© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n140
K11n140 		K11n142
K11n142
K11n141
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   The Knot K11n141

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Acknowledgement

K11n141 as Morse Link
DrawMorseLink

PD Presentation: X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X9,21,10,20 X11,19,12,18 X2,13,3,14 X15,6,16,7 X17,22,18,1 X19,11,20,10 X21,9,22,8

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

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

Alexander Polynomial: - 5t-1 + 11 - 5t

Conway Polynomial: 1 - 5z2

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

Determinant and Signature: {21, 0}

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

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

A2 (sl(3)) Invariant: q-20 + q-18 + q-14 - q-10 - q-6 - q-4 - q-2 - 1 + q2 + 2q6 + 2q8

HOMFLY-PT Polynomial: 2a-2 - 1 - 2z2 - a2 - 2a2z2 - a4z2 + a6

Kauffman Polynomial: - 2a-2 + 2a-2z2 + 3a-1z - 2a-1z3 + a-1z5 - 1 - 6z2 + 13z4 - 6z6 + z8 + 11az - 24az3 + 20az5 - 7az7 + az9 + a2 - 16a2z2 + 22a2z4 - 11a2z6 + 2a2z8 + 8a3z - 19a3z3 + 15a3z5 - 6a3z7 + a3z9 - 2a4z2 + 4a4z4 - 4a4z6 + a4z8 + 3a5z3 - 4a5z5 + a5z7 - a6 + 6a6z2 - 5a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {-5, 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=0 is the signature of 11141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 5        2
j = 3         
j = 1      32 
j = -1     21  
j = -3    12   
j = -5   22    
j = -7   1     
j = -9 12      
j = -11         
j = -131        

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, 141]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 141]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[5, 16, 6, 17], X[7, 14, 8, 15], 
 
>   X[9, 21, 10, 20], X[11, 19, 12, 18], X[2, 13, 3, 14], X[15, 6, 16, 7], 
 
>   X[17, 22, 18, 1], X[19, 11, 20, 10], X[21, 9, 22, 8]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 141]]
Out[4]=   
GaussCode[1, -7, 2, -1, -3, 8, -4, 11, -5, 10, -6, -2, 7, 4, -8, 3, -9, 6, -10, 
 
>   5, -11, 9]
In[5]:=
DTCode[Knot[11, NonAlternating, 141]]
Out[5]=   
DTCode[4, 12, -16, -14, -20, -18, 2, -6, -22, -10, -8]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 141]][t]
Out[6]=   
     5
11 - - - 5 t
     t
In[7]:=
Conway[Knot[11, NonAlternating, 141]][z]
Out[7]=   
       2
1 - 5 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 141]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 141]], KnotSignature[Knot[11, NonAlternating, 141]]}
Out[9]=   
{21, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 141]][q]
Out[10]=   
     -6    -5   2    3    3    4            2
3 + q   - q   + -- - -- + -- - - - 2 q + 2 q
                 4    3    2   q
                q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 141]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 141]][q]
Out[12]=   
      -20    -18    -14    -10    -6    -4    -2    2      6      8
-1 + q    + q    + q    - q    - q   - q   - q   + q  + 2 q  + 2 q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 141]][a, z]
Out[13]=   
     2     2    6      2      2  2    4  2
-1 + -- - a  + a  - 2 z  - 2 a  z  - a  z
      2
     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 141]][a, z]
Out[14]=   
                                                      2
     2     2    6   3 z               3        2   2 z        2  2      4  2
-1 - -- + a  - a  + --- + 11 a z + 8 a  z - 6 z  + ---- - 16 a  z  - 2 a  z  + 
      2              a                               2
     a                                              a
 
                 3
       6  2   2 z          3       3  3      5  3       4       2  4
>   6 a  z  - ---- - 24 a z  - 19 a  z  + 3 a  z  + 13 z  + 22 a  z  + 
               a
 
                         5
       4  4      6  4   z          5       3  5      5  5      6       2  6
>   4 a  z  - 5 a  z  + -- + 20 a z  + 15 a  z  - 4 a  z  - 6 z  - 11 a  z  - 
                        a
 
       4  6    6  6        7      3  7    5  7    8      2  8    4  8      9
>   4 a  z  + a  z  - 7 a z  - 6 a  z  + a  z  + z  + 2 a  z  + a  z  + a z  + 
 
     3  9
>   a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 141]], Vassiliev[3][Knot[11, NonAlternating, 141]]}
Out[15]=   
{-5, 4}
In[16]:=
Kh[Knot[11, NonAlternating, 141]][q, t]
Out[16]=   
1           1        1       2       1       2       2       1      2      2
- + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
q          13  6    9  5    9  4    7  3    5  3    5  2    3  2    3     q t
          q   t    q  t    q  t    q  t    q  t    q  t    q  t    q  t
 
               5  2
>   2 q t + 2 q  t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n141
K11n140
K11n140 		K11n142
K11n142