© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a233
K11a233
K11a235
K11a235
K11a234
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   The Knot K11a234

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Acknowledgement

K11a234 as Morse Link
DrawMorseLink

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

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

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

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

Conway Polynomial: 1 + 11z2 + 21z4 + 12z6 + 2z8

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

Determinant and Signature: {37, 8}

Jones Polynomial: q4 - q5 + 3q6 - 3q7 + 4q8 - 5q9 + 5q10 - 5q11 + 4q12 - 3q13 + 2q14 - q15

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

A2 (sl(3)) Invariant: q14 + 2q18 + q20 + 2q22 + q24 - 2q30 - q34 - q44

HOMFLY-PT Polynomial: - a-12 - 6a-12z2 - 5a-12z4 - a-12z6 - 2a-10 + 3a-10z2 + 10a-10z4 + 6a-10z6 + a-10z8 + 4a-8 + 14a-8z2 + 16a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial: - a-19z + a-19z3 - 2a-18z2 + 2a-18z4 - a-17z3 + 2a-17z5 + a-16z2 - 2a-16z4 + 2a-16z6 - a-15z + 3a-15z3 - 4a-15z5 + 2a-15z7 - 2a-14z2 + 5a-14z4 - 6a-14z6 + 2a-14z8 + 2a-13z - 11a-13z3 + 14a-13z5 - 9a-13z7 + 2a-13z9 - a-12 + 6a-12z2 - 7a-12z4 + 5a-12z6 - 4a-12z8 + a-12z10 + a-11z - 13a-11z3 + 25a-11z5 - 16a-11z7 + 3a-11z9 + 2a-10 - 3a-10z2 + 6a-10z6 - 5a-10z8 + a-10z10 - 3a-9z + 3a-9z3 + 5a-9z5 - 5a-9z7 + a-9z9 + 4a-8 - 14a-8z2 + 16a-8z4 - 7a-8z6 + a-8z8

V2 and V3, the type 2 and 3 Vassiliev invariants: {11, 35}

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=8 is the signature of 11234. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9r = 10r = 11
j = 31           1
j = 29          1 
j = 27         21 
j = 25        21  
j = 23       32   
j = 21      22    
j = 19     33     
j = 17    12      
j = 15   23       
j = 13  11        
j = 11  2         
j = 911          
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, Alternating, 234]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 234]]
Out[3]=   
PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], 
 
>   X[20, 10, 21, 9], X[22, 12, 1, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], 
 
>   X[6, 18, 7, 17], X[8, 20, 9, 19], X[10, 22, 11, 21]]
In[4]:=
GaussCode[Knot[11, Alternating, 234]]
Out[4]=   
GaussCode[1, -7, 2, -1, 3, -9, 4, -10, 5, -11, 6, -8, 7, -2, 8, -3, 9, -4, 10, 
 
>   -5, 11, -6]
In[5]:=
DTCode[Knot[11, Alternating, 234]]
Out[5]=   
DTCode[4, 14, 16, 18, 20, 22, 2, 12, 6, 8, 10]
In[6]:=
alex = Alexander[Knot[11, Alternating, 234]][t]
Out[6]=   
    2    4    5    5            2      3      4
5 + -- - -- + -- - - - 5 t + 5 t  - 4 t  + 2 t
     4    3    2   t
    t    t    t
In[7]:=
Conway[Knot[11, Alternating, 234]][z]
Out[7]=   
        2       4       6      8
1 + 11 z  + 21 z  + 12 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 234]}
In[9]:=
{KnotDet[Knot[11, Alternating, 234]], KnotSignature[Knot[11, Alternating, 234]]}
Out[9]=   
{37, 8}
In[10]:=
J=Jones[Knot[11, Alternating, 234]][q]
Out[10]=   
 4    5      6      7      8      9      10      11      12      13      14
q  - q  + 3 q  - 3 q  + 4 q  - 5 q  + 5 q   - 5 q   + 4 q   - 3 q   + 2 q   - 
 
     15
>   q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 234]}
In[12]:=
A2Invariant[Knot[11, Alternating, 234]][q]
Out[12]=   
 14      18    20      22    24      30    34    44
q   + 2 q   + q   + 2 q   + q   - 2 q   - q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 234]][a, z]
Out[13]=   
                      2      2       2      4       4       4    6       6
  -12    2    4    6 z    3 z    14 z    5 z    10 z    16 z    z     6 z
-a    - --- + -- - ---- + ---- + ----- - ---- + ----- + ----- - --- + ---- + 
         10    8    12     10      8      12      10      8      12    10
        a     a    a      a       a      a       a       a      a     a
 
       6    8     8
    7 z    z     z
>   ---- + --- + --
      8     10    8
     a     a     a
In[14]:=
Kauffman[Knot[11, Alternating, 234]][a, z]
Out[14]=   
                                                    2    2       2      2
  -12    2    4     z     z    2 z    z    3 z   2 z    z     2 z    6 z
-a    + --- + -- - --- - --- + --- + --- - --- - ---- + --- - ---- + ---- - 
         10    8    19    15    13    11    9     18     16    14     12
        a     a    a     a     a     a     a     a      a     a      a
 
       2       2    3     3       3       3       3      3      4      4
    3 z    14 z    z     z     3 z    11 z    13 z    3 z    2 z    2 z
>   ---- - ----- + --- - --- + ---- - ----- - ----- + ---- + ---- - ---- + 
     10      8      19    17    15      13      11      9     18     16
    a       a      a     a     a       a       a       a     a      a
 
       4      4       4      5      5       5       5      5      6      6
    5 z    7 z    16 z    2 z    4 z    14 z    25 z    5 z    2 z    6 z
>   ---- - ---- + ----- + ---- - ---- + ----- + ----- + ---- + ---- - ---- + 
     14     12      8      17     15      13      11      9     16     14
    a      a       a      a      a       a       a       a     a      a
 
       6      6      6      7      7       7      7      8      8      8    8
    5 z    6 z    7 z    2 z    9 z    16 z    5 z    2 z    4 z    5 z    z
>   ---- + ---- - ---- + ---- - ---- - ----- - ---- + ---- - ---- - ---- + -- + 
     12     10      8     15     13      11      9     14     12     10     8
    a      a       a     a      a       a       a     a      a      a      a
 
       9      9    9    10    10
    2 z    3 z    z    z     z
>   ---- + ---- + -- + --- + ---
     13     11     9    12    10
    a      a      a    a     a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 234]], Vassiliev[3][Knot[11, Alternating, 234]]}
Out[15]=   
{11, 35}
In[16]:=
Kh[Knot[11, Alternating, 234]][q, t]
Out[16]=   
 7    9    9        11  2    13  2    13  3      15  3      15  4    17  4
q  + q  + q  t + 2 q   t  + q   t  + q   t  + 2 q   t  + 3 q   t  + q   t  + 
 
       17  5      19  5      19  6      21  6      21  7      23  7
>   2 q   t  + 3 q   t  + 3 q   t  + 2 q   t  + 2 q   t  + 3 q   t  + 
 
       23  8      25  8    25  9      27  9    27  10    29  10    31  11
>   2 q   t  + 2 q   t  + q   t  + 2 q   t  + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a234
K11a233
K11a233
K11a235
K11a235