L9a37

From Knot Atlas
Revision as of 21:16, 28 August 2005 by ScottTestRobot (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

L9a36.gif

L9a36

L9a38.gif

L9a38

L9a37.gif Visit L9a37's page at Knotilus!

Visit L9a37's page at the original Knot Atlas!

L9a37 is [math]\displaystyle{ 9^2_{7} }[/math] in the Rolfsen table of links.


L9a37 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{41}{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:L9a37/V 2,1 Data:L9a37/V 3,1 Data:L9a37/V 4,1 Data:L9a37/V 4,2 Data:L9a37/V 4,3 Data:L9a37/V 5,1 Data:L9a37/V 5,2 Data:L9a37/V 5,3 Data:L9a37/V 5,4 Data:L9a37/V 6,1 Data:L9a37/V 6,2 Data:L9a37/V 6,3 Data:L9a37/V 6,4 Data:L9a37/V 6,5 Data:L9a37/V 6,6 Data:L9a37/V 6,7 Data:L9a37/V 6,8 Data:L9a37/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 L9a37. 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      42  2
10     44   0
8    43    1
6   24     2
4  24      -2
2 13       2
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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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, 37]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 37]]
Out[3]=  
PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[2, 9, 3, 10], 

 X[4, 18, 5, 17], X[16, 8, 17, 7], X[14, 6, 15, 5], X[6, 16, 7, 15], 



  X[8, 14, 1, 13]]
In[4]:=
GaussCode[Link[9, Alternating, 37]]
Out[4]=  
GaussCode[{1, -4, 2, -5, 7, -8, 6, -9}, 
  {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}]
In[5]:=
BR[Link[9, Alternating, 37]]
Out[5]=  
BR[Link[9, Alternating, 37]]
In[6]:=
alex = Alexander[Link[9, Alternating, 37]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 37]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 37]], KnotSignature[Link[9, Alternating, 37]]}
Out[9]=  
{Infinity, 3}
In[10]:=
J=Jones[Link[9, Alternating, 37]][q]
Out[10]=  
     1                      3/2      5/2      7/2      9/2      11/2

-(-------) + 2 Sqrt[q] - 4 q    + 6 q    - 8 q    + 7 q    - 7 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, 37]][q]
Out[12]=  
 -2    4      6      8    10      12      14      18    20    24    26
q   + q  - 2 q  + 2 q  + q   + 2 q   + 3 q   + 2 q   - q   + q   - q
In[13]:=
Kauffman[Link[9, Alternating, 37]][a, z]
Out[13]=  
                                            2     2      2      2

-4    1      1     z    5 z   2 z   2 z   z     z    3 z    4 z



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



      5      3      9    5     3     a     10    8     6      4
     a  z   a  z   a    a     a           a     a     a      a

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

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

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


   a      a     a    a
In[14]:=
{Vassiliev[2][Link[9, Alternating, 37]], Vassiliev[3][Link[9, Alternating, 37]]}
Out[14]=  
    41

{0, --}


    24
In[15]:=
Kh[Link[9, Alternating, 37]][q, t]
Out[15]=  
                           2

  2      4     1     1   q       4        6        6  2      8  2



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

  14  6      16  6    18  7


  q   t  + 2 q   t  + q   t
Retrieved from "https://katlas.org/index.php?title=L9a37&oldid=37263"