© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a35
K11a35
K11a37
K11a37
K11a36
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   The Knot K11a36

Visit K11a36's page at Knotilus!

Acknowledgement

K11a36 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - 2t-3 + 12t-2 - 28t-1 + 37 - 28t + 12t2 - 2t3

Conway Polynomial: 1 + 2z2 - 2z6

Other knots with the same Alexander/Conway Polynomial: {K11a169, ...}

Determinant and Signature: {121, 0}

Jones Polynomial: q-6 - 4q-5 + 8q-4 - 13q-3 + 17q-2 - 19q-1 + 20 - 16q + 12q2 - 7q3 + 3q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial: {K11a35, K11a316, ...}

A2 (sl(3)) Invariant: q-18 - 2q-16 + q-14 + q-12 - 4q-10 + 3q-8 - 2q-6 + 3q-2 - 1 + 5q2 - 2q4 + q6 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial: - a-4 - a-4z2 + a-2 + 3a-2z2 + 2a-2z4 + 2 + z2 - z4 - z6 - a2 - 2a2z2 - 2a2z4 - a2z6 + a4z2 + a4z4

Kauffman Polynomial: a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 - a-3z + 4a-3z3 - 7a-3z5 + 5a-3z7 - a-2 + 5a-2z2 - 5a-2z4 - 2a-2z6 + 5a-2z8 - 6a-1z + 20a-1z3 - 23a-1z5 + 8a-1z7 + 3a-1z9 + 2 + 7z4 - 19z6 + 11z8 + z10 - 8az + 25az3 - 25az5 + 7az9 + a2 - 3a2z2 + 15a2z4 - 28a2z6 + 12a2z8 + a2z10 - 5a3z + 18a3z3 - 20a3z5 + a3z7 + 4a3z9 + 6a4z4 - 13a4z6 + 6a4z8 - a5z + 7a5z3 - 10a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

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 1136. 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         51 
j = 5        72  
j = 3       95   
j = 1      117    
j = -1     910     
j = -3    810      
j = -5   59       
j = -7  38        
j = -9 15         
j = -11 3          
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, 36]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 36]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], 
 
>   X[16, 9, 17, 10], X[20, 11, 21, 12], X[6, 14, 7, 13], X[12, 15, 13, 16], 
 
>   X[22, 17, 1, 18], X[10, 19, 11, 20], X[18, 21, 19, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 36]]
Out[4]=   
GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -5, 9, -11, 10, 
 
>   -6, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 36]]
Out[5]=   
DTCode[4, 8, 14, 2, 16, 20, 6, 12, 22, 10, 18]
In[6]:=
alex = Alexander[Knot[11, Alternating, 36]][t]
Out[6]=   
     2    12   28              2      3
37 - -- + -- - -- - 28 t + 12 t  - 2 t
      3    2   t
     t    t
In[7]:=
Conway[Knot[11, Alternating, 36]][z]
Out[7]=   
       2      6
1 + 2 z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 36], Knot[11, Alternating, 169]}
In[9]:=
{KnotDet[Knot[11, Alternating, 36]], KnotSignature[Knot[11, Alternating, 36]]}
Out[9]=   
{121, 0}
In[10]:=
J=Jones[Knot[11, Alternating, 36]][q]
Out[10]=   
      -6   4    8    13   17   19              2      3      4    5
20 + q   - -- + -- - -- + -- - -- - 16 q + 12 q  - 7 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, 35], Knot[11, Alternating, 36], 
 
>   Knot[11, Alternating, 316]}
In[12]:=
A2Invariant[Knot[11, Alternating, 36]][q]
Out[12]=   
      -18    2     -14    -12    4    3    2    3       2      4    6      8
-1 + q    - --- + q    + q    - --- + -- - -- + -- + 5 q  - 2 q  + q  + 2 q  - 
             16                  10    8    6    2
            q                   q     q    q    q
 
       10    12    16
>   3 q   + q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 36]][a, z]
Out[13]=   
                           2      2                             4
     -4    -2    2    2   z    3 z       2  2    4  2    4   2 z       2  4
2 - a   + a   - a  + z  - -- + ---- - 2 a  z  + a  z  - z  + ---- - 2 a  z  + 
                           4     2                             2
                          a     a                             a
 
     4  4    6    2  6
>   a  z  - z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 36]][a, z]
Out[14]=   
                                                                2      2
     -4    -2    2   z    z    6 z              3      5     3 z    5 z
2 - a   - a   + a  + -- - -- - --- - 8 a z - 5 a  z - a  z + ---- + ---- - 
                      5    3    a                              4      2
                     a    a                                   a      a
 
                         3      3       3
       2  2    6  2   2 z    4 z    20 z          3       3  3      5  3
>   3 a  z  + a  z  - ---- + ---- + ----- + 25 a z  + 18 a  z  + 7 a  z  + 
                        5      3      a
                       a      a
 
              4      4                                   5      5       5
       4   5 z    5 z        2  4      4  4      6  4   z    7 z    23 z
>   7 z  - ---- - ---- + 15 a  z  + 6 a  z  - 2 a  z  + -- - ---- - ----- - 
             4      2                                    5     3      a
            a      a                                    a     a
 
                                               6      6
          5       3  5       5  5       6   3 z    2 z        2  6       4  6
>   25 a z  - 20 a  z  - 10 a  z  - 19 z  + ---- - ---- - 28 a  z  - 13 a  z  + 
                                              4      2
                                             a      a
 
               7      7                                8
     6  6   5 z    8 z     3  7      5  7       8   5 z        2  8      4  8
>   a  z  + ---- + ---- + a  z  + 4 a  z  + 11 z  + ---- + 12 a  z  + 6 a  z  + 
              3     a                                 2
             a                                       a
 
       9
    3 z         9      3  9    10    2  10
>   ---- + 7 a z  + 4 a  z  + z   + a  z
     a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 36]], Vassiliev[3][Knot[11, Alternating, 36]]}
Out[15]=   
{2, 1}
In[16]:=
Kh[Knot[11, Alternating, 36]][q, t]
Out[16]=   
10            1        3        1       5       3       8       5       9
-- + 11 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
 
      8      10     9               3        3  2      5  2      5  3
>   ----- + ---- + --- + 7 q t + 9 q  t + 5 q  t  + 7 q  t  + 2 q  t  + 
     3  2    3     q t
    q  t    q  t
 
       7  3    7  4      9  4    11  5
>   5 q  t  + q  t  + 2 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a36
K11a35
K11a35
K11a37
K11a37