© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a79
K11a79
K11a81
K11a81
K11a80
Knotscape
This page is passe. Go here instead!

   The Knot K11a80

Visit K11a80's page at Knotilus!

Acknowledgement

K11a80 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22

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

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

Alexander Polynomial: t-4 - 6t-3 + 16t-2 - 28t-1 + 35 - 28t + 16t2 - 6t3 + t4

Conway Polynomial: 1 - 2z2 + 2z6 + z8

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

Determinant and Signature: {137, 0}

Jones Polynomial: q-6 - 4q-5 + 9q-4 - 14q-3 + 19q-2 - 22q-1 + 22 - 19q + 14q2 - 8q3 + 4q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial: {K11a270, ...}

A2 (sl(3)) Invariant: q-18 - q-16 + 2q-12 - 3q-10 + 4q-8 - q-6 - q-4 + 2q-2 - 5 + 4q2 - 3q4 + 2q6 + 3q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial: a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + 5z2 + 9z4 + 5z6 + z8 - a2 - 7a2z2 - 7a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial: - a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 4a-3z3 - 10a-3z5 + 7a-3z7 - a-2 + 3a-2z2 - 9a-2z6 + 8a-2z8 - 4a-1z + 13a-1z3 - 12a-1z5 - a-1z7 + 6a-1z9 - 5z2 + 27z4 - 34z6 + 13z8 + 2z10 - 4az + 11az3 + az5 - 19az7 + 12az9 + a2 - 11a2z2 + 33a2z4 - 38a2z6 + 12a2z8 + 2a2z10 - 2a3z + 8a3z3 - 7a3z5 - 7a3z7 + 6a3z9 + a4 - 4a4z2 + 10a4z4 - 16a4z6 + 7a4z8 - a5z + 5a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {-2, 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=0 is the signature of 1180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 11           1
j = 9          3 
j = 7         51 
j = 5        93  
j = 3       105   
j = 1      129    
j = -1     1111     
j = -3    811      
j = -5   611       
j = -7  38        
j = -9 16         
j = -11 3          
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, Alternating, 80]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 80]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], 
 
>   X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 14, 19, 13], X[20, 15, 21, 16], 
 
>   X[6, 17, 7, 18], X[8, 20, 9, 19], X[16, 21, 17, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 80]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, 
 
>   -8, 11, -5]
In[5]:=
DTCode[Knot[11, Alternating, 80]]
Out[5]=   
DTCode[4, 10, 12, 14, 22, 2, 18, 20, 6, 8, 16]
In[6]:=
alex = Alexander[Knot[11, Alternating, 80]][t]
Out[6]=   
      -4   6    16   28              2      3    4
35 + t   - -- + -- - -- - 28 t + 16 t  - 6 t  + t
            3    2   t
           t    t
In[7]:=
Conway[Knot[11, Alternating, 80]][z]
Out[7]=   
       2      6    8
1 - 2 z  + 2 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 80]}
In[9]:=
{KnotDet[Knot[11, Alternating, 80]], KnotSignature[Knot[11, Alternating, 80]]}
Out[9]=   
{137, 0}
In[10]:=
J=Jones[Knot[11, Alternating, 80]][q]
Out[10]=   
      -6   4    9    14   19   22              2      3      4    5
22 + q   - -- + -- - -- + -- - -- - 19 q + 14 q  - 8 q  + 4 q  - q
            5    4    3    2   q
           q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 80], Knot[11, Alternating, 270]}
In[12]:=
A2Invariant[Knot[11, Alternating, 80]][q]
Out[12]=   
      -18    -16    2     3    4     -6    -4   2       2      4      6
-5 + q    - q    + --- - --- + -- - q   - q   + -- + 4 q  - 3 q  + 2 q  + 
                    12    10    8                2
                   q     q     q                q
 
       8      10      12    14
>   3 q  - 2 q   + 2 q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 80]][a, z]
Out[13]=   
                          2                                 4
 -2    2    4      2   2 z       2  2      4  2      4   3 z       2  4
a   - a  + a  + 5 z  - ---- - 7 a  z  + 2 a  z  + 9 z  - ---- - 7 a  z  + 
                         2                                 2
                        a                                 a
 
                    6
     4  4      6   z       2  6    8
>   a  z  + 5 z  - -- - 2 a  z  + z
                    2
                   a
In[14]:=
Kauffman[Knot[11, Alternating, 80]][a, z]
Out[14]=   
                                                              2      2
  -2    2    4   z    4 z              3      5        2   2 z    3 z
-a   + a  + a  - -- - --- - 4 a z - 2 a  z - a  z - 5 z  + ---- + ---- - 
                  3    a                                     4      2
                 a                                          a      a
 
                                  3      3       3
        2  2      4  2    6  2   z    4 z    13 z          3      3  3
>   11 a  z  - 4 a  z  + a  z  - -- + ---- + ----- + 11 a z  + 8 a  z  + 
                                  5     3      a
                                 a     a
 
                         4                                    5       5
       5  3       4   6 z        2  4       4  4      6  4   z    10 z
>   5 a  z  + 27 z  - ---- + 33 a  z  + 10 a  z  - 2 a  z  + -- - ----- - 
                        4                                     5     3
                       a                                     a     a
 
        5                                         6      6
    12 z       5      3  5      5  5       6   4 z    9 z        2  6
>   ----- + a z  - 7 a  z  - 9 a  z  - 34 z  + ---- - ---- - 38 a  z  - 
      a                                          4      2
                                                a      a
 
                          7    7                                            8
        4  6    6  6   7 z    z          7      3  7      5  7       8   8 z
>   16 a  z  + a  z  + ---- - -- - 19 a z  - 7 a  z  + 4 a  z  + 13 z  + ---- + 
                         3    a                                            2
                        a                                                 a
 
                            9
        2  8      4  8   6 z          9      3  9      10      2  10
>   12 a  z  + 7 a  z  + ---- + 12 a z  + 6 a  z  + 2 z   + 2 a  z
                          a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 80]], Vassiliev[3][Knot[11, Alternating, 80]]}
Out[15]=   
{-2, 1}
In[16]:=
Kh[Knot[11, Alternating, 80]][q, t]
Out[16]=   
11            1        3        1       6       3       8       6      11
-- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q            13  6    11  5    9  5    9  4    7  4    7  3    5  3    5  2
            q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
      8      11    11                3        3  2      5  2      5  3
>   ----- + ---- + --- + 9 q t + 10 q  t + 5 q  t  + 9 q  t  + 3 q  t  + 
     3  2    3     q t
    q  t    q  t
 
       7  3    7  4      9  4    11  5
>   5 q  t  + q  t  + 3 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a80
K11a79
K11a79
K11a81
K11a81