© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n28
K11n28
K11n30
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K11n29
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   The Knot K11n29

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Acknowledgement

K11n29 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X22,15,1,16 X20,17,21,18 X10,20,11,19 X16,21,17,22

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

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

Alexander Polynomial: - 3t-2 + 13t-1 - 19 + 13t - 3t2

Conway Polynomial: 1 + z2 - 3z4

Other knots with the same Alexander/Conway Polynomial: {1036, K11a230, ...}

Determinant and Signature: {51, 2}

Jones Polynomial: 2q-1 - 4 + 6q - 8q2 + 9q3 - 8q4 + 7q5 - 4q6 + 2q7 - q8

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

A2 (sl(3)) Invariant: 2q-4 + 2q2 - 3q4 - q8 + 2q12 + 3q16 - q20 + q22 - q24 - q26

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

Kauffman Polynomial: 2a-9z - 3a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 2a-8z6 + 2a-7z - 3a-7z3 - 2a-7z5 + 2a-7z7 - a-6 + 2a-6z4 - 3a-6z6 + 2a-6z8 + a-5z + a-5z9 + 2a-4 - 11a-4z2 + 18a-4z4 - 11a-4z6 + 4a-4z8 + 3a-3z - 5a-3z3 + 4a-3z5 - a-3z7 + a-3z9 + 3a-2 - 14a-2z2 + 14a-2z4 - 6a-2z6 + 2a-2z8 + 2a-1z - 5a-1z3 + a-1z5 + a-1z7 + 2 - 6z2 + 3z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 4}

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 1129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17         1
j = 15        1 
j = 13       31 
j = 11      41  
j = 9     43   
j = 7    54    
j = 5   34     
j = 3  35      
j = 1 24       
j = -1 2        
j = -32         


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, 29]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 29]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], 
 
>   X[14, 9, 15, 10], X[18, 12, 19, 11], X[13, 7, 14, 6], X[22, 15, 1, 16], 
 
>   X[20, 17, 21, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 29]]
Out[4]=   
GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, 
 
>   -9, 11, -8]
In[5]:=
DTCode[Knot[11, NonAlternating, 29]]
Out[5]=   
DTCode[4, 8, -12, 2, 14, 18, -6, 22, 20, 10, 16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 29]][t]
Out[6]=   
      3    13             2
-19 - -- + -- + 13 t - 3 t
       2   t
      t
In[7]:=
Conway[Knot[11, NonAlternating, 29]][z]
Out[7]=   
     2      4
1 + z  - 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[10, 36], Knot[11, Alternating, 230], Knot[11, NonAlternating, 29]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 29]], KnotSignature[Knot[11, NonAlternating, 29]]}
Out[9]=   
{51, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 29]][q]
Out[10]=   
     2            2      3      4      5      6      7    8
-4 + - + 6 q - 8 q  + 9 q  - 8 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, NonAlternating, 29]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 29]][q]
Out[12]=   
2       2      4    8      12      16    20    22    24    26
-- + 2 q  - 3 q  - q  + 2 q   + 3 q   - q   + q   - q   - q
 4
q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 29]][a, z]
Out[13]=   
                                    2    2      2    4      4
     -8    -6   2    3       2   2 z    z    4 z    z    2 z
2 - a   + a   + -- - -- + 2 z  + ---- + -- - ---- - -- - ----
                 4    2            6     4     2     4     2
                a    a            a     a     a     a     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 29]][a, z]
Out[14]=   
                                                                 2       2
     -8    -6   2    3    2 z   2 z   z    3 z   2 z      2   3 z    11 z
2 - a   - a   + -- + -- + --- + --- + -- + --- + --- - 6 z  + ---- - ----- - 
                 4    2    9     7     5    3     a             8      4
                a    a    a     a     a    a                   a      a
 
        2      3      3      3      3             4      4       4       4
    14 z    3 z    3 z    5 z    5 z       4   5 z    2 z    18 z    14 z
>   ----- - ---- - ---- - ---- - ---- + 3 z  - ---- + ---- + ----- + ----- + 
      2       9      7      3     a              8      6      4       2
     a       a      a      a                    a      a      a       a
 
     5      5      5    5      6      6       6      6      7    7    7
    z    2 z    4 z    z    2 z    3 z    11 z    6 z    2 z    z    z
>   -- - ---- + ---- + -- + ---- - ---- - ----- - ---- + ---- - -- + -- + 
     9     7      3    a      8      6      4       2      7     3   a
    a     a      a           a      a      a       a      a     a
 
       8      8      8    9    9
    2 z    4 z    2 z    z    z
>   ---- + ---- + ---- + -- + --
      6      4      2     5    3
     a      a      a     a    a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 29]], Vassiliev[3][Knot[11, NonAlternating, 29]]}
Out[15]=   
{1, 4}
In[16]:=
Kh[Knot[11, NonAlternating, 29]][q, t]
Out[16]=   
         3     2      2    2 q      3        5        5  2      7  2
4 q + 3 q  + ----- + --- + --- + 5 q  t + 3 q  t + 4 q  t  + 5 q  t  + 
              3  2   q t    t
             q  t
 
       7  3      9  3      9  4      11  4    11  5      13  5    13  6
>   4 q  t  + 4 q  t  + 3 q  t  + 4 q   t  + q   t  + 3 q   t  + q   t  + 
 
     15  6    17  7
>   q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n29
K11n28
K11n28
K11n30
K11n30