© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a266
K11a266
K11a268
K11a268
K11a267
Knotscape
This page is passe. Go here instead!

   The Knot K11a267

Visit K11a267's page at Knotilus!

Acknowledgement

K11a267 as Morse Link
DrawMorseLink

PD Presentation: X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X18,11,19,12 X22,13,1,14 X8,15,9,16 X4,18,5,17 X2,19,3,20 X16,21,17,22

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

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

Alexander Polynomial: - t-4 + 7t-3 - 22t-2 + 41t-1 - 49 + 41t - 22t2 + 7t3 - t4

Conway Polynomial: 1 - z6 - z8

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

Determinant and Signature: {191, -2}

Jones Polynomial: - q-8 + 5q-7 - 12q-6 + 20q-5 - 27q-4 + 31q-3 - 31q-2 + 27q-1 - 19 + 12q - 5q2 + q3

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

A2 (sl(3)) Invariant: - q-24 + 2q-22 - 3q-18 + 5q-16 - 5q-14 + 2q-12 + q-10 - 4q-8 + 6q-6 - 6q-4 + 6q-2 - 2q2 + 4q4 - 3q6 + q8

HOMFLY-PT Polynomial: 1 + z2 + 2z4 + z6 - 3a2z2 - 6a2z4 - 4a2z6 - a2z8 + 3a4z2 + 5a4z4 + 2a4z6 - a6z2 - a6z4

Kauffman Polynomial: - a-2z4 + a-2z6 + 2a-1z3 - 8a-1z5 + 5a-1z7 + 1 - 3z2 + 14z4 - 23z6 + 11z8 + 8az5 - 21az7 + 12az9 - 9a2z2 + 42a2z4 - 54a2z6 + 15a2z8 + 5a2z10 - 2a3z3 + 26a3z5 - 51a3z7 + 26a3z9 - 9a4z2 + 42a4z4 - 60a4z6 + 21a4z8 + 5a4z10 + 4a5z3 - 5a5z5 - 13a5z7 + 14a5z9 - 3a6z2 + 12a6z4 - 25a6z6 + 17a6z8 + 4a7z3 - 14a7z5 + 12a7z7 - 3a8z4 + 5a8z6 + a9z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {0, 0}

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 11267. 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 = 7           1
j = 5          4 
j = 3         81 
j = 1        114  
j = -1       168   
j = -3      1612    
j = -5     1515     
j = -7    1216      
j = -9   815       
j = -11  412        
j = -13 18         
j = -15 4          
j = -171           


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, 267]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 267]]
Out[3]=   
PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], 
 
>   X[20, 10, 21, 9], X[18, 11, 19, 12], X[22, 13, 1, 14], X[8, 15, 9, 16], 
 
>   X[4, 18, 5, 17], X[2, 19, 3, 20], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 267]]
Out[4]=   
GaussCode[1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -6, 10, 
 
>   -5, 11, -7]
In[5]:=
DTCode[Knot[11, Alternating, 267]]
Out[5]=   
DTCode[6, 10, 12, 14, 20, 18, 22, 8, 4, 2, 16]
In[6]:=
alex = Alexander[Knot[11, Alternating, 267]][t]
Out[6]=   
       -4   7    22   41              2      3    4
-49 - t   + -- - -- + -- + 41 t - 22 t  + 7 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 267]][z]
Out[7]=   
     6    8
1 - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 267]}
In[9]:=
{KnotDet[Knot[11, Alternating, 267]], KnotSignature[Knot[11, Alternating, 267]]}
Out[9]=   
{191, -2}
In[10]:=
J=Jones[Knot[11, Alternating, 267]][q]
Out[10]=   
       -8   5    12   20   27   31   31   27             2    3
-19 - q   + -- - -- + -- - -- + -- - -- + -- + 12 q - 5 q  + q
             7    6    5    4    3    2   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, 267]}
In[12]:=
A2Invariant[Knot[11, Alternating, 267]][q]
Out[12]=   
  -24    2     3     5     5     2     -10   4    6    6    6       2      4
-q    + --- - --- + --- - --- + --- + q    - -- + -- - -- + -- - 2 q  + 4 q  - 
         22    18    16    14    12           8    6    4    2
        q     q     q     q     q            q    q    q    q
 
       6    8
>   3 q  + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 267]][a, z]
Out[13]=   
     2      2  2      4  2    6  2      4      2  4      4  4    6  4    6
1 + z  - 3 a  z  + 3 a  z  - a  z  + 2 z  - 6 a  z  + 5 a  z  - a  z  + z  - 
 
       2  6      4  6    2  8
>   4 a  z  + 2 a  z  - a  z
In[14]:=
Kauffman[Knot[11, Alternating, 267]][a, z]
Out[14]=   
                                            3
       2      2  2      4  2      6  2   2 z       3  3      5  3      7  3
1 - 3 z  - 9 a  z  - 9 a  z  - 3 a  z  + ---- - 2 a  z  + 4 a  z  + 4 a  z  + 
                                          a
 
             4                                                 5
        4   z        2  4       4  4       6  4      8  4   8 z         5
>   14 z  - -- + 42 a  z  + 42 a  z  + 12 a  z  - 3 a  z  - ---- + 8 a z  + 
             2                                               a
            a
 
                                                     6
        3  5      5  5       7  5    9  5       6   z        2  6       4  6
>   26 a  z  - 5 a  z  - 14 a  z  + a  z  - 23 z  + -- - 54 a  z  - 60 a  z  - 
                                                     2
                                                    a
 
                            7
        6  6      8  6   5 z          7       3  7       5  7       7  7
>   25 a  z  + 5 a  z  + ---- - 21 a z  - 51 a  z  - 13 a  z  + 12 a  z  + 
                          a
 
        8       2  8       4  8       6  8         9       3  9       5  9
>   11 z  + 15 a  z  + 21 a  z  + 17 a  z  + 12 a z  + 26 a  z  + 14 a  z  + 
 
       2  10      4  10
>   5 a  z   + 5 a  z
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 267]], Vassiliev[3][Knot[11, Alternating, 267]]}
Out[15]=   
{0, 0}
In[16]:=
Kh[Knot[11, Alternating, 267]][q, t]
Out[16]=   
12   16     1        4        1        8        4        12       8      15
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 3   q     17  7    15  6    13  6    13  5    11  5    11  4    9  4    9  3
q         q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
     12      16      15      15     16    8 t                 2      3  2
>   ----- + ----- + ----- + ---- + ---- + --- + 11 q t + 4 q t  + 8 q  t  + 
     7  3    7  2    5  2    5      3      q
    q  t    q  t    q  t    q  t   q  t
 
     3  3      5  3    7  4
>   q  t  + 4 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a267
K11a266
K11a266
K11a268
K11a268