© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a65
K11a65
K11a67
K11a67
K11a66
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   The Knot K11a66

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Acknowledgement

K11a66 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X20,13,21,14 X22,15,1,16 X10,18,11,17 X6,19,7,20 X14,21,15,22

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

DT (Dowker-Thistlethwaite) Code: 4 8 16 12 2 18 20 22 10 6 14

Alexander Polynomial: - t-4 + 6t-3 - 15t-2 + 24t-1 - 27 + 24t - 15t2 + 6t3 - t4

Conway Polynomial: 1 + 2z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial: {K11a163, ...}

Determinant and Signature: {119, -2}

Jones Polynomial: - q-8 + 3q-7 - 7q-6 + 12q-5 - 16q-4 + 19q-3 - 19q-2 + 17q-1 - 12 + 8q - 4q2 + q3

Other knots (up to mirrors) with the same Jones Polynomial: {K11a121, ...}

A2 (sl(3)) Invariant: - q-24 - 2q-18 + 3q-16 - 2q-14 + 2q-12 + 2q-10 - 2q-8 + 4q-6 - 4q-4 + 3q-2 - q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial: 1 + 2z2 + 3z4 + z6 - 2a2 - 7a2z2 - 9a2z4 - 5a2z6 - a2z8 + 4a4 + 10a4z2 + 8a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial: - 2a-2z4 + a-2z6 + 4a-1z3 - 10a-1z5 + 4a-1z7 + 1 - 5z2 + 16z4 - 20z6 + 7z8 + az - az3 + 6az5 - 13az7 + 6az9 + 2a2 - 17a2z2 + 41a2z4 - 35a2z6 + 8a2z8 + 2a2z10 + 2a3z - 12a3z3 + 29a3z5 - 28a3z7 + 11a3z9 + 4a4 - 21a4z2 + 36a4z4 - 26a4z6 + 7a4z8 + 2a4z10 - 3a5z3 + 5a5z5 - 6a5z7 + 5a5z9 + 2a6 - 7a6z2 + 8a6z4 - 9a6z6 + 6a6z8 + 2a7z3 - 7a7z5 + 5a7z7 + 2a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, -4}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-2 is the signature of 1166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 7           1
j = 5          3 
j = 3         51 
j = 1        73  
j = -1       105   
j = -3      108    
j = -5     99     
j = -7    710      
j = -9   59       
j = -11  27        
j = -13 15         
j = -15 2          
j = -171           


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]:=
Show[DrawMorseLink[Knot[11, Alternating, 66]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 66]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], 
 
>   X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 13, 21, 14], X[22, 15, 1, 16], 
 
>   X[10, 18, 11, 17], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 66]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, 
 
>   -7, 11, -8]
In[5]:=
DTCode[Knot[11, Alternating, 66]]
Out[5]=   
DTCode[4, 8, 16, 12, 2, 18, 20, 22, 10, 6, 14]
In[6]:=
alex = Alexander[Knot[11, Alternating, 66]][t]
Out[6]=   
       -4   6    15   24              2      3    4
-27 - t   + -- - -- + -- + 24 t - 15 t  + 6 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 66]][z]
Out[7]=   
       2    4      6    8
1 + 2 z  + z  - 2 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 66], Knot[11, Alternating, 163]}
In[9]:=
{KnotDet[Knot[11, Alternating, 66]], KnotSignature[Knot[11, Alternating, 66]]}
Out[9]=   
{119, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 66]][q]
Out[10]=   
       -8   3    7    12   16   19   19   17            2    3
-12 - q   + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q  + q
             7    6    5    4    3    2   q
            q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 66], Knot[11, Alternating, 121]}
In[12]:=
A2Invariant[Knot[11, Alternating, 66]][q]
Out[12]=   
  -24    2     3     2     2     2    2    4    4    3     2      4      6    8
-q    - --- + --- - --- + --- + --- - -- + -- - -- + -- - q  + 2 q  - 2 q  + q
         18    16    14    12    10    8    6    4    2
        q     q     q     q     q     q    q    q    q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 66]][a, z]
Out[13]=   
       2      4      6      2      2  2       4  2      6  2      4      2  4
1 - 2 a  + 4 a  - 2 a  + 2 z  - 7 a  z  + 10 a  z  - 3 a  z  + 3 z  - 9 a  z  + 
 
       4  4    6  4    6      2  6      4  6    2  8
>   8 a  z  - a  z  + z  - 5 a  z  + 2 a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 66]][a, z]
Out[14]=   
       2      4      6            3      9        2       2  2       4  2
1 + 2 a  + 4 a  + 2 a  + a z + 2 a  z + a  z - 5 z  - 17 a  z  - 21 a  z  - 
 
                           3
       6  2      8  2   4 z       3       3  3      5  3      7  3      9  3
>   7 a  z  + 2 a  z  + ---- - a z  - 12 a  z  - 3 a  z  + 2 a  z  - 2 a  z  + 
                         a
 
               4                                                 5
        4   2 z        2  4       4  4      6  4      8  4   10 z         5
>   16 z  - ---- + 41 a  z  + 36 a  z  + 8 a  z  - 5 a  z  - ----- + 6 a z  + 
              2                                                a
             a
 
                                                    6
        3  5      5  5      7  5    9  5       6   z        2  6       4  6
>   29 a  z  + 5 a  z  - 7 a  z  + a  z  - 20 z  + -- - 35 a  z  - 26 a  z  - 
                                                    2
                                                   a
 
                           7
       6  6      8  6   4 z          7       3  7      5  7      7  7      8
>   9 a  z  + 3 a  z  + ---- - 13 a z  - 28 a  z  - 6 a  z  + 5 a  z  + 7 z  + 
                         a
 
       2  8      4  8      6  8        9       3  9      5  9      2  10
>   8 a  z  + 7 a  z  + 6 a  z  + 6 a z  + 11 a  z  + 5 a  z  + 2 a  z   + 
 
       4  10
>   2 a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 66]], Vassiliev[3][Knot[11, Alternating, 66]]}
Out[15]=   
{2, -4}
In[16]:=
Kh[Knot[11, Alternating, 66]][q, t]
Out[16]=   
8    10     1        2        1        5        2        7        5       9
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 3   q     17  7    15  6    13  6    13  5    11  5    11  4    9  4    9  3
q         q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      7      10       9      9      10    5 t                2      3  2
>   ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t  + 5 q  t  + 
     7  3    7  2    5  2    5      3      q
    q  t    q  t    q  t    q  t   q  t
 
     3  3      5  3    7  4
>   q  t  + 3 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a66
K11a65
K11a65
K11a67
K11a67