L9a17

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L9a16.gif

L9a16

L9a18.gif

L9a18

L9a17.gif Visit L9a17's page at Knotilus!

Visit L9a17's page at the original Knot Atlas!

L9a17 is [math]\displaystyle{ 9^2_{27} }[/math] in the Rolfsen table of links.


L9a17 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X12,4,13,3 X16,8,17,7 X18,10,5,9 X8,18,9,17 X14,12,15,11 X10,16,11,15 X2536 X4,14,1,13
Gauss code {1, -8, 2, -9}, {8, -1, 3, -5, 4, -7, 6, -2, 9, -6, 7, -3, 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{ 7 q^{9/2}-7 q^{7/2}+5 q^{5/2}-4 q^{3/2}+q^{17/2}-3 q^{15/2}+5 q^{13/2}-6 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature 3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^5 a^{-3} -z^5 a^{-5} +z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} +3 z a^{-1} -4 z a^{-3} +z a^{-7} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} -z^6 a^{-2} +z^6 a^{-4} -2 z^6 a^{-6} -4 z^6 a^{-8} -z^5 a^{-1} -z^5 a^{-3} +7 z^5 a^{-5} +4 z^5 a^{-7} -3 z^5 a^{-9} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} +4 z^3 a^{-1} +7 z^3 a^{-3} -5 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +3 z^2 a^{-2} +7 z^2 a^{-4} +z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-1} -7 z a^{-3} +2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{7}{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:L9a17/V 2,1 Data:L9a17/V 3,1 Data:L9a17/V 4,1 Data:L9a17/V 4,2 Data:L9a17/V 4,3 Data:L9a17/V 5,1 Data:L9a17/V 5,2 Data:L9a17/V 5,3 Data:L9a17/V 5,4 Data:L9a17/V 6,1 Data:L9a17/V 6,2 Data:L9a17/V 6,3 Data:L9a17/V 6,4 Data:L9a17/V 6,5 Data:L9a17/V 6,6 Data:L9a17/V 6,7 Data:L9a17/V 6,8 Data:L9a17/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 L9a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-101234567χ
18         1-1
16        2 2
14       31 -2
12      32  1
10     43   -1
8    33    0
6   24     2
4  23      -1
2 14       3
0          0
-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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 17]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 17]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[18, 10, 5, 9], 

 X[8, 18, 9, 17], X[14, 12, 15, 11], X[10, 16, 11, 15], X[2, 5, 3, 6], 



  X[4, 14, 1, 13]]
In[4]:=
GaussCode[Link[9, Alternating, 17]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 3, -5, 4, -7, 6, -2, 9, -6, 7, -3, 5, 
   -4}]
In[5]:=
BR[Link[9, Alternating, 17]]
Out[5]=  
BR[Link[9, Alternating, 17]]
In[6]:=
alex = Alexander[Link[9, Alternating, 17]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 17]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 17]], KnotSignature[Link[9, Alternating, 17]]}
Out[9]=  
{Infinity, 3}
In[10]:=
J=Jones[Link[9, Alternating, 17]][q]
Out[10]=  
     1                    3/2      5/2      7/2      9/2      11/2

-(-------) + Sqrt[q] - 4 q    + 5 q    - 7 q    + 7 q    - 6 q     + 



 Sqrt[q]

    13/2      15/2    17/2


  5 q     - 3 q     + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 17]][q]
Out[12]=  
     -2      2      4    6      8    12      16    18    20    24    26
1 + q   + 2 q  + 4 q  + q  + 3 q  - q   - 2 q   + q   - q   + q   - q
In[13]:=
Kauffman[Link[9, Alternating, 17]][a, z]
Out[13]=  
                                                        2       2

 -6   3    3     1      3      2    2 z   7 z   5 z   z     2 z



-a   - -- - -- + ---- + ---- + --- + --- - --- - --- + --- - ---- + 



       4    2    5      3     a z    7     3     a     10     8
      a    a    a  z   a  z         a     a           a      a

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

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

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


  a     a      a     a    a    a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 17]], Vassiliev[3][Link[9, Alternating, 17]]}
Out[14]=  
      7

{0, -(-)}


      2
In[15]:=
Kh[Link[9, Alternating, 17]][q, t]
Out[15]=  
                       2

  2      4     1     q       4        6        6  2      8  2



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



              2  2   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  + 2 q   t  + 3 q   t  + 

  14  6      16  6    18  7


  q   t  + 2 q   t  + q   t
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