© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a56
K11a56
K11a58
K11a58
K11a57
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   The Knot K11a57

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Acknowledgement

K11a57 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X16,6,17,5 X2837 X20,9,21,10 X22,11,1,12 X18,13,19,14 X6,16,7,15 X12,17,13,18 X14,19,15,20 X10,21,11,22

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

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

Alexander Polynomial: - t-4 + 5t-3 - 12t-2 + 20t-1 - 23 + 20t - 12t2 + 5t3 - t4

Conway Polynomial: 1 + z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial: {K11a108, K11a139, K11a231, ...}

Determinant and Signature: {99, -2}

Jones Polynomial: q-7 - 3q-6 + 7q-5 - 12q-4 + 14q-3 - 16q-2 + 16q-1 - 12 + 10q - 5q2 + 2q3 - q4

Other knots (up to mirrors) with the same Jones Polynomial: {K11a231, ...}

A2 (sl(3)) Invariant: q-20 - q-18 + 3q-16 - q-14 - q-12 - 6q-8 + q-6 - 3q-4 + 4q-2 + 5 + 2q2 + 4q4 - 2q6 - q8 - q10 - q12

HOMFLY-PT Polynomial: - 4a-2 - 4a-2z2 - a-2z4 + 12 + 18z2 + 10z4 + 2z6 - 10a2 - 19a2z2 - 15a2z4 - 6a2z6 - a2z8 + 3a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial: - 4a-3z + 8a-3z3 - 5a-3z5 + a-3z7 + 4a-2 - 8a-2z2 + 11a-2z4 - 8a-2z6 + 2a-2z8 - 11a-1z + 20a-1z3 - 9a-1z5 - 3a-1z7 + 2a-1z9 + 12 - 24z2 + 28z4 - 21z6 + 4z8 + z10 - 17az + 29az3 - 14az5 - 9az7 + 6az9 + 10a2 - 22a2z2 + 30a2z4 - 31a2z6 + 10a2z8 + a2z10 - 16a3z + 35a3z3 - 29a3z5 + 4a3z7 + 4a3z9 + 3a4 - 2a4z2 + 6a4z4 - 12a4z6 + 8a4z8 - 6a5z + 16a5z3 - 16a5z5 + 9a5z7 + 3a6z2 - 6a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 3}

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 1157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 9           1
j = 7          1 
j = 5         41 
j = 3        61  
j = 1       64   
j = -1      106    
j = -3     77     
j = -5    79      
j = -7   57       
j = -9  27        
j = -11 15         
j = -13 2          
j = -151           


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, 57]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 57]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], 
 
>   X[20, 9, 21, 10], X[22, 11, 1, 12], X[18, 13, 19, 14], X[6, 16, 7, 15], 
 
>   X[12, 17, 13, 18], X[14, 19, 15, 20], X[10, 21, 11, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 57]]
Out[4]=   
GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, 
 
>   -5, 11, -6]
In[5]:=
DTCode[Knot[11, Alternating, 57]]
Out[5]=   
DTCode[4, 8, 16, 2, 20, 22, 18, 6, 12, 14, 10]
In[6]:=
alex = Alexander[Knot[11, Alternating, 57]][t]
Out[6]=   
       -4   5    12   20              2      3    4
-23 - t   + -- - -- + -- + 20 t - 12 t  + 5 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 57]][z]
Out[7]=   
     2      4      6    8
1 + z  - 2 z  - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 57], Knot[11, Alternating, 108], 
 
>   Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
In[9]:=
{KnotDet[Knot[11, Alternating, 57]], KnotSignature[Knot[11, Alternating, 57]]}
Out[9]=   
{99, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 57]][q]
Out[10]=   
       -7   3    7    12   14   16   16             2      3    4
-12 + q   - -- + -- - -- + -- - -- + -- + 10 q - 5 q  + 2 q  - q
             6    5    4    3    2   q
            q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 57], Knot[11, Alternating, 231]}
In[12]:=
A2Invariant[Knot[11, Alternating, 57]][q]
Out[12]=   
     -20    -18    3     -14    -12   6     -6   3    4       2      4      6
5 + q    - q    + --- - q    - q    - -- + q   - -- + -- + 2 q  + 4 q  - 2 q  - 
                   16                  8          4    2
                  q                   q          q    q
 
     8    10    12
>   q  - q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 57]][a, z]
Out[13]=   
                                    2                                 4
     4        2      4       2   4 z        2  2      4  2       4   z
12 - -- - 10 a  + 3 a  + 18 z  - ---- - 19 a  z  + 6 a  z  + 10 z  - -- - 
      2                            2                                  2
     a                            a                                  a
 
        2  4      4  4      6      2  6    4  6    2  8
>   15 a  z  + 4 a  z  + 2 z  - 6 a  z  + a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 57]][a, z]
Out[14]=   
     4        2      4   4 z   11 z                3        5         2
12 + -- + 10 a  + 3 a  - --- - ---- - 17 a z - 16 a  z - 6 a  z - 24 z  - 
      2                   3     a
     a                   a
 
       2                                             3       3
    8 z        2  2      4  2      6  2    8  2   8 z    20 z          3
>   ---- - 22 a  z  - 2 a  z  + 3 a  z  - a  z  + ---- + ----- + 29 a z  + 
      2                                             3      a
     a                                             a
 
                                                4
        3  3       5  3      7  3       4   11 z        2  4      4  4
>   35 a  z  + 16 a  z  - 2 a  z  + 28 z  + ----- + 30 a  z  + 6 a  z  - 
                                              2
                                             a
 
                         5      5
       6  4    8  4   5 z    9 z          5       3  5       5  5      7  5
>   6 a  z  + a  z  - ---- - ---- - 14 a z  - 29 a  z  - 16 a  z  + 3 a  z  - 
                        3     a
                       a
 
               6                                    7      7
        6   8 z        2  6       4  6      6  6   z    3 z         7
>   21 z  - ---- - 31 a  z  - 12 a  z  + 6 a  z  + -- - ---- - 9 a z  + 
              2                                     3    a
             a                                     a
 
                                  8                           9
       3  7      5  7      8   2 z        2  8      4  8   2 z         9
>   4 a  z  + 9 a  z  + 4 z  + ---- + 10 a  z  + 8 a  z  + ---- + 6 a z  + 
                                 2                          a
                                a
 
       3  9    10    2  10
>   4 a  z  + z   + a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 57]], Vassiliev[3][Knot[11, Alternating, 57]]}
Out[15]=   
{1, 3}
In[16]:=
Kh[Knot[11, Alternating, 57]][q, t]
Out[16]=   
7    10     1        2        1        5        2       7       5       7
-- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
 3   q     15  6    13  5    11  5    11  4    9  4    9  3    7  3    7  2
q         q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      7      9      7     6 t                2      3  2    3  3      5  3
>   ----- + ---- + ---- + --- + 6 q t + 4 q t  + 6 q  t  + q  t  + 4 q  t  + 
     5  2    5      3      q
    q  t    q  t   q  t
 
     5  4    7  4    9  5
>   q  t  + q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a57
K11a56
K11a56
K11a58
K11a58