© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n159
K11n159
K11n161
K11n161
K11n160
Knotscape
This page is passe. Go here instead!

   The Knot K11n160

Visit K11n160's page at Knotilus!

Acknowledgement

K11n160 as Morse Link
DrawMorseLink

PD Presentation: X6271 X3,11,4,10 X12,6,13,5 X20,8,21,7 X16,10,17,9 X11,19,12,18 X22,13,1,14 X8,16,9,15 X17,4,18,5 X2,19,3,20 X14,21,15,22

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

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

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, K11n131, ...}

Determinant and Signature: {67, 2}

Jones Polynomial: 2q-1 - 5 + 8q - 10q2 + 12q3 - 11q4 + 9q5 - 6q6 + 3q7 - q8

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

A2 (sl(3)) Invariant: 2q-4 - 1 + 2q2 - 3q4 + q6 + 3q12 - q14 + 3q16 - q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial: - 2a-6 - 2a-6z2 - a-6z4 + 5a-4 + 8a-4z2 + 4a-4z4 + a-4z6 - 4a-2 - 7a-2z2 - 3a-2z4 + 2 + 2z2

Kauffman Polynomial: - 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 - 3a-7z + 9a-7z3 - 12a-7z5 + 5a-7z7 + 2a-6 - 8a-6z2 + 15a-6z4 - 13a-6z6 + 5a-6z8 - 4a-5z + 14a-5z3 - 10a-5z5 + a-5z7 + 2a-5z9 + 5a-4 - 21a-4z2 + 35a-4z4 - 24a-4z6 + 8a-4z8 - 4a-3z3 + 6a-3z5 - 3a-3z7 + 2a-3z9 + 4a-2 - 17a-2z2 + 17a-2z4 - 8a-2z6 + 3a-2z8 + a-1z - 7a-1z3 + 3a-1z5 + a-1z7 + 2 - 5z2 + 3z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 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=2 is the signature of 11160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17         1
j = 15        2 
j = 13       41 
j = 11      52  
j = 9     64   
j = 7    65    
j = 5   46     
j = 3  46      
j = 1 25       
j = -1 3        
j = -32         


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, 160]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 160]]
Out[3]=   
PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[12, 6, 13, 5], X[20, 8, 21, 7], 
 
>   X[16, 10, 17, 9], X[11, 19, 12, 18], X[22, 13, 1, 14], X[8, 16, 9, 15], 
 
>   X[17, 4, 18, 5], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 160]]
Out[4]=   
GaussCode[1, -10, -2, 9, 3, -1, 4, -8, 5, 2, -6, -3, 7, -11, 8, -5, -9, 6, 10, 
 
>   -4, 11, -7]
In[5]:=
DTCode[Knot[11, NonAlternating, 160]]
Out[5]=   
DTCode[6, -10, 12, 20, 16, -18, 22, 8, -4, 2, 14]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 160]][t]
Out[6]=   
       -3   6    16             2    3
-21 + t   - -- + -- + 16 t - 6 t  + t
             2   t
            t
In[7]:=
Conway[Knot[11, NonAlternating, 160]][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, 160]], KnotSignature[Knot[11, NonAlternating, 160]]}
Out[9]=   
{67, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 160]][q]
Out[10]=   
     2             2       3       4      5      6      7    8
-5 + - + 8 q - 10 q  + 12 q  - 11 q  + 9 q  - 6 q  + 3 q  - q
     q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 160]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 160]][q]
Out[12]=   
     2       2      4    6      12    14      16    18      20    22    24
-1 + -- + 2 q  - 3 q  + q  + 3 q   - q   + 3 q   - q   - 2 q   + q   - q
      4
     q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 160]][a, z]
Out[13]=   
                             2      2      2    4      4      4    6
    2    5    4       2   2 z    8 z    7 z    z    4 z    3 z    z
2 - -- + -- - -- + 2 z  - ---- + ---- - ---- - -- + ---- - ---- + --
     6    4    2            6      4      2     6     4      2     4
    a    a    a            a      a      a     a     a      a     a
In[14]:=
Kauffman[Knot[11, NonAlternating, 160]][a, z]
Out[14]=   
                                           2      2       2       2      3
    2    5    4    3 z   4 z   z      2   z    8 z    21 z    17 z    2 z
2 + -- + -- + -- - --- - --- + - - 5 z  + -- - ---- - ----- - ----- - ---- + 
     6    4    2    7     5    a           8     6      4       2       9
    a    a    a    a     a                a     a      a       a       a
 
       3       3      3      3             4       4       4       4    5
    9 z    14 z    4 z    7 z       4   6 z    15 z    35 z    17 z    z
>   ---- + ----- - ---- - ---- + 3 z  - ---- + ----- + ----- + ----- + -- - 
      7      5       3     a              8      6       4       2      9
     a      a       a                    a      a       a       a      a
 
        5       5      5      5      6       6       6      6      7    7
    12 z    10 z    6 z    3 z    3 z    13 z    24 z    8 z    5 z    z
>   ----- - ----- + ---- + ---- + ---- - ----- - ----- - ---- + ---- + -- - 
      7       5       3     a       8      6       4       2      7     5
     a       a       a             a      a       a       a      a     a
 
       7    7      8      8      8      9      9
    3 z    z    5 z    8 z    3 z    2 z    2 z
>   ---- + -- + ---- + ---- + ---- + ---- + ----
      3    a      6      4      2      5      3
     a           a      a      a      a      a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 160]], Vassiliev[3][Knot[11, NonAlternating, 160]]}
Out[15]=   
{1, 3}
In[16]:=
Kh[Knot[11, NonAlternating, 160]][q, t]
Out[16]=   
         3     2      3    2 q      3        5        5  2      7  2
5 q + 4 q  + ----- + --- + --- + 6 q  t + 4 q  t + 6 q  t  + 6 q  t  + 
              3  2   q t    t
             q  t
 
       7  3      9  3      9  4      11  4      11  5      13  5    13  6
>   5 q  t  + 6 q  t  + 4 q  t  + 5 q   t  + 2 q   t  + 4 q   t  + q   t  + 
 
       15  6    17  7
>   2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n160
K11n159
K11n159
K11n161
K11n161