© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n99
K11n99
K11n101
K11n101
K11n100
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   The Knot K11n100

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Acknowledgement

K11n100 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X9,19,10,18 X2,11,3,12 X6,14,7,13 X15,22,16,1 X17,20,18,21 X19,9,20,8 X21,16,22,17

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

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

Alexander Polynomial: 2t-2 - 11t-1 + 19 - 11t + 2t2

Conway Polynomial: 1 - 3z2 + 2z4

Other knots with the same Alexander/Conway Polynomial: {937, ...}

Determinant and Signature: {45, 0}

Jones Polynomial: - q-3 + 4q-2 - 5q-1 + 7 - 8q + 7q2 - 6q3 + 4q4 - 2q5 + q6

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

A2 (sl(3)) Invariant: - q-10 + 2q-8 + 2q-6 - q-4 + 2q-2 - 1 - q6 + q8 - 2q10 + q14 - q16 + q18 + q20

HOMFLY-PT Polynomial: a-6 - a-4 - 2a-4z2 + a-2z4 + z4 + a2 - a2z2

Kauffman Polynomial: - a-6 + 4a-6z2 - 4a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 7a-5z5 + 2a-5z7 - a-4 + 3a-4z2 - 5a-4z6 + 2a-4z8 + 4a-3z3 - 6a-3z5 + a-3z9 - 6a-2z2 + 13a-2z4 - 12a-2z6 + 4a-2z8 + 3a-1z - 4a-1z3 + 2a-1z5 - a-1z7 + a-1z9 - 8z2 + 13z4 - 6z6 + 2z8 + az - az3 + az5 + az7 - a2 - 3a2z2 + 4a2z4 + a3z3

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

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=0 is the signature of 11100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 13         1
j = 11        1 
j = 9       31 
j = 7      31  
j = 5     43   
j = 3    43    
j = 1   34     
j = -1  35      
j = -3 12       
j = -5 3        
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, NonAlternating, 100]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 100]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[12, 8, 13, 7], 
 
>   X[9, 19, 10, 18], X[2, 11, 3, 12], X[6, 14, 7, 13], X[15, 22, 16, 1], 
 
>   X[17, 20, 18, 21], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 100]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -7, 4, 10, -5, -2, 6, -4, 7, -3, -8, 11, -9, 5, -10, 
 
>   9, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 100]]
Out[5]=   
DTCode[4, 10, 14, 12, -18, 2, 6, -22, -20, -8, -16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 100]][t]
Out[6]=   
     2    11             2
19 + -- - -- - 11 t + 2 t
      2   t
     t
In[7]:=
Conway[Knot[11, NonAlternating, 100]][z]
Out[7]=   
       2      4
1 - 3 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[9, 37], Knot[11, NonAlternating, 100]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 100]], KnotSignature[Knot[11, NonAlternating, 100]]}
Out[9]=   
{45, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 100]][q]
Out[10]=   
     -3   4    5            2      3      4      5    6
7 - q   + -- - - - 8 q + 7 q  - 6 q  + 4 q  - 2 q  + q
           2   q
          q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 100]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 100]][q]
Out[12]=   
      -10   2    2     -4   2     6    8      10    14    16    18    20
-1 - q    + -- + -- - q   + -- - q  + q  - 2 q   + q   - q   + q   + q
             8    6          2
            q    q          q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 100]][a, z]
Out[13]=   
                    2                 4
 -6    -4    2   2 z     2  2    4   z
a   - a   + a  - ---- - a  z  + z  + --
                   4                  2
                  a                  a
In[14]:=
Kauffman[Knot[11, NonAlternating, 100]][a, z]
Out[14]=   
                                              2      2      2
  -6    -4    2   2 z   3 z            2   4 z    3 z    6 z       2  2
-a   - a   - a  - --- + --- + a z - 8 z  + ---- + ---- - ---- - 3 a  z  + 
                   5     a                   6      4      2
                  a                         a      a      a
 
       3      3      3                             4       4                5
    6 z    4 z    4 z       3    3  3       4   4 z    13 z       2  4   7 z
>   ---- + ---- - ---- - a z  + a  z  + 13 z  - ---- + ----- + 4 a  z  - ---- - 
      5      3     a                              6      2                 5
     a      a                                    a      a                 a
 
       5      5                  6      6       6      7    7
    6 z    2 z       5      6   z    5 z    12 z    2 z    z       7      8
>   ---- + ---- + a z  - 6 z  + -- - ---- - ----- + ---- - -- + a z  + 2 z  + 
      3     a                    6     4      2       5    a
     a                          a     a      a       a
 
       8      8    9    9
    2 z    4 z    z    z
>   ---- + ---- + -- + --
      4      2     3   a
     a      a     a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 100]], Vassiliev[3][Knot[11, NonAlternating, 100]]}
Out[15]=   
{-3, -3}
In[16]:=
Kh[Knot[11, NonAlternating, 100]][q, t]
Out[16]=   
5           1       3       1      2      3               3        3  2
- + 3 q + ----- + ----- + ----- + ---- + --- + 4 q t + 4 q  t + 3 q  t  + 
q          7  3    5  2    3  2    3     q t
          q  t    q  t    q  t    q  t
 
       5  2      5  3      7  3    7  4      9  4    9  5    11  5    13  6
>   4 q  t  + 3 q  t  + 3 q  t  + q  t  + 3 q  t  + q  t  + q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n100
K11n99
K11n99
K11n101
K11n101