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K11a51
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   The Knot K11a52

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Acknowledgement

K11a52 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - 2t-3 + 13t-2 - 32t-1 + 43 - 32t + 13t2 - 2t3

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

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

Determinant and Signature: {137, 0}

Jones Polynomial: - q-5 + 3q-4 - 7q-3 + 13q-2 - 18q-1 + 22 - 22q + 20q2 - 15q3 + 10q4 - 5q5 + q6

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

A2 (sl(3)) Invariant: - q-16 + q-12 - 3q-10 + 3q-8 + 2q-6 - 3q-4 + 4q-2 - 3 + 2q2 + q4 - q6 + 5q8 - 4q10 + q12 + q14 - 3q16 + q18

HOMFLY-PT Polynomial: - a-4 + a-4z4 + 2a-2 + a-2z2 - a-2z4 - a-2z6 - 1 - z2 - z4 - z6 + 2a2 + 3a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial: - a-6z4 + a-6z6 + a-5z + 3a-5z3 - 10a-5z5 + 5a-5z7 - a-4 + 11a-4z4 - 21a-4z6 + 9a-4z8 + 7a-3z3 - 9a-3z5 - 8a-3z7 + 7a-3z9 - 2a-2 - 2a-2z2 + 27a-2z4 - 41a-2z6 + 14a-2z8 + 2a-2z10 - 3a-1z + 10a-1z3 - a-1z5 - 18a-1z7 + 12a-1z9 - 1 - z2 + 19z4 - 27z6 + 11z8 + 2z10 - 3az + 10az3 - 9az5 + 5az9 - 2a2 + 4a2z2 - a2z4 - 5a2z6 + 6a2z8 + 2a3z3 - 6a3z5 + 5a3z7 - a4 + 3a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 0}

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=0 is the signature of 1152. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 13           1
j = 11          4 
j = 9         61 
j = 7        94  
j = 5       116   
j = 3      119    
j = 1     1111     
j = -1    812      
j = -3   510       
j = -5  28        
j = -7 15         
j = -9 2          
j = -111           


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, 52]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 52]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[12, 8, 13, 7], 
 
>   X[2, 9, 3, 10], X[18, 12, 19, 11], X[22, 13, 1, 14], X[20, 16, 21, 15], 
 
>   X[10, 18, 11, 17], X[16, 20, 17, 19], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 52]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, 
 
>   -8, 11, -7]
In[5]:=
DTCode[Knot[11, Alternating, 52]]
Out[5]=   
DTCode[4, 8, 14, 12, 2, 18, 22, 20, 10, 16, 6]
In[6]:=
alex = Alexander[Knot[11, Alternating, 52]][t]
Out[6]=   
     2    13   32              2      3
43 - -- + -- - -- - 32 t + 13 t  - 2 t
      3    2   t
     t    t
In[7]:=
Conway[Knot[11, Alternating, 52]][z]
Out[7]=   
       2    4      6
1 + 2 z  + z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 52]}
In[9]:=
{KnotDet[Knot[11, Alternating, 52]], KnotSignature[Knot[11, Alternating, 52]]}
Out[9]=   
{137, 0}
In[10]:=
J=Jones[Knot[11, Alternating, 52]][q]
Out[10]=   
      -5   3    7    13   18              2       3       4      5    6
22 - q   + -- - -- + -- - -- - 22 q + 20 q  - 15 q  + 10 q  - 5 q  + q
            4    3    2   q
           q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 52]}
In[12]:=
A2Invariant[Knot[11, Alternating, 52]][q]
Out[12]=   
      -16    -12    3    3    2    3    4       2    4    6      8      10
-3 - q    + q    - --- + -- + -- - -- + -- + 2 q  + q  - q  + 5 q  - 4 q   + 
                    10    8    6    4    2
                   q     q    q    q    q
 
     12    14      16    18
>   q   + q   - 3 q   + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 52]][a, z]
Out[13]=   
                                  2                           4    4
      -4   2       2    4    2   z       2  2    4  2    4   z    z
-1 - a   + -- + 2 a  - a  - z  + -- + 3 a  z  - a  z  - z  + -- - -- + 
            2                     2                           4    2
           a                     a                           a    a
 
                    6
       2  4    6   z
>   2 a  z  - z  - --
                    2
                   a
In[14]:=
Kauffman[Knot[11, Alternating, 52]][a, z]
Out[14]=   
                                                              2
      -4   2       2    4   z    3 z            5      2   2 z       2  2
-1 - a   - -- - 2 a  - a  + -- - --- - 3 a z + a  z - z  - ---- + 4 a  z  + 
            2                5    a                          2
           a                a                               a
 
                 3      3       3                                          4
       4  2   3 z    7 z    10 z          3      3  3      5  3       4   z
>   3 a  z  + ---- + ---- + ----- + 10 a z  + 2 a  z  - 2 a  z  + 19 z  - -- + 
                5      3      a                                            6
               a      a                                                   a
 
        4       4                         5      5    5
    11 z    27 z     2  4      4  4   10 z    9 z    z         5      3  5
>   ----- + ----- - a  z  - 5 a  z  - ----- - ---- - -- - 9 a z  - 6 a  z  + 
      4       2                         5       3    a
     a       a                         a       a
 
                     6       6       6                          7      7
     5  5       6   z    21 z    41 z       2  6      4  6   5 z    8 z
>   a  z  - 27 z  + -- - ----- - ----- - 5 a  z  + 3 a  z  + ---- - ---- - 
                     6     4       2                           5      3
                    a     a       a                           a      a
 
        7                        8       8                9       9
    18 z       3  7       8   9 z    14 z       2  8   7 z    12 z         9
>   ----- + 5 a  z  + 11 z  + ---- + ----- + 6 a  z  + ---- + ----- + 5 a z  + 
      a                         4      2                 3      a
                               a      a                 a
 
               10
       10   2 z
>   2 z   + -----
              2
             a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 52]], Vassiliev[3][Knot[11, Alternating, 52]]}
Out[15]=   
{2, 0}
In[16]:=
Kh[Knot[11, Alternating, 52]][q, t]
Out[16]=   
12            1        2       1       5       2       8       5      10
-- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 
q            11  5    9  4    7  4    7  3    5  3    5  2    3  2    3
            q   t    q  t    q  t    q  t    q  t    q  t    q  t    q  t
 
     8                 3        3  2       5  2      5  3      7  3      7  4
>   --- + 11 q t + 11 q  t + 9 q  t  + 11 q  t  + 6 q  t  + 9 q  t  + 4 q  t  + 
    q t
 
       9  4    9  5      11  5    13  6
>   6 q  t  + q  t  + 4 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a52
K11a51
K11a51
K11a53
K11a53