© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n131
K11n131 		K11n133
K11n133
K11n132
Knotscape 		This page is passe. Go here instead!

   The Knot K11n132

Visit K11n132's page at Knotilus!

Acknowledgement

K11n132 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X5,19,6,18 X7,16,8,17 X9,14,10,15 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6

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: 2t-2 - 6t-1 + 9 - 6t + 2t2

Conway Polynomial: 1 + 2z2 + 2z4

Other knots with the same Alexander/Conway Polynomial: {88, 10129, K11n39, K11n45, K11n50, ...}

Determinant and Signature: {25, 0}

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

Other knots (up to mirrors) with the same Jones Polynomial: {K11n50, K11n133, ...}

A2 (sl(3)) Invariant: - q-22 - q-16 + q-14 + q-10 + q-8 + q-4 + 1 + q2 - q4

HOMFLY-PT Polynomial: - z2 + a2 + 2a2z2 + a2z4 + a4 + 2a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial: a-1z + 2z2 + 3az - 5az3 + 2az5 - a2 + a2z2 + a2z4 - 3a2z6 + a2z8 + 3a3z - 8a3z3 + 6a3z5 - 4a3z7 + a3z9 + a4 - 7a4z2 + 15a4z4 - 13a4z6 + 3a4z8 - a5z + 4a5z3 - a5z5 - 3a5z7 + a5z9 + a6 - 6a6z2 + 14a6z4 - 10a6z6 + 2a6z8 - 2a7z + 7a7z3 - 5a7z5 + a7z7

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

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 11132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 3        1
j = 1       2 
j = -1      22 
j = -3     21  
j = -5    22   
j = -7   22    
j = -9  12     
j = -11 12      
j = -13 1       
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, 132]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 132]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 19, 6, 18], X[7, 16, 8, 17], 
 
>   X[9, 14, 10, 15], X[2, 11, 3, 12], X[13, 20, 14, 21], X[15, 8, 16, 9], 
 
>   X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 7, 22, 6]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 132]]
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, 132]]
Out[5]=   
DTCode[4, 10, -18, -16, -14, 2, -20, -8, -22, -12, -6]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 132]][t]
Out[6]=   
    2    6            2
9 + -- - - - 6 t + 2 t
     2   t
    t
In[7]:=
Conway[Knot[11, NonAlternating, 132]][z]
Out[7]=   
       2      4
1 + 2 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[8, 8], Knot[10, 129], Knot[11, NonAlternating, 39], 
 
>   Knot[11, NonAlternating, 45], Knot[11, NonAlternating, 50], 
 
>   Knot[11, NonAlternating, 132]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 132]], KnotSignature[Knot[11, NonAlternating, 132]]}
Out[9]=   
{25, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 132]][q]
Out[10]=   
     -7   2    3    4    4    4    3
3 - 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, 50], Knot[11, NonAlternating, 132], 
 
>   Knot[11, NonAlternating, 133]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 132]][q]
Out[12]=   
     -22    -16    -14    -10    -8    -4    2    4
1 - q    - q    + q    + q    + q   + q   + q  - q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 132]][a, z]
Out[13]=   
 2    4    6    2      2  2      4  2    6  2    2  4    4  4
a  + a  - a  - z  + 2 a  z  + 2 a  z  - a  z  + a  z  + a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 132]][a, z]
Out[14]=   
  2    4    6   z              3      5        7        2    2  2      4  2
-a  + a  + a  + - + 3 a z + 3 a  z - a  z - 2 a  z + 2 z  + a  z  - 7 a  z  - 
                a
 
       6  2        3      3  3      5  3      7  3    2  4       4  4
>   6 a  z  - 5 a z  - 8 a  z  + 4 a  z  + 7 a  z  + a  z  + 15 a  z  + 
 
        6  4        5      3  5    5  5      7  5      2  6       4  6
>   14 a  z  + 2 a z  + 6 a  z  - a  z  - 5 a  z  - 3 a  z  - 13 a  z  - 
 
        6  6      3  7      5  7    7  7    2  8      4  8      6  8    3  9
>   10 a  z  - 4 a  z  - 3 a  z  + a  z  + a  z  + 3 a  z  + 2 a  z  + a  z  + 
 
     5  9
>   a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 132]], Vassiliev[3][Knot[11, NonAlternating, 132]]}
Out[15]=   
{2, -3}
In[16]:=
Kh[Knot[11, NonAlternating, 132]][q, t]
Out[16]=   
2           1        1        1        2        1       2       2       2
- + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
q          15  7    13  6    11  6    11  5    9  5    9  4    7  4    7  3
          q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      2       2       2      1      2     3
>   ----- + ----- + ----- + ---- + --- + q  t
     5  3    5  2    3  2    3     q t
    q  t    q  t    q  t    q  t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n132
K11n131
K11n131 		K11n133
K11n133