© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a76
L11a76
L11a78
L11a78
L11a77
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a77

Visit L11a77's page at Knotilus!

Acknowledgement

L11a77 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,4,13,3 X14,12,15,11 X22,15,5,16 X16,7,17,8 X20,17,21,18 X18,10,19,9 X8,20,9,19 X10,21,11,22 X2536 X4,14,1,13

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

Jones Polynomial: - q-13/2 + 3q-11/2 - 7q-9/2 + 12q-7/2 - 16q-5/2 + 19q-3/2 - 20q-1/2 + 16q1/2 - 13q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant: q-20 + 3q-14 - 3q-12 + q-10 - 4q-6 + 3q-4 - 2q-2 + 5 + 2q2 + 4q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial: a-3z + a-3z3 - 2a-1z-1 - 4a-1z - 5a-1z3 - 2a-1z5 + 4az-1 + 8az + 8az3 + 4az5 + az7 - 3a3z-1 - 7a3z - 6a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial: - a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 10a-3z5 - 4a-3z7 - a-2z2 - 5a-2z4 + 13a-2z6 - 6a-2z8 - 2a-1z-1 + 9a-1z - 23a-1z3 + 22a-1z5 - a-1z7 - 4a-1z9 + 2 - z2 - 15z4 + 29z6 - 12z8 - z10 - 4az-1 + 17az - 36az3 + 29az5 - 7az9 + 3a2 - 5a2z2 - 7a2z4 + 20a2z6 - 11a2z8 - a2z10 - 3a3z-1 + 13a3z - 26a3z3 + 25a3z5 - 8a3z7 - 3a3z9 + 3a4 - 7a4z2 + 6a4z4 + 2a4z6 - 5a4z8 - a5z-1 + 2a5z - 4a5z3 + 7a5z5 - 5a5z7 + a6 - 3a6z2 + 5a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 10           1
j = 8          3 
j = 6         51 
j = 4        83  
j = 2       85   
j = 0      128    
j = -2     910     
j = -4    710      
j = -6   59       
j = -8  27        
j = -10 15         
j = -12 2          
j = -141           


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, 77]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 77]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[22, 15, 5, 16], 
 
>   X[16, 7, 17, 8], X[20, 17, 21, 18], X[18, 10, 19, 9], X[8, 20, 9, 19], 
 
>   X[10, 21, 11, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 5, -8, 7, -9, 3, -2, 11, -3, 4, -5, 6, -7, 
 
>    8, -6, 9, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(13/2)     3      7      12     16     19      20
-q        + ----- - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 
             11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q      q      q      q
 
        3/2      5/2      7/2    9/2
>   13 q    + 8 q    - 4 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -20    3     3     -10   4    3    2       2      6      8    10    12
5 + q    + --- - --- + q    - -- + -- - -- + 2 q  + 4 q  - 2 q  + q   + q   - 
            14    12           6    4    2
           q     q            q    q    q
 
     14
>   q
In[8]:=
HOMFLYPT[Link[11, Alternating, 77]][a, z]
Out[8]=   
               3    5                                         3      3
-2    4 a   3 a    a    z    4 z              3        5     z    5 z
--- + --- - ---- + -- + -- - --- + 8 a z - 7 a  z + 2 a  z + -- - ---- + 
a z    z     z     z     3    a                               3    a
                        a                                    a
 
                                  5
         3      3  3    5  3   2 z         5      3  5      7
>   8 a z  - 6 a  z  + a  z  - ---- + 4 a z  - 2 a  z  + a z
                                a
In[9]:=
Kauffman[Link[11, Alternating, 77]][a, z]
Out[9]=   
                                      3    5
       2      4    6    2    4 a   3 a    a    2 z   9 z                3
2 + 3 a  + 3 a  + a  - --- - --- - ---- - -- + --- + --- + 17 a z + 13 a  z + 
                       a z    z     z     z     3     a
                                               a
 
                          2    2                                    3       3
       5      7      2   z    z       2  2      4  2      6  2   7 z    23 z
>   2 a  z - a  z - z  - -- - -- - 5 a  z  - 7 a  z  - 3 a  z  - ---- - ----- - 
                          4    2                                   3      a
                         a    a                                   a
 
                                                        4      4
          3       3  3      5  3      7  3       4   2 z    5 z       2  4
>   36 a z  - 26 a  z  - 4 a  z  + 2 a  z  - 15 z  + ---- - ---- - 7 a  z  + 
                                                       4      2
                                                      a      a
 
                            5       5
       4  4      6  4   10 z    22 z          5       3  5      5  5    7  5
>   6 a  z  + 5 a  z  + ----- + ----- + 29 a z  + 25 a  z  + 7 a  z  - a  z  + 
                          3       a
                         a
 
             6       6                                     7    7
        6   z    13 z        2  6      4  6      6  6   4 z    z       3  7
>   29 z  - -- + ----- + 20 a  z  + 2 a  z  - 3 a  z  - ---- - -- - 8 a  z  - 
             4     2                                      3    a
            a     a                                      a
 
                         8                           9
       5  7       8   6 z        2  8      4  8   4 z         9      3  9
>   5 a  z  - 12 z  - ---- - 11 a  z  - 5 a  z  - ---- - 7 a z  - 3 a  z  - 
                        2                          a
                       a
 
     10    2  10
>   z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     10     1        2        1        5        2       7       5       9
12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
      2    14  6    12  5    10  5    10  4    8  4    8  3    6  3    6  2
     q    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      7      10     9              2        2  2      4  2      4  3
>   ----- + ---- + ---- + 8 t + 8 q  t + 5 q  t  + 8 q  t  + 3 q  t  + 
     4  2    4      2
    q  t    q  t   q  t
 
       6  3    6  4      8  4    10  5
>   5 q  t  + q  t  + 3 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a77
L11a76
L11a76
L11a78
L11a78