© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a155
K11a155
K11a157
K11a157
K11a156
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   The Knot K11a156

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Acknowledgement

K11a156 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X18,5,19,6 X12,8,13,7 X14,10,15,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,17,1,18 X6,19,7,20 X16,21,17,22

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

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

Alexander Polynomial: - t-4 + 5t-3 - 11t-2 + 18t-1 - 21 + 18t - 11t2 + 5t3 - t4

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

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

Determinant and Signature: {91, -2}

Jones Polynomial: - q-8 + 2q-7 - 5q-6 + 9q-5 - 12q-4 + 15q-3 - 14q-2 + 13q-1 - 10 + 6q - 3q2 + q3

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

A2 (sl(3)) Invariant: - q-24 - q-22 - q-20 - 2q-18 + 3q-16 + 3q-12 + 3q-10 - q-8 + 3q-6 - 4q-4 + q-2 - 1 - q2 + 2q4 - q6 + q8

HOMFLY-PT Polynomial: 2 + 5z2 + 4z4 + z6 - 6a2 - 15a2z2 - 14a2z4 - 6a2z6 - a2z8 + 9a4 + 17a4z2 + 10a4z4 + 2a4z6 - 4a6 - 4a6z2 - a6z4

Kauffman Polynomial: 2a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 7a-1z3 - 9a-1z5 + 3a-1z7 + 2 - 4z2 + 6z4 - 10z6 + 4z8 - 5az + 10az3 - 8az5 - 3az7 + 3az9 + 6a2 - 22a2z2 + 32a2z4 - 24a2z6 + 6a2z8 + a2z10 - 6a3z + 8a3z3 + 7a3z5 - 13a3z7 + 6a3z9 + 9a4 - 30a4z2 + 43a4z4 - 25a4z6 + 6a4z8 + a4z10 - 6a5z + 10a5z3 - 4a5z7 + 3a5z9 + 4a6 - 13a6z2 + 16a6z4 - 10a6z6 + 4a6z8 - a7z + 2a7z3 - 5a7z5 + 3a7z7 + a8z2 - 4a8z4 + 2a8z6 + 2a9z - 3a9z3 + a9z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {3, -7}

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 11156. 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          2 
j = 3         41 
j = 1        62  
j = -1       74   
j = -3      87    
j = -5     76     
j = -7    58      
j = -9   47       
j = -11  15        
j = -13 14         
j = -15 1          
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, 156]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 156]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[12, 8, 13, 7], 
 
>   X[14, 10, 15, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], 
 
>   X[22, 17, 1, 18], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 156]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -4, 7, -5, 8, -11, 9, -3, 10, 
 
>   -8, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 156]]
Out[5]=   
DTCode[4, 10, 18, 12, 14, 2, 8, 20, 22, 6, 16]
In[6]:=
alex = Alexander[Knot[11, Alternating, 156]][t]
Out[6]=   
       -4   5    11   18              2      3    4
-21 - t   + -- - -- + -- + 18 t - 11 t  + 5 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 156]][z]
Out[7]=   
       2    4      6    8
1 + 3 z  - z  - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 156]}
In[9]:=
{KnotDet[Knot[11, Alternating, 156]], KnotSignature[Knot[11, Alternating, 156]]}
Out[9]=   
{91, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 156]][q]
Out[10]=   
       -8   2    5    9    12   15   14   13            2    3
-10 - q   + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 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, 156]}
In[12]:=
A2Invariant[Knot[11, Alternating, 156]][q]
Out[12]=   
      -24    -22    -20    2     3     3     3     -8   3    4     -2    2
-1 - q    - q    - q    - --- + --- + --- + --- - q   + -- - -- + q   - q  + 
                           18    16    12    10          6    4
                          q     q     q     q           q    q
 
       4    6    8
>   2 q  - q  + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 156]][a, z]
Out[13]=   
       2      4      6      2       2  2       4  2      6  2      4
2 - 6 a  + 9 a  - 4 a  + 5 z  - 15 a  z  + 17 a  z  - 4 a  z  + 4 z  - 
 
        2  4       4  4    6  4    6      2  6      4  6    2  8
>   14 a  z  + 10 a  z  - a  z  + z  - 6 a  z  + 2 a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 156]][a, z]
Out[14]=   
       2      4      6   2 z              3        5      7        9        2
2 + 6 a  + 9 a  + 4 a  - --- - 5 a z - 6 a  z - 6 a  z - a  z + 2 a  z - 4 z  + 
                          a
 
       2                                               3
    2 z        2  2       4  2       6  2    8  2   7 z          3      3  3
>   ---- - 22 a  z  - 30 a  z  - 13 a  z  + a  z  + ---- + 10 a z  + 8 a  z  + 
      2                                              a
     a
 
                                             4
        5  3      7  3      9  3      4   3 z        2  4       4  4
>   10 a  z  + 2 a  z  - 3 a  z  + 6 z  - ---- + 32 a  z  + 43 a  z  + 
                                            2
                                           a
 
                            5
        6  4      8  4   9 z         5      3  5      7  5    9  5       6
>   16 a  z  - 4 a  z  - ---- - 8 a z  + 7 a  z  - 5 a  z  + a  z  - 10 z  + 
                          a
 
     6                                                 7
    z        2  6       4  6       6  6      8  6   3 z         7       3  7
>   -- - 24 a  z  - 25 a  z  - 10 a  z  + 2 a  z  + ---- - 3 a z  - 13 a  z  - 
     2                                               a
    a
 
       5  7      7  7      8      2  8      4  8      6  8        9      3  9
>   4 a  z  + 3 a  z  + 4 z  + 6 a  z  + 6 a  z  + 4 a  z  + 3 a z  + 6 a  z  + 
 
       5  9    2  10    4  10
>   3 a  z  + a  z   + a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 156]], Vassiliev[3][Knot[11, Alternating, 156]]}
Out[15]=   
{3, -7}
In[16]:=
Kh[Knot[11, Alternating, 156]][q, t]
Out[16]=   
7    7     1        1        1        4        1        5        4       7
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 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
 
      5       8       7      6      8     4 t                2      3  2
>   ----- + ----- + ----- + ---- + ---- + --- + 6 q t + 2 q t  + 4 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  + 2 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a156
K11a155
K11a155
K11a157
K11a157