© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n35
K11n35
K11n37
K11n37
K11n36
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   The Knot K11n36

Visit K11n36's page at Knotilus!

Acknowledgement

K11n36 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X5,13,6,12 X2837 X9,17,10,16 X11,18,12,19 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,14,20,15 X21,10,22,11

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

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

Alexander Polynomial: - t-4 + 4t-3 - 8t-2 + 13t-1 - 15 + 13t - 8t2 + 4t3 - t4

Conway Polynomial: 1 + z2 - 4z4 - 4z6 - z8

Other knots with the same Alexander/Conway Polynomial: {K11n44, ...}

Determinant and Signature: {67, 2}

Jones Polynomial: q-3 - 3q-2 + 6q-1 - 9 + 11q - 11q2 + 11q3 - 8q4 + 5q5 - 2q6

Other knots (up to mirrors) with the same Jones Polynomial: {K11n7, K11n44, ...}

A2 (sl(3)) Invariant: q-8 - q-6 + 2q-4 - q-2 + q2 - 3q4 + 3q6 - 2q8 + 3q10 + q12 + 2q16 - 2q18 - q22

HOMFLY-PT Polynomial: - 2a-6 - a-6z2 + 5a-4 + 9a-4z2 + 5a-4z4 + a-4z6 - 4a-2 - 12a-2z2 - 13a-2z4 - 6a-2z6 - a-2z8 + 2 + 5z2 + 4z4 + z6

Kauffman Polynomial: - 3a-7z + 3a-7z3 + 2a-6 - 5a-6z2 + 4a-6z4 + a-6z6 - 6a-5z + 13a-5z3 - 8a-5z5 + 4a-5z7 + 5a-4 - 16a-4z2 + 23a-4z4 - 14a-4z6 + 5a-4z8 - 6a-3z + 17a-3z3 - 14a-3z5 + 2a-3z7 + 2a-3z9 + 4a-2 - 18a-2z2 + 30a-2z4 - 27a-2z6 + 9a-2z8 - 5a-1z + 14a-1z3 - 15a-1z5 + a-1z7 + 2a-1z9 + 2 - 5z2 + 8z4 - 11z6 + 4z8 - 2az + 7az3 - 9az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

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 1136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 13         2
j = 11        3 
j = 9       52 
j = 7      63  
j = 5     55   
j = 3    66    
j = 1   46     
j = -1  25      
j = -3 14       
j = -5 2        
j = -71         


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, NonAlternating, 36]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 36]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], 
 
>   X[9, 17, 10, 16], X[11, 18, 12, 19], X[13, 7, 14, 6], X[15, 20, 16, 21], 
 
>   X[17, 1, 18, 22], X[19, 14, 20, 15], X[21, 10, 22, 11]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 36]]
Out[4]=   
GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 5, -9, 6, -10, 
 
>   8, -11, 9]
In[5]:=
DTCode[Knot[11, NonAlternating, 36]]
Out[5]=   
DTCode[4, 8, -12, 2, -16, -18, -6, -20, -22, -14, -10]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 36]][t]
Out[6]=   
       -4   4    8    13             2      3    4
-15 - t   + -- - -- + -- + 13 t - 8 t  + 4 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, NonAlternating, 36]][z]
Out[7]=   
     2      4      6    8
1 + z  - 4 z  - 4 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 36], Knot[11, NonAlternating, 44]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 36]], KnotSignature[Knot[11, NonAlternating, 36]]}
Out[9]=   
{67, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 36]][q]
Out[10]=   
      -3   3    6              2       3      4      5      6
-9 + q   - -- + - + 11 q - 11 q  + 11 q  - 8 q  + 5 q  - 2 q
            2   q
           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 36], 
 
>   Knot[11, NonAlternating, 44]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 36]][q]
Out[12]=   
 -8    -6   2     -2    2      4      6      8      10    12      16      18
q   - q   + -- - q   + q  - 3 q  + 3 q  - 2 q  + 3 q   + q   + 2 q   - 2 q   - 
             4
            q
 
     22
>   q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 36]][a, z]
Out[13]=   
                           2      2       2             4       4         6
    2    5    4       2   z    9 z    12 z       4   5 z    13 z     6   z
2 - -- + -- - -- + 5 z  - -- + ---- - ----- + 4 z  + ---- - ----- + z  + -- - 
     6    4    2           6     4      2              4      2           4
    a    a    a           a     a      a              a      a           a
 
       6    8
    6 z    z
>   ---- - --
      2     2
     a     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 36]][a, z]
Out[14]=   
                                                             2       2
    2    5    4    3 z   6 z   6 z   5 z              2   5 z    16 z
2 + -- + -- + -- - --- - --- - --- - --- - 2 a z - 5 z  - ---- - ----- - 
     6    4    2    7     5     3     a                     6      4
    a    a    a    a     a     a                           a      a
 
        2                3       3       3       3                      4
    18 z       2  2   3 z    13 z    17 z    14 z         3      4   4 z
>   ----- + 2 a  z  + ---- + ----- + ----- + ----- + 7 a z  + 8 z  + ---- + 
      2                 7      5       3       a                       6
     a                 a      a       a                               a
 
        4       4                5       5       5                     6
    23 z    30 z       2  4   8 z    14 z    15 z         5       6   z
>   ----- + ----- - 3 a  z  - ---- - ----- - ----- - 9 a z  - 11 z  + -- - 
      4       2                 5      3       a                       6
     a       a                 a      a                               a
 
        6       6              7      7    7                      8      8
    14 z    27 z     2  6   4 z    2 z    z         7      8   5 z    9 z
>   ----- - ----- + a  z  + ---- + ---- + -- + 3 a z  + 4 z  + ---- + ---- + 
      4       2               5      3    a                      4      2
     a       a               a      a                           a      a
 
       9      9
    2 z    2 z
>   ---- + ----
      3     a
     a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 36]], Vassiliev[3][Knot[11, NonAlternating, 36]]}
Out[15]=   
{1, 3}
In[16]:=
Kh[Knot[11, NonAlternating, 36]][q, t]
Out[16]=   
         3     1       2       1       4      2      5    4 q      3
6 q + 6 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q  t + 
              7  4    5  3    3  3    3  2      2   q t    t
             q  t    q  t    q  t    q  t    q t
 
       5        5  2      7  2      7  3      9  3      9  4      11  4
>   5 q  t + 5 q  t  + 6 q  t  + 3 q  t  + 5 q  t  + 2 q  t  + 3 q   t  + 
 
       13  5
>   2 q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n36
K11n35
K11n35
K11n37
K11n37