© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a200
K11a200
K11a202
K11a202
K11a201
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   The Knot K11a201

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Acknowledgement

K11a201 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: 4t-2 - 20t-1 + 33 - 20t + 4t2

Conway Polynomial: 1 - 4z2 + 4z4

Other knots with the same Alexander/Conway Polynomial: {K11a103, ...}

Determinant and Signature: {81, 0}

Jones Polynomial: q-6 - 2q-5 + 5q-4 - 8q-3 + 10q-2 - 13q-1 + 13 - 11q + 9q2 - 5q3 + 3q4 - q5

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

A2 (sl(3)) Invariant: q-20 + q-18 - q-16 + 2q-14 + q-12 - 3q-10 + q-8 - 2q-6 - 2q-4 + q-2 - 1 + 3q2 - q4 + q6 + 3q8 - 2q10 + q12 + q14 - q16

HOMFLY-PT Polynomial: - a-4z2 + a-2 + a-2z4 + 1 + z2 + 2z4 - 2a2 - 2a2z2 + a2z4 - 2a4z2 + a6

Kauffman Polynomial: - 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 + 3a-3z3 - 8a-3z5 + 4a-3z7 - a-2 - a-2z2 + 6a-2z4 - 8a-2z6 + 4a-2z8 + 4a-1z - 6a-1z3 + 8a-1z5 - 6a-1z7 + 3a-1z9 + 1 - 18z2 + 35z4 - 20z6 + 4z8 + z10 + 12az - 32az3 + 39az5 - 22az7 + 6az9 + 2a2 - 21a2z2 + 33a2z4 - 19a2z6 + 3a2z8 + a2z10 + 8a3z - 18a3z3 + 16a3z5 - 10a3z7 + 3a3z9 - 3a4z2 + 7a4z4 - 9a4z6 + 3a4z8 + 3a5z3 - 6a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {-4, 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=0 is the signature of 11201. 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 = 11           1
j = 9          2 
j = 7         31 
j = 5        62  
j = 3       53   
j = 1      86    
j = -1     66     
j = -3    47      
j = -5   46       
j = -7  14        
j = -9 14         
j = -11 1          
j = -131           


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, 201]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 201]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[14, 8, 15, 7], 
 
>   X[20, 9, 21, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[6, 16, 7, 15], 
 
>   X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 201]]
Out[4]=   
GaussCode[1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -6, 10, 
 
>   -5, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 201]]
Out[5]=   
DTCode[4, 12, 16, 14, 20, 18, 2, 6, 22, 10, 8]
In[6]:=
alex = Alexander[Knot[11, Alternating, 201]][t]
Out[6]=   
     4    20             2
33 + -- - -- - 20 t + 4 t
      2   t
     t
In[7]:=
Conway[Knot[11, Alternating, 201]][z]
Out[7]=   
       2      4
1 - 4 z  + 4 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 103], Knot[11, Alternating, 201]}
In[9]:=
{KnotDet[Knot[11, Alternating, 201]], KnotSignature[Knot[11, Alternating, 201]]}
Out[9]=   
{81, 0}
In[10]:=
J=Jones[Knot[11, Alternating, 201]][q]
Out[10]=   
      -6   2    5    8    10   13             2      3      4    5
13 + q   - -- + -- - -- + -- - -- - 11 q + 9 q  - 5 q  + 3 q  - q
            5    4    3    2   q
           q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 201]}
In[12]:=
A2Invariant[Knot[11, Alternating, 201]][q]
Out[12]=   
      -20    -18    -16    2     -12    3     -8   2    2     -2      2    4
-1 + q    + q    - q    + --- + q    - --- + q   - -- - -- + q   + 3 q  - q  + 
                           14           10          6    4
                          q            q           q    q
 
     6      8      10    12    14    16
>   q  + 3 q  - 2 q   + q   + q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 201]][a, z]
Out[13]=   
                            2                               4
     -2      2    6    2   z       2  2      4  2      4   z     2  4
1 + a   - 2 a  + a  + z  - -- - 2 a  z  - 2 a  z  + 2 z  + -- + a  z
                            4                               2
                           a                               a
In[14]:=
Kauffman[Knot[11, Alternating, 201]][a, z]
Out[14]=   
                                                         2    2
     -2      2    6   4 z               3         2   3 z    z        2  2
1 - a   + 2 a  - a  + --- + 12 a z + 8 a  z - 18 z  + ---- - -- - 21 a  z  - 
                       a                                4     2
                                                       a     a
 
                           3      3      3
       4  2      6  2   2 z    3 z    6 z          3       3  3      5  3
>   3 a  z  + 4 a  z  - ---- + ---- - ---- - 32 a z  - 18 a  z  + 3 a  z  + 
                          5      3     a
                         a      a
 
               4      4                                   5      5      5
        4   7 z    6 z        2  4      4  4      6  4   z    8 z    8 z
>   35 z  - ---- + ---- + 33 a  z  + 7 a  z  - 4 a  z  + -- - ---- + ---- + 
              4      2                                    5     3     a
             a      a                                    a     a
 
                                              6      6
          5       3  5      5  5       6   3 z    8 z        2  6      4  6
>   39 a z  + 16 a  z  - 6 a  z  - 20 z  + ---- - ---- - 19 a  z  - 9 a  z  + 
                                             4      2
                                            a      a
 
               7      7                                            8
     6  6   4 z    6 z          7       3  7      5  7      8   4 z
>   a  z  + ---- - ---- - 22 a z  - 10 a  z  + 2 a  z  + 4 z  + ---- + 
              3     a                                             2
             a                                                   a
 
                           9
       2  8      4  8   3 z         9      3  9    10    2  10
>   3 a  z  + 3 a  z  + ---- + 6 a z  + 3 a  z  + z   + a  z
                         a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 201]], Vassiliev[3][Knot[11, Alternating, 201]]}
Out[15]=   
{-4, 4}
In[16]:=
Kh[Knot[11, Alternating, 201]][q, t]
Out[16]=   
6           1        1        1       4       1       4       4       6
- + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          13  6    11  5    9  5    9  4    7  4    7  3    5  3    5  2
          q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
      4      7      6               3        3  2      5  2      5  3
>   ----- + ---- + --- + 6 q t + 5 q  t + 3 q  t  + 6 q  t  + 2 q  t  + 
     3  2    3     q t
    q  t    q  t
 
       7  3    7  4      9  4    11  5
>   3 q  t  + q  t  + 2 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a201
K11a200
K11a200
K11a202
K11a202