© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n25
K11n25
K11n27
K11n27
K11n26
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   The Knot K11n26

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Acknowledgement

K11n26 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 X20,15,21,16 X22,17,1,18 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, -8, 11, -9}

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

Alexander Polynomial: - t-3 + 5t-2 - 9t-1 + 11 - 9t + 5t2 - t3

Conway Polynomial: 1 + 2z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial: {920, 10149, ...}

Determinant and Signature: {41, 0}

Jones Polynomial: q-2 - 3q-1 + 5 - 6q + 7q2 - 6q3 + 6q4 - 4q5 + 2q6 - q7

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

A2 (sl(3)) Invariant: q-6 - q-4 + q-2 - q2 + q4 - q6 + 2q8 + q10 + q12 + 2q14 - q16 - q20 - q22

HOMFLY-PT Polynomial: - 2a-6 - a-6z2 + 4a-4 + 6a-4z2 + 2a-4z4 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 1 + 2z2 + z4

Kauffman Polynomial: - 4a-7z + 8a-7z3 - 5a-7z5 + a-7z7 + 2a-6 - 6a-6z2 + 12a-6z4 - 9a-6z6 + 2a-6z8 - 7a-5z + 19a-5z3 - 13a-5z5 + a-5z9 + 4a-4 - 11a-4z2 + 20a-4z4 - 19a-4z6 + 5a-4z8 - 5a-3z + 16a-3z3 - 16a-3z5 + 2a-3z7 + a-3z9 + 2a-2 - 7a-2z2 + 9a-2z4 - 9a-2z6 + 3a-2z8 - 3a-1z + 8a-1z3 - 8a-1z5 + 3a-1z7 + 1 - z2 + z4 + z6 - az + 3az3 + a2z2

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 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=0 is the signature of 1126. 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 = 15         1
j = 13        1 
j = 11       31 
j = 9      31  
j = 7     33   
j = 5    43    
j = 3   23     
j = 1  34      
j = -1 13       
j = -3 2        
j = -51         


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, 26]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 26]]
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[20, 15, 21, 16], 
 
>   X[22, 17, 1, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 26]]
Out[4]=   
GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, 
 
>   -8, 11, -9]
In[5]:=
DTCode[Knot[11, NonAlternating, 26]]
Out[5]=   
DTCode[4, 8, -12, 2, 14, 18, -6, 20, 22, 10, 16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 26]][t]
Out[6]=   
      -3   5    9            2    3
11 - t   + -- - - - 9 t + 5 t  - t
            2   t
           t
In[7]:=
Conway[Knot[11, NonAlternating, 26]][z]
Out[7]=   
       2    4    6
1 + 2 z  - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[9, 20], Knot[10, 149], Knot[11, NonAlternating, 26]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 26]], KnotSignature[Knot[11, NonAlternating, 26]]}
Out[9]=   
{41, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 26]][q]
Out[10]=   
     -2   3            2      3      4      5      6    7
5 + q   - - - 6 q + 7 q  - 6 q  + 6 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, 26]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 26]][q]
Out[12]=   
 -6    -4    -2    2    4    6      8    10    12      14    16    20    22
q   - q   + q   - q  + q  - q  + 2 q  + q   + q   + 2 q   - q   - q   - q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 26]][a, z]
Out[13]=   
                           2      2      2           4      4    6
    2    4    2       2   z    6 z    5 z     4   2 z    4 z    z
1 - -- + -- - -- + 2 z  - -- + ---- - ---- + z  + ---- - ---- - --
     6    4    2           6     4      2           4      2     2
    a    a    a           a     a      a           a      a     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 26]][a, z]
Out[14]=   
                                                         2       2      2
    2    4    2    4 z   7 z   5 z   3 z          2   6 z    11 z    7 z
1 + -- + -- + -- - --- - --- - --- - --- - a z - z  - ---- - ----- - ---- + 
     6    4    2    7     5     3     a                 6      4       2
    a    a    a    a     a     a                       a      a       a
 
               3       3       3      3                     4       4      4
     2  2   8 z    19 z    16 z    8 z         3    4   12 z    20 z    9 z
>   a  z  + ---- + ----- + ----- + ---- + 3 a z  + z  + ----- + ----- + ---- - 
              7      5       3      a                     6       4       2
             a      a       a                            a       a       a
 
       5       5       5      5           6       6      6    7      7      7
    5 z    13 z    16 z    8 z     6   9 z    19 z    9 z    z    2 z    3 z
>   ---- - ----- - ----- - ---- + z  - ---- - ----- - ---- + -- + ---- + ---- + 
      7      5       3      a            6      4       2     7     3     a
     a      a       a                   a      a       a     a     a
 
       8      8      8    9    9
    2 z    5 z    3 z    z    z
>   ---- + ---- + ---- + -- + --
      6      4      2     5    3
     a      a      a     a    a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 26]], Vassiliev[3][Knot[11, NonAlternating, 26]]}
Out[15]=   
{2, 4}
In[16]:=
Kh[Knot[11, NonAlternating, 26]][q, t]
Out[16]=   
3           1      2      1               3        3  2      5  2      5  3
- + 3 q + ----- + ---- + --- + 4 q t + 2 q  t + 3 q  t  + 4 q  t  + 3 q  t  + 
q          5  2    3     q t
          q  t    q  t
 
       7  3      7  4      9  4    9  5      11  5    11  6    13  6    15  7
>   3 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 K11n26
K11n25
K11n25
K11n27
K11n27