© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n130
K11n130
K11n132
K11n132
K11n131
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   The Knot K11n131

Visit K11n131's page at Knotilus!

Acknowledgement

K11n131 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X18,5,19,6 X7,16,8,17 X9,14,10,15 X2,11,3,12 X20,14,21,13 X15,8,16,9 X22,17,1,18 X12,20,13,19 X6,21,7,22

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

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

Alexander Polynomial: t-3 - 6t-2 + 16t-1 - 21 + 16t - 6t2 + t3

Conway Polynomial: 1 + z2 + z6

Other knots with the same Alexander/Conway Polynomial: {K11n7, K11n160, ...}

Determinant and Signature: {67, -2}

Jones Polynomial: q-7 - 4q-6 + 7q-5 - 10q-4 + 12q-3 - 11q-2 + 10q-1 - 7 + 4q - q2

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

A2 (sl(3)) Invariant: q-22 - q-20 - 2q-18 + 2q-16 - 2q-14 + q-12 + q-10 - q-8 + 3q-6 - 2q-4 + 3q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial: - z2 - z4 + 2a2 + 4a2z2 + 3a2z4 + a2z6 - a4 - 3a4z2 - 2a4z4 + a6z2

Kauffman Polynomial: - a-1z3 + a-1z5 + 2z2 - 7z4 + 4z6 + 3az3 - 11az5 + 6az7 - 2a2 + 6a2z2 - 8a2z4 - 2a2z6 + 4a2z8 + a3z + 5a3z3 - 14a3z5 + 7a3z7 + a3z9 - a4 + 4a4z2 - a4z4 - 5a4z6 + 5a4z8 + a5z - 3a5z3 + 2a5z5 + a5z7 + a5z9 - a6z2 + a6z4 + a6z6 + a6z8 - 4a7z3 + 4a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, -1}

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 11131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 5         1
j = 3        3 
j = 1       41 
j = -1      63  
j = -3     65   
j = -5    65    
j = -7   46     
j = -9  36      
j = -11 14       
j = -13 3        
j = -151         


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, 131]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 131]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[7, 16, 8, 17], 
 
>   X[9, 14, 10, 15], X[2, 11, 3, 12], X[20, 14, 21, 13], X[15, 8, 16, 9], 
 
>   X[22, 17, 1, 18], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 131]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -11, -4, 8, -5, -2, 6, -10, 7, 5, -8, 4, 9, -3, 10, 
 
>   -7, 11, -9]
In[5]:=
DTCode[Knot[11, NonAlternating, 131]]
Out[5]=   
DTCode[4, 10, 18, -16, -14, 2, 20, -8, 22, 12, 6]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 131]][t]
Out[6]=   
       -3   6    16             2    3
-21 + t   - -- + -- + 16 t - 6 t  + t
             2   t
            t
In[7]:=
Conway[Knot[11, NonAlternating, 131]][z]
Out[7]=   
     2    6
1 + z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 131], 
 
>   Knot[11, NonAlternating, 160]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 131]], KnotSignature[Knot[11, NonAlternating, 131]]}
Out[9]=   
{67, -2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 131]][q]
Out[10]=   
      -7   4    7    10   12   11   10          2
-7 + q   - -- + -- - -- + -- - -- + -- + 4 q - q
            6    5    4    3    2   q
           q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 131]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 131]][q]
Out[12]=   
 -22    -20    2     2     2     -12    -10    -8   3    2    3     2      4
q    - q    - --- + --- - --- + q    + q    - q   + -- - -- + -- - q  + 2 q  - 
               18    16    14                        6    4    2
              q     q     q                         q    q    q
 
     6
>   q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 131]][a, z]
Out[13]=   
   2    4    2      2  2      4  2    6  2    4      2  4      4  4    2  6
2 a  - a  - z  + 4 a  z  - 3 a  z  + a  z  - z  + 3 a  z  - 2 a  z  + a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 131]][a, z]
Out[14]=   
                                                                       3
    2    4    3      5        2      2  2      4  2    6  2    8  2   z
-2 a  - a  + a  z + a  z + 2 z  + 6 a  z  + 4 a  z  - a  z  - a  z  - -- + 
                                                                      a
 
         3      3  3      5  3      7  3      4      2  4    4  4    6  4
>   3 a z  + 5 a  z  - 3 a  z  - 4 a  z  - 7 z  - 8 a  z  - a  z  + a  z  + 
 
             5
     8  4   z          5       3  5      5  5      7  5      6      2  6
>   a  z  + -- - 11 a z  - 14 a  z  + 2 a  z  + 4 a  z  + 4 z  - 2 a  z  - 
            a
 
       4  6    6  6        7      3  7    5  7      2  8      4  8    6  8
>   5 a  z  + a  z  + 6 a z  + 7 a  z  + a  z  + 4 a  z  + 5 a  z  + a  z  + 
 
     3  9    5  9
>   a  z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 131]], Vassiliev[3][Knot[11, NonAlternating, 131]]}
Out[15]=   
{1, -1}
In[16]:=
Kh[Knot[11, NonAlternating, 131]][q, t]
Out[16]=   
5    6     1        3        1        4        3       6       4       6
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
 3   q    15  6    13  5    11  5    11  4    9  4    9  3    7  3    7  2
q        q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      6      5      6     3 t              2      3  2    5  3
>   ----- + ---- + ---- + --- + 4 q t + q t  + 3 q  t  + q  t
     5  2    5      3      q
    q  t    q  t   q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n131
K11n130
K11n130
K11n132
K11n132