© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n94
K11n94
K11n96
K11n96
K11n95
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   The Knot K11n95

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Acknowledgement

K11n95 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,4,11,3 X14,6,15,5 X7,13,8,12 X2,10,3,9 X11,21,12,20 X18,14,19,13 X15,22,16,1 X6,18,7,17 X19,9,20,8 X21,16,22,17

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

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

Alexander Polynomial: - t-3 + 5t-2 - 7t-1 + 7 - 7t + 5t2 - t3

Conway Polynomial: 1 + 4z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial: {911, ...}

Determinant and Signature: {33, 4}

Jones Polynomial: 2q2 - 3q3 + 5q4 - 6q5 + 6q6 - 5q7 + 4q8 - 2q9

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

A2 (sl(3)) Invariant: 2q6 + 2q10 + q16 - 2q18 + q20 - q22 + q24 + q26 - q28 - q32

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

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

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, 9}

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 1195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 19       2
j = 17      2 
j = 15     32 
j = 13    32  
j = 11   33   
j = 9  23    
j = 7 13     
j = 512      
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, 95]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 95]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[7, 13, 8, 12], 
 
>   X[2, 10, 3, 9], X[11, 21, 12, 20], X[18, 14, 19, 13], X[15, 22, 16, 1], 
 
>   X[6, 18, 7, 17], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 95]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -9, -4, 10, 5, -2, -6, 4, 7, -3, -8, 11, 9, -7, -10, 
 
>   6, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 95]]
Out[5]=   
DTCode[4, 10, 14, -12, 2, -20, 18, -22, 6, -8, -16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 95]][t]
Out[6]=   
     -3   5    7            2    3
7 - t   + -- - - - 7 t + 5 t  - t
           2   t
          t
In[7]:=
Conway[Knot[11, NonAlternating, 95]][z]
Out[7]=   
       2    4    6
1 + 4 z  - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[9, 11], Knot[11, NonAlternating, 95]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 95]], KnotSignature[Knot[11, NonAlternating, 95]]}
Out[9]=   
{33, 4}
In[10]:=
J=Jones[Knot[11, NonAlternating, 95]][q]
Out[10]=   
   2      3      4      5      6      7      8      9
2 q  - 3 q  + 5 q  - 6 q  + 6 q  - 5 q  + 4 q  - 2 q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 95]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 95]][q]
Out[12]=   
   6      10    16      18    20    22    24    26    28    32
2 q  + 2 q   + q   - 2 q   + q   - q   + q   + q   - q   - q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 95]][a, z]
Out[13]=   
                          2      2      2    4      4      4    6
  -10   2    3    3    3 z    5 z    6 z    z    4 z    2 z    z
-a    + -- - -- + -- + ---- - ---- + ---- + -- - ---- + ---- - --
         8    6    4     8      6      4     8     6      4     6
        a    a    a     a      a      a     a     a      a     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 95]][a, z]
Out[14]=   
                                               2      2      2      2      3
 -10   2    3    3    3 z   3 z   z    z    4 z    6 z    9 z    7 z    3 z
a    + -- + -- + -- - --- - --- - -- - -- - ---- - ---- - ---- - ---- + ---- + 
        8    6    4    11    9     7    5    10      8      6      4     11
       a    a    a    a     a     a    a    a       a      a      a     a
 
       3      3      4      4      4      4      5      5    5    6     6
    7 z    4 z    2 z    5 z    6 z    3 z    4 z    5 z    z    z     z
>   ---- + ---- + ---- + ---- + ---- + ---- - ---- - ---- - -- + --- - -- - 
      9      7     10      8      6      4      9      7     5    10    8
     a      a     a       a      a      a      a      a     a    a     a
 
       6      7      7    7    8    8
    2 z    2 z    3 z    z    z    z
>   ---- + ---- + ---- + -- + -- + --
      6      9      7     5    8    6
     a      a      a     a    a    a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 95]], Vassiliev[3][Knot[11, NonAlternating, 95]]}
Out[15]=   
{4, 9}
In[16]:=
Kh[Knot[11, NonAlternating, 95]][q, t]
Out[16]=   
   3    5      5      7        7  2      9  2      9  3      11  3      11  4
2 q  + q  + 2 q  t + q  t + 3 q  t  + 2 q  t  + 3 q  t  + 3 q   t  + 3 q   t  + 
 
       13  4      13  5      15  5      15  6      17  6      19  7
>   3 q   t  + 2 q   t  + 3 q   t  + 2 q   t  + 2 q   t  + 2 q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n95
K11n94
K11n94
K11n96
K11n96