© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a154
K11a154
K11a156
K11a156
K11a155
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   The Knot K11a155

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Acknowledgement

K11a155 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: 3t-3 - 16t-2 + 40t-1 - 53 + 40t - 16t2 + 3t3

Conway Polynomial: 1 + 3z2 + 2z4 + 3z6

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

Determinant and Signature: {171, -2}

Jones Polynomial: q-9 - 4q-8 + 9q-7 - 17q-6 + 23q-5 - 27q-4 + 29q-3 - 24q-2 + 19q-1 - 12 + 5q - q2

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

A2 (sl(3)) Invariant: q-28 - 2q-24 + 3q-22 - 5q-20 - 3q-18 + 4q-16 - 4q-14 + 6q-12 + 2q-8 + 4q-6 - 5q-4 + 5q-2 - 3 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial: - z4 - a2 - 2a2z2 + a2z6 + 6a4 + 10a4z2 + 6a4z4 + 2a4z6 - 5a6 - 6a6z2 - 3a6z4 + a8 + a8z2

Kauffman Polynomial: a-1z5 - 3z4 + 5z6 - az + 4az3 - 15az5 + 12az7 + a2 - 4a2z2 + 13a2z4 - 25a2z6 + 16a2z8 - 4a3z + 13a3z3 - 18a3z5 - 4a3z7 + 11a3z9 + 6a4 - 16a4z2 + 45a4z4 - 61a4z6 + 24a4z8 + 3a4z10 - 10a5z + 22a5z3 - 7a5z5 - 25a5z7 + 18a5z9 + 5a6 - 15a6z2 + 38a6z4 - 45a6z6 + 15a6z8 + 3a6z10 - 9a7z + 20a7z3 - 14a7z5 - 5a7z7 + 7a7z9 + a8 - 2a8z2 + 7a8z4 - 13a8z6 + 7a8z8 - 2a9z + 7a9z3 - 9a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {3, -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=-2 is the signature of 11155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 5           1
j = 3          4 
j = 1         81 
j = -1        114  
j = -3       149   
j = -5      1510    
j = -7     1214     
j = -9    1115      
j = -11   612       
j = -13  311        
j = -15 16         
j = -17 3          
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, Alternating, 155]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 155]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[12, 7, 13, 8], 
 
>   X[2, 10, 3, 9], X[16, 11, 17, 12], X[20, 13, 21, 14], X[8, 15, 9, 16], 
 
>   X[22, 17, 1, 18], X[14, 19, 15, 20], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 155]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -11, 4, -8, 5, -2, 6, -4, 7, -10, 8, -6, 9, -3, 10, 
 
>   -7, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 155]]
Out[5]=   
DTCode[4, 10, 18, 12, 2, 16, 20, 8, 22, 14, 6]
In[6]:=
alex = Alexander[Knot[11, Alternating, 155]][t]
Out[6]=   
      3    16   40              2      3
-53 + -- - -- + -- + 40 t - 16 t  + 3 t
       3    2   t
      t    t
In[7]:=
Conway[Knot[11, Alternating, 155]][z]
Out[7]=   
       2      4      6
1 + 3 z  + 2 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 155]}
In[9]:=
{KnotDet[Knot[11, Alternating, 155]], KnotSignature[Knot[11, Alternating, 155]]}
Out[9]=   
{171, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 155]][q]
Out[10]=   
       -9   4    9    17   23   27   29   24   19          2
-12 + q   - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - 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, Alternating, 155]}
In[12]:=
A2Invariant[Knot[11, Alternating, 155]][q]
Out[12]=   
      -28    2     3     5     3     4     4     6    2    4    5    5
-3 + q    - --- + --- - --- - --- + --- - --- + --- + -- + -- - -- + -- - 
             24    22    20    18    16    14    12    8    6    4    2
            q     q     q     q     q     q     q     q    q    q    q
 
       2      4    6
>   2 q  + 3 q  - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 155]][a, z]
Out[13]=   
  2      4      6    8      2  2       4  2      6  2    8  2    4      4  4
-a  + 6 a  - 5 a  + a  - 2 a  z  + 10 a  z  - 6 a  z  + a  z  - z  + 6 a  z  - 
 
       6  4    2  6      4  6
>   3 a  z  + a  z  + 2 a  z
In[14]:=
Kauffman[Knot[11, Alternating, 155]][a, z]
Out[14]=   
 2      4      6    8            3         5        7        9        2  2
a  + 6 a  + 5 a  + a  - a z - 4 a  z - 10 a  z - 9 a  z - 2 a  z - 4 a  z  - 
 
        4  2       6  2      8  2    10  2        3       3  3       5  3
>   16 a  z  - 15 a  z  - 2 a  z  + a   z  + 4 a z  + 13 a  z  + 22 a  z  + 
 
        7  3      9  3      4       2  4       4  4       6  4      8  4
>   20 a  z  + 7 a  z  - 3 z  + 13 a  z  + 45 a  z  + 38 a  z  + 7 a  z  - 
 
                5
       10  4   z          5       3  5      5  5       7  5      9  5      6
>   2 a   z  + -- - 15 a z  - 18 a  z  - 7 a  z  - 14 a  z  - 9 a  z  + 5 z  - 
               a
 
        2  6       4  6       6  6       8  6    10  6         7      3  7
>   25 a  z  - 61 a  z  - 45 a  z  - 13 a  z  + a   z  + 12 a z  - 4 a  z  - 
 
        5  7      7  7      9  7       2  8       4  8       6  8      8  8
>   25 a  z  - 5 a  z  + 4 a  z  + 16 a  z  + 24 a  z  + 15 a  z  + 7 a  z  + 
 
        3  9       5  9      7  9      4  10      6  10
>   11 a  z  + 18 a  z  + 7 a  z  + 3 a  z   + 3 a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 155]], Vassiliev[3][Knot[11, Alternating, 155]]}
Out[15]=   
{3, -4}
In[16]:=
Kh[Knot[11, Alternating, 155]][q, t]
Out[16]=   
9    11     1        3        1        6        3        11       6
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 3   q     19  8    17  7    15  7    15  6    13  6    13  5    11  5
q         q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      12      11      15      12      14      15      10     14    4 t
>   ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 
     11  4    9  4    9  3    7  3    7  2    5  2    5      3      q
    q   t    q  t    q  t    q  t    q  t    q  t    q  t   q  t
 
               2      3  2    5  3
>   8 q t + q t  + 4 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a155
K11a154
K11a154
K11a156
K11a156