ----- + ---- + ---- - ---- + ---- + ----- + ---- - -- - ---- - 
   8      6       4      2      9      7      5       3    a      8
  a      a       a      a      a      a      a       a           a

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


   a      a      a      a      a      a     a    a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 15]], Vassiliev[3][Link[9, Alternating, 15]]}
Out[14]=  
      15

{0, -(--)}


      2
In[15]:=
Kh[Link[9, Alternating, 15]][q, t]
Out[15]=  
                           2

  2      4     1     1   q       4        6        6  2      8  2



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



              2  2   t   t
             q  t

    8  3      10  3      10  4      12  4    12  5      14  5
 3 q  t  + 4 q   t  + 3 q   t  + 3 q   t  + q   t  + 3 q   t  + 

  14  6    16  6    18  7


  q   t  + q   t  + q   t
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