© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a216
K11a216
K11a218
K11a218
K11a217
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   The Knot K11a217

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Acknowledgement

K11a217 as Morse Link
DrawMorseLink

PD Presentation: X4251 X12,3,13,4 X18,5,19,6 X14,8,15,7 X20,9,21,10 X16,12,17,11 X2,13,3,14 X8,16,9,15 X22,17,1,18 X6,19,7,20 X10,21,11,22

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

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

Alexander Polynomial: - t-4 + 6t-3 - 16t-2 + 29t-1 - 35 + 29t - 16t2 + 6t3 - t4

Conway Polynomial: 1 + 3z2 - 2z6 - z8

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

Determinant and Signature: {139, -2}

Jones Polynomial: - q-8 + 3q-7 - 8q-6 + 14q-5 - 19q-4 + 23q-3 - 22q-2 + 20q-1 - 15 + 9q - 4q2 + q3

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

A2 (sl(3)) Invariant: - q-24 - q-20 - 3q-18 + 4q-16 - 2q-14 + 3q-12 + 3q-10 - 2q-8 + 5q-6 - 5q-4 + 3q-2 - 1 - 2q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial: 1 + 3z2 + 3z4 + z6 - 3a2 - 9a2z2 - 10a2z4 - 5a2z6 - a2z8 + 6a4 + 12a4z2 + 8a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial: a-2z2 - 2a-2z4 + a-2z6 - a-1z + 6a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 3z2 + 9z4 - 15z6 + 7z8 - 3az + 10az3 - 11az5 - 5az7 + 6az9 + 3a2 - 14a2z2 + 31a2z4 - 38a2z6 + 13a2z8 + 2a2z10 - 5a3z + 13a3z3 - 5a3z5 - 16a3z7 + 12a3z9 + 6a4 - 20a4z2 + 38a4z4 - 38a4z6 + 14a4z8 + 2a4z10 - 7a5z + 18a5z3 - 13a5z5 - a5z7 + 6a5z9 + 3a6 - 9a6z2 + 14a6z4 - 13a6z6 + 8a6z8 - 3a7z + 7a7z3 - 9a7z5 + 6a7z7 + a8z2 - 4a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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=-2 is the signature of 11217. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 7           1
j = 5          3 
j = 3         61 
j = 1        93  
j = -1       116   
j = -3      1210    
j = -5     1110     
j = -7    812      
j = -9   611       
j = -11  28        
j = -13 16         
j = -15 2          
j = -171           


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, Alternating, 217]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 217]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], 
 
>   X[20, 9, 21, 10], X[16, 12, 17, 11], X[2, 13, 3, 14], X[8, 16, 9, 15], 
 
>   X[22, 17, 1, 18], X[6, 19, 7, 20], X[10, 21, 11, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 217]]
Out[4]=   
GaussCode[1, -7, 2, -1, 3, -10, 4, -8, 5, -11, 6, -2, 7, -4, 8, -6, 9, -3, 10, 
 
>   -5, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 217]]
Out[5]=   
DTCode[4, 12, 18, 14, 20, 16, 2, 8, 22, 6, 10]
In[6]:=
alex = Alexander[Knot[11, Alternating, 217]][t]
Out[6]=   
       -4   6    16   29              2      3    4
-35 - t   + -- - -- + -- + 29 t - 16 t  + 6 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 217]][z]
Out[7]=   
       2      6    8
1 + 3 z  - 2 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 217]}
In[9]:=
{KnotDet[Knot[11, Alternating, 217]], KnotSignature[Knot[11, Alternating, 217]]}
Out[9]=   
{139, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 217]][q]
Out[10]=   
       -8   3    8    14   19   23   22   20            2    3
-15 - q   + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q  + q
             7    6    5    4    3    2   q
            q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 217]}
In[12]:=
A2Invariant[Knot[11, Alternating, 217]][q]
Out[12]=   
      -24    -20    3     4     2     3     3    2    5    5    3       2
-1 - q    - q    - --- + --- - --- + --- + --- - -- + -- - -- + -- - 2 q  + 
                    18    16    14    12    10    8    6    4    2
                   q     q     q     q     q     q    q    q    q
 
       4      6    8
>   3 q  - 2 q  + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 217]][a, z]
Out[13]=   
       2      4      6      2      2  2       4  2      6  2      4
1 - 3 a  + 6 a  - 3 a  + 3 z  - 9 a  z  + 12 a  z  - 3 a  z  + 3 z  - 
 
        2  4      4  4    6  4    6      2  6      4  6    2  8
>   10 a  z  + 8 a  z  - a  z  + z  - 5 a  z  + 2 a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 217]][a, z]
Out[14]=   
       2      4      6   z              3        5        7      9        2
1 + 3 a  + 6 a  + 3 a  - - - 3 a z - 5 a  z - 7 a  z - 3 a  z + a  z - 3 z  + 
                         a
 
     2                                              3
    z        2  2       4  2      6  2    8  2   6 z          3       3  3
>   -- - 14 a  z  - 20 a  z  - 9 a  z  + a  z  + ---- + 10 a z  + 13 a  z  + 
     2                                            a
    a
 
                                             4
        5  3      7  3      9  3      4   2 z        2  4       4  4
>   18 a  z  + 7 a  z  - 2 a  z  + 9 z  - ---- + 31 a  z  + 38 a  z  + 
                                            2
                                           a
 
                            5
        6  4      8  4   9 z          5      3  5       5  5      7  5
>   14 a  z  - 4 a  z  - ---- - 11 a z  - 5 a  z  - 13 a  z  - 9 a  z  + 
                          a
 
                     6                                                 7
     9  5       6   z        2  6       4  6       6  6      8  6   4 z
>   a  z  - 15 z  + -- - 38 a  z  - 38 a  z  - 13 a  z  + 3 a  z  + ---- - 
                     2                                               a
                    a
 
         7       3  7    5  7      7  7      8       2  8       4  8
>   5 a z  - 16 a  z  - a  z  + 6 a  z  + 7 z  + 13 a  z  + 14 a  z  + 
 
       6  8        9       3  9      5  9      2  10      4  10
>   8 a  z  + 6 a z  + 12 a  z  + 6 a  z  + 2 a  z   + 2 a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 217]], Vassiliev[3][Knot[11, Alternating, 217]]}
Out[15]=   
{3, -6}
In[16]:=
Kh[Knot[11, Alternating, 217]][q, t]
Out[16]=   
10   11     1        2        1        6        2        8        6      11
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 3   q     17  7    15  6    13  6    13  5    11  5    11  4    9  4    9  3
q         q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      8      12      11      10     12    6 t                2      3  2
>   ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 3 q t  + 6 q  t  + 
     7  3    7  2    5  2    5      3      q
    q  t    q  t    q  t    q  t   q  t
 
     3  3      5  3    7  4
>   q  t  + 3 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a217
K11a216
K11a216
K11a218
K11a218