© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a174
L11a174
L11a176
L11a176
L11a175
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a175

Visit L11a175's page at Knotilus!

Acknowledgement

L11a175 as Morse Link
DrawMorseLink

PD Presentation: X8192 X2,9,3,10 X10,3,11,4 X20,12,21,11 X14,6,15,5 X4,21,5,22 X16,7,17,8 X22,15,7,16 X18,14,19,13 X6,18,1,17 X12,20,13,19

Gauss Code: {{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -11, 9, -5, 8, -7, 10, -9, 11, -4, 6, -8}}

Jones Polynomial: - q-15/2 + 3q-13/2 - 6q-11/2 + 9q-9/2 - 12q-7/2 + 13q-5/2 - 13q-3/2 + 11q-1/2 - 9q1/2 + 5q3/2 - 3q5/2 + q7/2

A2 (sl(3)) Invariant: q-22 - q-20 + q-18 + q-16 + 3q-12 - 2q-10 + q-8 - q-6 - q-4 + 3q-2 + 3q2 + q4 + q8 - q10

HOMFLY-PT Polynomial: - a-1z-1 + a-1z + 3a-1z3 + a-1z5 + 2az-1 - 4az3 - 4az5 - az7 - 2a3z-1 - 4a3z - 5a3z3 - 4a3z5 - a3z7 + a5z-1 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial: 4a-2z2 - 8a-2z4 + 5a-2z6 - a-2z8 - a-1z-1 - a-1z + 17a-1z3 - 28a-1z5 + 16a-1z7 - 3a-1z9 + 10z2 - 22z4 + 7z6 + 5z8 - 2z10 - 2az-1 + 4az + 12az3 - 45az5 + 38az7 - 9az9 - a2 + 13a2z2 - 43a2z4 + 36a2z6 - 4a2z8 - 2a2z10 - 2a3z-1 + 10a3z - 22a3z3 + 12a3z5 + 11a3z7 - 6a3z9 + 4a4z2 - 14a4z4 + 25a4z6 - 10a4z8 - a5z-1 + 4a5z - 12a5z3 + 23a5z5 - 11a5z7 - 3a6z2 + 12a6z4 - 9a6z6 - a7z + 4a7z3 - 6a7z5 - 3a8z4 - a9z3

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 8           1
j = 6          2 
j = 4         31 
j = 2        62  
j = 0       53   
j = -2      86    
j = -4     66     
j = -6    67      
j = -8   47       
j = -10  25        
j = -12 14         
j = -14 2          
j = -161           


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
Length[Skeleton[L]]
Out[2]=   
2
In[3]:=
Show[DrawMorseLink[Link[11, Alternating, 175]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 175]]
Out[4]=   
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[20, 12, 21, 11], 
 
>   X[14, 6, 15, 5], X[4, 21, 5, 22], X[16, 7, 17, 8], X[22, 15, 7, 16], 
 
>   X[18, 14, 19, 13], X[6, 18, 1, 17], X[12, 20, 13, 19]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -6, 5, -10}, 
 
>   {7, -1, 2, -3, 4, -11, 9, -5, 8, -7, 10, -9, 11, -4, 6, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(15/2)     3       6      9      12     13     13      11
-q        + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 
             13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q       q      q      q      q
 
       3/2      5/2    7/2
>   5 q    - 3 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -22    -20    -18    -16    3     2     -8    -6    -4   3       2    4    8
q    - q    + q    + q    + --- - --- + q   - q   - q   + -- + 3 q  + q  + q  - 
                             12    10                      2
                            q     q                       q
 
     10
>   q
In[8]:=
HOMFLYPT[Link[11, Alternating, 175]][a, z]
Out[8]=   
                  3    5                            3
   1     2 a   2 a    a    z      3        5     3 z         3      3  3
-(---) + --- - ---- + -- + - - 4 a  z + 2 a  z + ---- - 4 a z  - 5 a  z  + 
  a z     z     z     z    a                      a
 
               5
       5  3   z         5      3  5    5  5      7    3  7
>   3 a  z  + -- - 4 a z  - 4 a  z  + a  z  - a z  - a  z
              a
In[9]:=
Kauffman[Link[11, Alternating, 175]][a, z]
Out[9]=   
                     3    5
  2    1    2 a   2 a    a    z               3        5      7         2
-a  - --- - --- - ---- - -- - - + 4 a z + 10 a  z + 4 a  z - a  z + 10 z  + 
      a z    z     z     z    a
 
       2                                      3
    4 z        2  2      4  2      6  2   17 z          3       3  3
>   ---- + 13 a  z  + 4 a  z  - 3 a  z  + ----- + 12 a z  - 22 a  z  - 
      2                                     a
     a
 
                                            4
        5  3      7  3    9  3       4   8 z        2  4       4  4
>   12 a  z  + 4 a  z  - a  z  - 22 z  - ---- - 43 a  z  - 14 a  z  + 
                                           2
                                          a
 
                             5
        6  4      8  4   28 z          5       3  5       5  5      7  5
>   12 a  z  - 3 a  z  - ----- - 45 a z  + 12 a  z  + 23 a  z  - 6 a  z  + 
                           a
 
              6                                       7
       6   5 z        2  6       4  6      6  6   16 z          7       3  7
>   7 z  + ---- + 36 a  z  + 25 a  z  - 9 a  z  + ----- + 38 a z  + 11 a  z  - 
             2                                      a
            a
 
                       8                           9
        5  7      8   z       2  8       4  8   3 z         9      3  9
>   11 a  z  + 5 z  - -- - 4 a  z  - 10 a  z  - ---- - 9 a z  - 6 a  z  - 
                       2                         a
                      a
 
       10      2  10
>   2 z   - 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
6    8      1        2        1        4        2        5        4       7
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 4    2    16  6    14  5    12  5    12  4    10  4    10  3    8  3    8  2
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      6      7      6           6 t      2      2  2      2  3      4  3
>   ----- + ---- + ---- + 5 t + --- + 3 t  + 6 q  t  + 2 q  t  + 3 q  t  + 
     6  2    6      4            2
    q  t    q  t   q  t         q
 
     4  4      6  4    8  5
>   q  t  + 2 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a175
L11a174
L11a174
L11a176
L11a176