© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a228
K11a228
K11a230
K11a230
K11a229
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   The Knot K11a229

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Acknowledgement

K11a229 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - 4t-2 + 18t-1 - 27 + 18t - 4t2

Conway Polynomial: 1 + 2z2 - 4z4

Other knots with the same Alexander/Conway Polynomial: {K11a226, ...}

Determinant and Signature: {71, 2}

Jones Polynomial: q-1 - 2 + 5q - 8q2 + 10q3 - 11q4 + 11q5 - 9q6 + 7q7 - 4q8 + 2q9 - q10

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

A2 (sl(3)) Invariant: q-4 + 3q2 - 2q4 + q8 - 2q10 + q12 - q14 + q16 + q18 - q20 + 3q22 - q26 + q28 - q30 - q32

HOMFLY-PT Polynomial: - a-10 + a-8 + 2a-8z2 + a-6 + a-6z2 - a-6z4 - a-4 - 2a-4z2 - 2a-4z4 - a-2z4 + 1 + z2

Kauffman Polynomial: - 2a-11z + 7a-11z3 - 5a-11z5 + a-11z7 + a-10 - 5a-10z2 + 12a-10z4 - 9a-10z6 + 2a-10z8 - a-9z3 + 6a-9z5 - 7a-9z7 + 2a-9z9 + a-8 - 2a-8z2 + 2a-8z4 - 3a-8z6 - a-8z8 + a-8z10 + 2a-7z - 12a-7z3 + 14a-7z5 - 12a-7z7 + 4a-7z9 - a-6 + 8a-6z2 - 12a-6z4 + 2a-6z6 + a-6z10 - 2a-5z + 5a-5z3 - 5a-5z5 + 2a-5z9 - a-4 + 7a-4z2 - 5a-4z4 - a-4z6 + 3a-4z8 - 2a-3z + 7a-3z3 - 6a-3z5 + 4a-3z7 - 2a-2z4 + 3a-2z6 - 2a-1z3 + 2a-1z5 + 1 - 2z2 + z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 7}

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 11229. 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 = 21           1
j = 19          1 
j = 17         31 
j = 15        41  
j = 13       53   
j = 11      64    
j = 9     55     
j = 7    56      
j = 5   35       
j = 3  25        
j = 1 14         
j = -1 1          
j = -31           


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, 229]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 229]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 6, 21, 5], X[18, 8, 19, 7], 
 
>   X[14, 10, 15, 9], X[2, 12, 3, 11], X[10, 14, 11, 13], X[22, 15, 1, 16], 
 
>   X[8, 18, 9, 17], X[6, 20, 7, 19], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 229]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -7, 6, -2, 7, -5, 8, -11, 9, -4, 10, 
 
>   -3, 11, -8]
In[5]:=
DTCode[Knot[11, Alternating, 229]]
Out[5]=   
DTCode[4, 12, 20, 18, 14, 2, 10, 22, 8, 6, 16]
In[6]:=
alex = Alexander[Knot[11, Alternating, 229]][t]
Out[6]=   
      4    18             2
-27 - -- + -- + 18 t - 4 t
       2   t
      t
In[7]:=
Conway[Knot[11, Alternating, 229]][z]
Out[7]=   
       2      4
1 + 2 z  - 4 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 226], Knot[11, Alternating, 229]}
In[9]:=
{KnotDet[Knot[11, Alternating, 229]], KnotSignature[Knot[11, Alternating, 229]]}
Out[9]=   
{71, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 229]][q]
Out[10]=   
     1            2       3       4       5      6      7      8      9    10
-2 + - + 5 q - 8 q  + 10 q  - 11 q  + 11 q  - 9 q  + 7 q  - 4 q  + 2 q  - q
     q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 229]}
In[12]:=
A2Invariant[Knot[11, Alternating, 229]][q]
Out[12]=   
 -4      2      4    8      10    12    14    16    18    20      22    26
q   + 3 q  - 2 q  + q  - 2 q   + q   - q   + q   + q   - q   + 3 q   - q   + 
 
     28    30    32
>   q   - q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 229]][a, z]
Out[13]=   
                                     2    2      2    4      4    4
     -10    -8    -6    -4    2   2 z    z    2 z    z    2 z    z
1 - a    + a   + a   - a   + z  + ---- + -- - ---- - -- - ---- - --
                                    8     6     4     6     4     2
                                   a     a     a     a     a     a
In[14]:=
Kauffman[Knot[11, Alternating, 229]][a, z]
Out[14]=   
                                                               2      2
     -10    -8    -6    -4   2 z   2 z   2 z   2 z      2   5 z    2 z
1 + a    + a   - a   - a   - --- + --- - --- - --- - 2 z  - ---- - ---- + 
                              11    7     5     3            10      8
                             a     a     a     a            a       a
 
       2      2      3    3       3      3      3      3            4      4
    8 z    7 z    7 z    z    12 z    5 z    7 z    2 z     4   12 z    2 z
>   ---- + ---- + ---- - -- - ----- + ---- + ---- - ---- + z  + ----- + ---- - 
      6      4     11     9     7       5      3     a            10      8
     a      a     a      a     a       a      a                  a       a
 
        4      4      4      5      5       5      5      5      5      6
    12 z    5 z    2 z    5 z    6 z    14 z    5 z    6 z    2 z    9 z
>   ----- - ---- - ---- - ---- + ---- + ----- - ---- - ---- + ---- - ---- - 
      6       4      2     11      9      7       5      3     a      10
     a       a      a     a       a      a       a      a            a
 
       6      6    6      6    7       7       7      7      8    8      8
    3 z    2 z    z    3 z    z     7 z    12 z    4 z    2 z    z    3 z
>   ---- + ---- - -- + ---- + --- - ---- - ----- + ---- + ---- - -- + ---- + 
      8      6     4     2     11     9      7       3     10     8     4
     a      a     a     a     a      a      a       a     a      a     a
 
       9      9      9    10    10
    2 z    4 z    2 z    z     z
>   ---- + ---- + ---- + --- + ---
      9      7      5     8     6
     a      a      a     a     a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 229]], Vassiliev[3][Knot[11, Alternating, 229]]}
Out[15]=   
{2, 7}
In[16]:=
Kh[Knot[11, Alternating, 229]][q, t]
Out[16]=   
         3     1      1    q      3        5        5  2      7  2      7  3
4 q + 2 q  + ----- + --- + - + 5 q  t + 3 q  t + 5 q  t  + 5 q  t  + 6 q  t  + 
              3  2   q t   t
             q  t
 
       9  3      9  4      11  4      11  5      13  5      13  6      15  6
>   5 q  t  + 5 q  t  + 6 q   t  + 4 q   t  + 5 q   t  + 3 q   t  + 4 q   t  + 
 
     15  7      17  7    17  8    19  8    21  9
>   q   t  + 3 q   t  + q   t  + q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a229
K11a228
K11a228
K11a230
K11a230