© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n64
K11n64
K11n66
K11n66
K11n65
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   The Knot K11n65

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Acknowledgement

K11n65 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X13,20,14,21 X6,15,7,16 X10,17,11,18 X19,1,20,22 X21,12,22,13

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

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

Alexander Polynomial: 3t-2 - 8t-1 + 11 - 8t + 3t2

Conway Polynomial: 1 + 4z2 + 3z4

Other knots with the same Alexander/Conway Polynomial: {815, ...}

Determinant and Signature: {33, 0}

Jones Polynomial: - q-7 + 2q-6 - 4q-5 + 5q-4 - 5q-3 + 6q-2 - 4q-1 + 4 - 2q

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

A2 (sl(3)) Invariant: - q-22 - 2q-16 - q-12 + q-10 + 3q-8 + 2q-6 + 3q-4 - 2q4

HOMFLY-PT Polynomial: - 2 - 2z2 + 5a2 + 6a2z2 + 2a2z4 - a4 + a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial: 2a-1z - 2 + 2z2 + z4 + 4az - 3az3 - az5 + az7 - 5a2 + 5a2z2 + 2a2z4 - 6a2z6 + 2a2z8 + 2a3z + 2a3z3 - 5a3z5 - a3z7 + a3z9 - a4 - a4z2 + 12a4z4 - 15a4z6 + 4a4z8 - 4a5z + 13a5z3 - 9a5z5 - a5z7 + a5z9 + a6 - 4a6z2 + 11a6z4 - 9a6z6 + 2a6z8 - 4a7z + 8a7z3 - 5a7z5 + a7z7

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, -5}

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 1165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 3        2
j = 1       2 
j = -1      33 
j = -3     31  
j = -5    23   
j = -7   33    
j = -9  12     
j = -11 13      
j = -13 1       
j = -151        


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, 65]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 65]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], 
 
>   X[18, 9, 19, 10], X[16, 11, 17, 12], X[13, 20, 14, 21], X[6, 15, 7, 16], 
 
>   X[10, 17, 11, 18], X[19, 1, 20, 22], X[21, 12, 22, 13]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 65]]
Out[4]=   
GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, 11, -7, -3, 8, -6, 9, -5, -10, 
 
>   7, -11, 10]
In[5]:=
DTCode[Knot[11, NonAlternating, 65]]
Out[5]=   
DTCode[4, 8, 14, 2, 18, 16, -20, 6, 10, -22, -12]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 65]][t]
Out[6]=   
     3    8            2
11 + -- - - - 8 t + 3 t
      2   t
     t
In[7]:=
Conway[Knot[11, NonAlternating, 65]][z]
Out[7]=   
       2      4
1 + 4 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[8, 15], Knot[11, NonAlternating, 65]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 65]], KnotSignature[Knot[11, NonAlternating, 65]]}
Out[9]=   
{33, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 65]][q]
Out[10]=   
     -7   2    4    5    5    6    4
4 - q   + -- - -- + -- - -- + -- - - - 2 q
           6    5    4    3    2   q
          q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 65]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 65]][q]
Out[12]=   
  -22    2     -12    -10   3    2    3       4
-q    - --- - q    + q    + -- + -- + -- - 2 q
         16                  8    6    4
        q                   q    q    q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 65]][a, z]
Out[13]=   
        2    4    6      2      2  2    4  2    6  2      2  4    4  4
-2 + 5 a  - a  - a  - 2 z  + 6 a  z  + a  z  - a  z  + 2 a  z  + a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 65]][a, z]
Out[14]=   
        2    4    6   2 z              3        5        7        2      2  2
-2 - 5 a  - a  + a  + --- + 4 a z + 2 a  z - 4 a  z - 4 a  z + 2 z  + 5 a  z  - 
                       a
 
     4  2      6  2        3      3  3       5  3      7  3    4      2  4
>   a  z  - 4 a  z  - 3 a z  + 2 a  z  + 13 a  z  + 8 a  z  + z  + 2 a  z  + 
 
        4  4       6  4      5      3  5      5  5      7  5      2  6
>   12 a  z  + 11 a  z  - a z  - 5 a  z  - 9 a  z  - 5 a  z  - 6 a  z  - 
 
        4  6      6  6      7    3  7    5  7    7  7      2  8      4  8
>   15 a  z  - 9 a  z  + a z  - a  z  - a  z  + a  z  + 2 a  z  + 4 a  z  + 
 
       6  8    3  9    5  9
>   2 a  z  + a  z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 65]], Vassiliev[3][Knot[11, NonAlternating, 65]]}
Out[15]=   
{4, -5}
In[16]:=
Kh[Knot[11, NonAlternating, 65]][q, t]
Out[16]=   
3           1        1        1        3        1       2       3       3
- + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
q          15  7    13  6    11  6    11  5    9  5    9  4    7  4    7  3
          q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      2       3       3      1      3       3
>   ----- + ----- + ----- + ---- + --- + 2 q  t
     5  3    5  2    3  2    3     q t
    q  t    q  t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n65
K11n64
K11n64
K11n66
K11n66