© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a196
K11a196
K11a198
K11a198
K11a197
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   The Knot K11a197

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Acknowledgement

K11a197 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: 3t-3 - 14t-2 + 33t-1 - 43 + 33t - 14t2 + 3t3

Conway Polynomial: 1 + 4z2 + 4z4 + 3z6

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

Determinant and Signature: {143, 2}

Jones Polynomial: - q-2 + 3q-1 - 7 + 14q - 19q2 + 23q3 - 23q4 + 21q5 - 16q6 + 10q7 - 5q8 + q9

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

A2 (sl(3)) Invariant: - q-6 + q-4 - q-2 - 2 + 5q2 - 3q4 + 3q6 + 3q8 - 2q10 + 4q12 - 4q14 + 3q16 - 3q20 + 3q22 - 3q24 - q26 + q28

HOMFLY-PT Polynomial: a-8z2 - 2a-6 - 5a-6z2 - 3a-6z4 + 2a-4 + 7a-4z2 + 6a-4z4 + 2a-4z6 + 2a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial: - a-10z4 + a-10z6 + a-9z + 4a-9z3 - 10a-9z5 + 5a-9z7 - 2a-8z2 + 11a-8z4 - 20a-8z6 + 9a-8z8 - 3a-7z + 14a-7z3 - 17a-7z5 - 5a-7z7 + 7a-7z9 + 2a-6 - 15a-6z2 + 42a-6z4 - 51a-6z6 + 17a-6z8 + 2a-6z10 - 6a-5z + 16a-5z3 - 6a-5z5 - 18a-5z7 + 13a-5z9 + 2a-4 - 15a-4z2 + 41a-4z4 - 42a-4z6 + 15a-4z8 + 2a-4z10 - 2a-3z + 8a-3z3 - 5a-3z5 - 3a-3z7 + 6a-3z9 - 2a-2 + a-2z2 + 6a-2z4 - 9a-2z6 + 7a-2z8 + a-1z - 5a-1z5 + 5a-1z7 - 1 + 3z2 - 5z4 + 3z6 + az - 2az3 + az5

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, 6}

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 11197. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 19           1
j = 17          4 
j = 15         61 
j = 13        104  
j = 11       116   
j = 9      1210    
j = 7     1111     
j = 5    812      
j = 3   611       
j = 1  29        
j = -1 15         
j = -3 2          
j = -51           


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, 197]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 197]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], 
 
>   X[18, 10, 19, 9], X[2, 12, 3, 11], X[8, 14, 9, 13], X[22, 15, 1, 16], 
 
>   X[20, 18, 21, 17], X[10, 20, 11, 19], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 197]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -11, 4, -7, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, 
 
>   -9, 11, -8]
In[5]:=
DTCode[Knot[11, Alternating, 197]]
Out[5]=   
DTCode[4, 12, 16, 14, 18, 2, 8, 22, 20, 10, 6]
In[6]:=
alex = Alexander[Knot[11, Alternating, 197]][t]
Out[6]=   
      3    14   33              2      3
-43 + -- - -- + -- + 33 t - 14 t  + 3 t
       3    2   t
      t    t
In[7]:=
Conway[Knot[11, Alternating, 197]][z]
Out[7]=   
       2      4      6
1 + 4 z  + 4 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 197]}
In[9]:=
{KnotDet[Knot[11, Alternating, 197]], KnotSignature[Knot[11, Alternating, 197]]}
Out[9]=   
{143, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 197]][q]
Out[10]=   
      -2   3              2       3       4       5       6       7      8    9
-7 - q   + - + 14 q - 19 q  + 23 q  - 23 q  + 21 q  - 16 q  + 10 q  - 5 q  + q
           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 197]}
In[12]:=
A2Invariant[Knot[11, Alternating, 197]][q]
Out[12]=   
      -6    -4    -2      2      4      6      8      10      12      14
-2 - q   + q   - q   + 5 q  - 3 q  + 3 q  + 3 q  - 2 q   + 4 q   - 4 q   + 
 
       16      20      22      24    26    28
>   3 q   - 3 q   + 3 q   - 3 q   - q   + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 197]][a, z]
Out[13]=   
                            2      2      2      2           4      4      4
     2    2    2       2   z    5 z    7 z    3 z     4   3 z    6 z    2 z
-1 - -- + -- + -- - 2 z  + -- - ---- + ---- + ---- - z  - ---- + ---- + ---- + 
      6    4    2           8     6      4      2           6      4      2
     a    a    a           a     a      a      a           a      a      a
 
       6    6
    2 z    z
>   ---- + --
      4     2
     a     a
In[14]:=
Kauffman[Knot[11, Alternating, 197]][a, z]
Out[14]=   
                                                               2       2
     2    2    2    z    3 z   6 z   2 z   z            2   2 z    15 z
-1 + -- + -- - -- + -- - --- - --- - --- + - + a z + 3 z  - ---- - ----- - 
      6    4    2    9    7     5     3    a                  8      6
     a    a    a    a    a     a     a                       a      a
 
        2    2      3       3       3      3                    4        4
    15 z    z    4 z    14 z    16 z    8 z         3      4   z     11 z
>   ----- + -- + ---- + ----- + ----- + ---- - 2 a z  - 5 z  - --- + ----- + 
      4      2     9      7       5       3                     10     8
     a      a     a      a       a       a                     a      a
 
        4       4      4       5       5      5      5      5
    42 z    41 z    6 z    10 z    17 z    6 z    5 z    5 z       5      6
>   ----- + ----- + ---- - ----- - ----- - ---- - ---- - ---- + a z  + 3 z  + 
      6       4       2      9       7       5      3     a
     a       a       a      a       a       a      a
 
     6        6       6       6      6      7      7       7      7      7
    z     20 z    51 z    42 z    9 z    5 z    5 z    18 z    3 z    5 z
>   --- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + ---- + 
     10     8       6       4       2      9      7      5       3     a
    a      a       a       a       a      a      a      a       a
 
       8       8       8      8      9       9      9      10      10
    9 z    17 z    15 z    7 z    7 z    13 z    6 z    2 z     2 z
>   ---- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + -----
      8      6       4       2      7      5       3      6       4
     a      a       a       a      a      a       a      a       a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 197]], Vassiliev[3][Knot[11, Alternating, 197]]}
Out[15]=   
{4, 6}
In[16]:=
Kh[Knot[11, Alternating, 197]][q, t]
Out[16]=   
         3     1       2      1      5    2 q       3        5         5  2
9 q + 6 q  + ----- + ----- + ---- + --- + --- + 11 q  t + 8 q  t + 12 q  t  + 
              5  3    3  2      2   q t    t
             q  t    q  t    q t
 
        7  2       7  3       9  3       9  4       11  4      11  5
>   11 q  t  + 11 q  t  + 12 q  t  + 10 q  t  + 11 q   t  + 6 q   t  + 
 
        13  5      13  6      15  6    15  7      17  7    19  8
>   10 q   t  + 4 q   t  + 6 q   t  + q   t  + 4 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a197
K11a196
K11a196
K11a198
K11a198