© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a212
K11a212
K11a214
K11a214
K11a213
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   The Knot K11a213

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Acknowledgement

K11a213 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - 3t-3 + 16t-2 - 33t-1 + 41 - 33t + 16t2 - 3t3

Conway Polynomial: 1 + 4z2 - 2z4 - 3z6

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

Determinant and Signature: {145, 4}

Jones Polynomial: 1 - 3q + 8q2 - 14q3 + 20q4 - 23q5 + 24q6 - 21q7 + 16q8 - 10q9 + 4q10 - q11

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

A2 (sl(3)) Invariant: 1 - q2 + q4 + 3q6 - 4q8 + 4q10 - 2q12 - q14 + 4q16 - 3q18 + 5q20 - 3q22 + 2q26 - 4q28 + 2q30 - q34

HOMFLY-PT Polynomial: - a-10 - a-10z2 + a-8 + 5a-8z2 + 3a-8z4 - 3a-6z2 - 5a-6z4 - 2a-6z6 + a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial: - a-13z3 + a-13z5 + a-12z2 - 4a-12z4 + 4a-12z6 - 4a-11z + 10a-11z3 - 14a-11z5 + 9a-11z7 + a-10 - 4a-10z2 + 9a-10z4 - 16a-10z6 + 11a-10z8 - 5a-9z + 21a-9z3 - 25a-9z5 + 3a-9z7 + 7a-9z9 + a-8 - 7a-8z2 + 26a-8z4 - 39a-8z6 + 17a-8z8 + 2a-8z10 - 3a-7z + 12a-7z3 - 9a-7z5 - 13a-7z7 + 12a-7z9 - 5a-6z2 + 20a-6z4 - 29a-6z6 + 11a-6z8 + 2a-6z10 - 3a-5z + 7a-5z3 - 6a-5z5 - 4a-5z7 + 5a-5z9 + 4a-4z4 - 9a-4z6 + 5a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, 10}

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=4 is the signature of 11213. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 23           1
j = 21          3 
j = 19         71 
j = 17        93  
j = 15       127   
j = 13      129    
j = 11     1112     
j = 9    912      
j = 7   511       
j = 5  39        
j = 3 16         
j = 1 2          
j = -11           


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, 213]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 213]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], 
 
>   X[18, 10, 19, 9], X[2, 12, 3, 11], X[20, 13, 21, 14], X[10, 16, 11, 15], 
 
>   X[6, 18, 7, 17], X[14, 19, 15, 20], X[8, 22, 9, 21]]
In[4]:=
GaussCode[Knot[11, Alternating, 213]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -9, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -5, 10, 
 
>   -7, 11, -4]
In[5]:=
DTCode[Knot[11, Alternating, 213]]
Out[5]=   
DTCode[4, 12, 16, 22, 18, 2, 20, 10, 6, 14, 8]
In[6]:=
alex = Alexander[Knot[11, Alternating, 213]][t]
Out[6]=   
     3    16   33              2      3
41 - -- + -- - -- - 33 t + 16 t  - 3 t
      3    2   t
     t    t
In[7]:=
Conway[Knot[11, Alternating, 213]][z]
Out[7]=   
       2      4      6
1 + 4 z  - 2 z  - 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 213]}
In[9]:=
{KnotDet[Knot[11, Alternating, 213]], KnotSignature[Knot[11, Alternating, 213]]}
Out[9]=   
{145, 4}
In[10]:=
J=Jones[Knot[11, Alternating, 213]][q]
Out[10]=   
             2       3       4       5       6       7       8       9
1 - 3 q + 8 q  - 14 q  + 20 q  - 23 q  + 24 q  - 21 q  + 16 q  - 10 q  + 
 
       10    11
>   4 q   - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 213]}
In[12]:=
A2Invariant[Knot[11, Alternating, 213]][q]
Out[12]=   
     2    4      6      8      10      12    14      16      18      20
1 - q  + q  + 3 q  - 4 q  + 4 q   - 2 q   - q   + 4 q   - 3 q   + 5 q   - 
 
       22      26      28      30    34
>   3 q   + 2 q   - 4 q   + 2 q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 213]][a, z]
Out[13]=   
                     2       2      2    2      2      4      4    4    4
  -10    -8    -2   z     5 z    3 z    z    2 z    3 z    5 z    z    z
-a    + a   + a   - --- + ---- - ---- + -- + ---- + ---- - ---- - -- + -- - 
                     10     8      6     4     2      8      6     4    2
                    a      a      a     a     a      a      a     a    a
 
       6    6
    2 z    z
>   ---- - --
      6     4
     a     a
In[14]:=
Kauffman[Knot[11, Alternating, 213]][a, z]
Out[14]=   
                                                 2       2      2      2
 -10    -8    -2   4 z   5 z   3 z   3 z   z    z     4 z    7 z    5 z
a    + a   - a   - --- - --- - --- - --- - -- + --- - ---- - ---- - ---- + 
                    11    9     7     5     3    12    10      8      6
                   a     a     a     a     a    a     a       a      a
 
       2    3        3       3       3      3      3      4      4       4
    3 z    z     10 z    21 z    12 z    7 z    5 z    4 z    9 z    26 z
>   ---- - --- + ----- + ----- + ----- + ---- + ---- - ---- + ---- + ----- + 
      2     13     11      9       7       5      3     12     10      8
     a     a      a       a       a       a      a     a      a       a
 
        4      4      4    5        5       5      5      5      5      6
    20 z    4 z    3 z    z     14 z    25 z    9 z    6 z    7 z    4 z
>   ----- + ---- - ---- + --- - ----- - ----- - ---- - ---- - ---- + ---- - 
      6       4      2     13     11      9       7      5      3     12
     a       a      a     a      a       a       a      a      a     a
 
        6       6       6      6    6      7      7       7      7      7
    16 z    39 z    29 z    9 z    z    9 z    3 z    13 z    4 z    3 z
>   ----- - ----- - ----- - ---- + -- + ---- + ---- - ----- - ---- + ---- + 
      10      8       6       4     2    11      9      7       5      3
     a       a       a       a     a    a       a      a       a      a
 
        8       8       8      8      9       9      9      10      10
    11 z    17 z    11 z    5 z    7 z    12 z    5 z    2 z     2 z
>   ----- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + -----
      10      8       6       4      9      7       5      8       6
     a       a       a       a      a      a       a      a       a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 213]], Vassiliev[3][Knot[11, Alternating, 213]]}
Out[15]=   
{4, 10}
In[16]:=
Kh[Knot[11, Alternating, 213]][q, t]
Out[16]=   
                            3
   3      5    1     2 q   q       5        7         7  2      9  2
6 q  + 3 q  + ---- + --- + -- + 9 q  t + 5 q  t + 11 q  t  + 9 q  t  + 
                 2    t    t
              q t
 
        9  3       11  3       11  4       13  4      13  5       15  5
>   12 q  t  + 11 q   t  + 12 q   t  + 12 q   t  + 9 q   t  + 12 q   t  + 
 
       15  6      17  6      17  7      19  7    19  8      21  8    23  9
>   7 q   t  + 9 q   t  + 3 q   t  + 7 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a213
K11a212
K11a212
K11a214
K11a214