© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n7
K11n7 		K11n9
K11n9
K11n8
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   The Knot K11n8

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Acknowledgement

K11n8 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,18,12,19 X13,6,14,7 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17

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

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

Alexander Polynomial: - t-3 + 6t-2 - 12t-1 + 15 - 12t + 6t2 - t3

Conway Polynomial: 1 + 3z2 - z6

Other knots with the same Alexander/Conway Polynomial: {K11n59, ...}

Determinant and Signature: {53, -4}

Jones Polynomial: - q-9 + 3q-8 - 6q-7 + 8q-6 - 9q-5 + 9q-4 - 7q-3 + 6q-2 - 3q-1 + 1

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

A2 (sl(3)) Invariant: - q-28 + q-24 - 2q-22 + q-20 - q-18 + 2q-14 - q-12 + 3q-10 - q-8 + q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial: a2 + 2a2z2 + a2z4 - 2a4z2 - 3a4z4 - a4z6 + a6 + 4a6z2 + 2a6z4 - a8 - a8z2

Kauffman Polynomial: - a2 + 3a2z2 - 3a2z4 + a2z6 - a3z + 6a3z3 - 9a3z5 + 3a3z7 + 4a4z2 - 3a4z4 - 6a4z6 + 3a4z8 - 4a5z + 14a5z3 - 18a5z5 + 4a5z7 + a5z9 - a6 + 3a6z2 + 3a6z4 - 11a6z6 + 5a6z8 - 3a7z + 10a7z3 - 9a7z5 + 2a7z7 + a7z9 - a8 + 6a8z4 - 4a8z6 + 2a8z8 - a9z + 3a9z3 + a9z7 - 2a10z2 + 3a10z4 - a11z + a11z3

V2 and V3, the type 2 and 3 Vassiliev invariants: {3, -6}

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 118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 1         1
j = -1        2 
j = -3       41 
j = -5      43  
j = -7     53   
j = -9    44    
j = -11   45     
j = -13  24      
j = -15 14       
j = -17 2        
j = -191         

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, 8]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 8]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 14, 8, 15], 
 
>   X[2, 9, 3, 10], X[11, 18, 12, 19], X[13, 6, 14, 7], X[15, 20, 16, 21], 
 
>   X[17, 12, 18, 13], X[19, 22, 20, 1], X[21, 16, 22, 17]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 8]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 9, -7, 4, -8, 11, -9, 6, -10, 
 
>   8, -11, 10]
In[5]:=
DTCode[Knot[11, NonAlternating, 8]]
Out[5]=   
DTCode[4, 8, 10, -14, 2, -18, -6, -20, -12, -22, -16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 8]][t]
Out[6]=   
      -3   6    12             2    3
15 - t   + -- - -- - 12 t + 6 t  - t
            2   t
           t
In[7]:=
Conway[Knot[11, NonAlternating, 8]][z]
Out[7]=   
       2    6
1 + 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 8], Knot[11, NonAlternating, 59]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 8]], KnotSignature[Knot[11, NonAlternating, 8]]}
Out[9]=   
{53, -4}
In[10]:=
J=Jones[Knot[11, NonAlternating, 8]][q]
Out[10]=   
     -9   3    6    8    9    9    7    6    3
1 - q   + -- - -- + -- - -- + -- - -- + -- - -
           8    7    6    5    4    3    2   q
          q    q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 8]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 8]][q]
Out[12]=   
     -28    -24    2     -20    -18    2     -12    3     -8    -6    -4    -2
1 - q    + q    - --- + q    - q    + --- - q    + --- - q   + q   + q   - q
                   22                  14           10
                  q                   q            q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 8]][a, z]
Out[13]=   
 2    6    8      2  2      4  2      6  2    8  2    2  4      4  4
a  + a  - a  + 2 a  z  - 2 a  z  + 4 a  z  - a  z  + a  z  - 3 a  z  + 
 
       6  4    4  6
>   2 a  z  - a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 8]][a, z]
Out[14]=   
  2    6    8    3        5        7      9      11        2  2      4  2
-a  - a  - a  - a  z - 4 a  z - 3 a  z - a  z - a   z + 3 a  z  + 4 a  z  + 
 
       6  2      10  2      3  3       5  3       7  3      9  3    11  3
>   3 a  z  - 2 a   z  + 6 a  z  + 14 a  z  + 10 a  z  + 3 a  z  + a   z  - 
 
       2  4      4  4      6  4      8  4      10  4      3  5       5  5
>   3 a  z  - 3 a  z  + 3 a  z  + 6 a  z  + 3 a   z  - 9 a  z  - 18 a  z  - 
 
       7  5    2  6      4  6       6  6      8  6      3  7      5  7
>   9 a  z  + a  z  - 6 a  z  - 11 a  z  - 4 a  z  + 3 a  z  + 4 a  z  + 
 
       7  7    9  7      4  8      6  8      8  8    5  9    7  9
>   2 a  z  + a  z  + 3 a  z  + 5 a  z  + 2 a  z  + a  z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 8]], Vassiliev[3][Knot[11, NonAlternating, 8]]}
Out[15]=   
{3, -6}
In[16]:=
Kh[Knot[11, NonAlternating, 8]][q, t]
Out[16]=   
3    4      1        2        1        4        2        4        4
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    19  7    17  6    15  6    15  5    13  5    13  4    11  4
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      5        4       4       5      3      4     t    2 t      2
>   ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + q t
     11  3    9  3    9  2    7  2    7      5      3    q
    q   t    q  t    q  t    q  t    q  t   q  t   q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n8
K11n7
K11n7 		K11n9
K11n9