© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a160
K11a160
K11a162
K11a162
K11a161
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   The Knot K11a161

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Acknowledgement

K11a161 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - 2t-3 + 8t-2 - 12t-1 + 13 - 12t + 8t2 - 2t3

Conway Polynomial: 1 + 2z2 - 4z4 - 2z6

Other knots with the same Alexander/Conway Polynomial: {1014, K11n2, ...}

Determinant and Signature: {57, -4}

Jones Polynomial: - q-9 + 2q-8 - 4q-7 + 6q-6 - 7q-5 + 9q-4 - 8q-3 + 7q-2 - 6q-1 + 4 - 2q + q2

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

A2 (sl(3)) Invariant: - q-28 - q-26 - q-22 + q-20 + q-18 + q-16 + 3q-14 + q-10 - q-8 - 2q-6 - q-2 + 1 + q2 + q6

HOMFLY-PT Polynomial: 2 + 3z2 + z4 - 3a2 - 5a2z2 - 4a2z4 - a2z6 + a4 - a4z2 - 3a4z4 - a4z6 + 3a6 + 6a6z2 + 2a6z4 - 2a8 - a8z2

Kauffman Polynomial: 2 - 9z2 + 12z4 - 6z6 + z8 + 3az - 12az3 + 19az5 - 11az7 + 2az9 + 3a2 - 20a2z2 + 32a2z4 - 14a2z6 - a2z8 + a2z10 + 5a3z - 16a3z3 + 31a3z5 - 23a3z7 + 5a3z9 + a4 - 13a4z2 + 29a4z4 - 21a4z6 + 2a4z8 + a4z10 + 2a5z - 2a5z3 + 3a5z5 - 8a5z7 + 3a5z9 - 3a6 + 4a6z2 + a6z4 - 9a6z6 + 4a6z8 + 2a7z - a7z3 - 6a7z5 + 4a7z7 - 2a8 + 5a8z2 - 6a8z4 + 4a8z6 + a9z - 2a9z3 + 3a9z5 - a10z2 + 2a10z4 - a11z + a11z3

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, -7}

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 11161. 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 = 5           1
j = 3          1 
j = 1         31 
j = -1        31  
j = -3       43   
j = -5      54    
j = -7     43     
j = -9    35      
j = -11   34       
j = -13  13        
j = -15 13         
j = -17 1          
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, 161]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 161]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], 
 
>   X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], 
 
>   X[22, 17, 1, 18], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 161]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -4, 8, -11, 9, -3, 10, 
 
>   -8, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 161]]
Out[5]=   
DTCode[4, 10, 18, 14, 12, 2, 8, 20, 22, 6, 16]
In[6]:=
alex = Alexander[Knot[11, Alternating, 161]][t]
Out[6]=   
     2    8    12             2      3
13 - -- + -- - -- - 12 t + 8 t  - 2 t
      3    2   t
     t    t
In[7]:=
Conway[Knot[11, Alternating, 161]][z]
Out[7]=   
       2      4      6
1 + 2 z  - 4 z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[10, 14], Knot[11, Alternating, 161], Knot[11, NonAlternating, 2]}
In[9]:=
{KnotDet[Knot[11, Alternating, 161]], KnotSignature[Knot[11, Alternating, 161]]}
Out[9]=   
{57, -4}
In[10]:=
J=Jones[Knot[11, Alternating, 161]][q]
Out[10]=   
     -9   2    4    6    7    9    8    7    6          2
4 - q   + -- - -- + -- - -- + -- - -- + -- - - - 2 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, 161]}
In[12]:=
A2Invariant[Knot[11, Alternating, 161]][q]
Out[12]=   
     -28    -26    -22    -20    -18    -16    3     -10    -8   2     -2
1 - q    - q    - q    + q    + q    + q    + --- + q    - q   - -- - q   + 
                                               14                 6
                                              q                  q
 
     2    6
>   q  + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 161]][a, z]
Out[13]=   
       2    4      6      8      2      2  2    4  2      6  2    8  2    4
2 - 3 a  + a  + 3 a  - 2 a  + 3 z  - 5 a  z  - a  z  + 6 a  z  - a  z  + z  - 
 
       2  4      4  4      6  4    2  6    4  6
>   4 a  z  - 3 a  z  + 2 a  z  - a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 161]][a, z]
Out[14]=   
       2    4      6      8              3        5        7      9      11
2 + 3 a  + a  - 3 a  - 2 a  + 3 a z + 5 a  z + 2 a  z + 2 a  z + a  z - a   z - 
 
       2       2  2       4  2      6  2      8  2    10  2         3
>   9 z  - 20 a  z  - 13 a  z  + 4 a  z  + 5 a  z  - a   z  - 12 a z  - 
 
        3  3      5  3    7  3      9  3    11  3       4       2  4
>   16 a  z  - 2 a  z  - a  z  - 2 a  z  + a   z  + 12 z  + 32 a  z  + 
 
        4  4    6  4      8  4      10  4         5       3  5      5  5
>   29 a  z  + a  z  - 6 a  z  + 2 a   z  + 19 a z  + 31 a  z  + 3 a  z  - 
 
       7  5      9  5      6       2  6       4  6      6  6      8  6
>   6 a  z  + 3 a  z  - 6 z  - 14 a  z  - 21 a  z  - 9 a  z  + 4 a  z  - 
 
          7       3  7      5  7      7  7    8    2  8      4  8      6  8
>   11 a z  - 23 a  z  - 8 a  z  + 4 a  z  + z  - a  z  + 2 a  z  + 4 a  z  + 
 
         9      3  9      5  9    2  10    4  10
>   2 a z  + 5 a  z  + 3 a  z  + a  z   + a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 161]], Vassiliev[3][Knot[11, Alternating, 161]]}
Out[15]=   
{2, -7}
In[16]:=
Kh[Knot[11, Alternating, 161]][q, t]
Out[16]=   
4    4      1        1        1        3        1        3        3
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 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
 
                                                                2
      4        3       5       4      3      5     3 t   3 t   t         2
>   ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + -- + 3 q t  + 
     11  3    9  3    9  2    7  2    7      5      3     q    q
    q   t    q  t    q  t    q  t    q  t   q  t   q
 
       3    3  3    5  4
>   q t  + q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a161
K11a160
K11a160
K11a162
K11a162