© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a88
K11a88
K11a90
K11a90
K11a89
Knotscape
This page is passe. Go here instead!

   The Knot K11a89

Visit K11a89's page at Knotilus!

Acknowledgement

K11a89 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: 2t-3 - 12t-2 + 28t-1 - 35 + 28t - 12t2 + 2t3

Conway Polynomial: 1 - 2z2 + 2z6

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

Determinant and Signature: {119, 2}

Jones Polynomial: q-3 - 3q-2 + 7q-1 - 11 + 16q - 19q2 + 19q3 - 17q4 + 13q5 - 8q6 + 4q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial: {K11a34, ...}

A2 (sl(3)) Invariant: q-10 - q-6 + 3q-4 - q-2 + 3q2 - 4q4 + 2q6 - 2q8 + 2q12 - 3q14 + 4q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial: - a-6z2 - a-6z4 + a-4 + 2a-4z2 + 2a-4z4 + a-4z6 - a-2 - a-2z2 + a-2z4 + a-2z6 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial: - a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 5a-7z3 - 11a-7z5 + 7a-7z7 + 3a-6z2 - 2a-6z4 - 7a-6z6 + 7a-6z8 - 4a-5z + 15a-5z3 - 19a-5z5 + 5a-5z7 + 4a-5z9 + a-4 - 4a-4z2 + 15a-4z4 - 24a-4z6 + 12a-4z8 + a-4z10 - 4a-3z + 13a-3z3 - 12a-3z5 - 4a-3z7 + 7a-3z9 + a-2 - 5a-2z2 + 14a-2z4 - 21a-2z6 + 9a-2z8 + a-2z10 - 2a-1z + 10a-1z3 - 13a-1z5 + a-1z7 + 3a-1z9 + 3z2 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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=2 is the signature of 1189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17           1
j = 15          3 
j = 13         51 
j = 11        83  
j = 9       95   
j = 7      108    
j = 5     99     
j = 3    710      
j = 1   510       
j = -1  26        
j = -3 15         
j = -5 2          
j = -71           


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, 89]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 89]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], 
 
>   X[18, 9, 19, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[6, 16, 7, 15], 
 
>   X[20, 18, 21, 17], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=
GaussCode[Knot[11, Alternating, 89]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, 
 
>   -9, 11, -7]
In[5]:=
DTCode[Knot[11, Alternating, 89]]
Out[5]=   
DTCode[4, 10, 12, 16, 18, 2, 22, 6, 20, 8, 14]
In[6]:=
alex = Alexander[Knot[11, Alternating, 89]][t]
Out[6]=   
      2    12   28              2      3
-35 + -- - -- + -- + 28 t - 12 t  + 2 t
       3    2   t
      t    t
In[7]:=
Conway[Knot[11, Alternating, 89]][z]
Out[7]=   
       2      6
1 - 2 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 89]}
In[9]:=
{KnotDet[Knot[11, Alternating, 89]], KnotSignature[Knot[11, Alternating, 89]]}
Out[9]=   
{119, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 89]][q]
Out[10]=   
       -3   3    7              2       3       4       5      6      7    8
-11 + q   - -- + - + 16 q - 19 q  + 19 q  - 17 q  + 13 q  - 8 q  + 4 q  - q
             2   q
            q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 34], Knot[11, Alternating, 89]}
In[12]:=
A2Invariant[Knot[11, Alternating, 89]][q]
Out[12]=   
 -10    -6   3     -2      2      4      6      8      12      14      16
q    - q   + -- - q   + 3 q  - 4 q  + 2 q  - 2 q  + 2 q   - 3 q   + 4 q   - 
              4
             q
 
     18    20      22    24
>   q   - q   + 2 q   - q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 89]][a, z]
Out[13]=   
                         2      2    2                   4      4    4    6    6
 -4    -2    2      2   z    2 z    z     2  2      4   z    2 z    z    z    z
a   - a   + a  - 3 z  - -- + ---- - -- + a  z  - 2 z  - -- + ---- + -- + -- + --
                         6     4     2                   6     4     2    4    2
                        a     a     a                   a     a     a    a    a
In[14]:=
Kauffman[Knot[11, Alternating, 89]][a, z]
Out[14]=   
                                                        2      2      2
 -4    -2    2   z    4 z   4 z   2 z            2   2 z    3 z    4 z
a   + a   - a  - -- - --- - --- - --- - a z + 3 z  + ---- + ---- - ---- - 
                  7    5     3     a                   8      6      4
                 a    a     a                         a      a      a
 
       2              3      3       3       3       3               4      4
    5 z       2  2   z    5 z    15 z    13 z    10 z         3   6 z    2 z
>   ---- + 3 a  z  - -- + ---- + ----- + ----- + ----- + 6 a z  - ---- - ---- + 
      2               9     7      5       3       a                8      6
     a               a     a      a       a                        a      a
 
        4       4              5       5       5       5       5
    15 z    14 z       2  4   z    11 z    19 z    12 z    13 z         5
>   ----- + ----- - 3 a  z  + -- - ----- - ----- - ----- - ----- - 8 a z  - 
      4       2                9     7       5       3       a
     a       a                a     a       a       a
 
              6      6       6       6              7      7      7    7
       6   4 z    7 z    24 z    21 z     2  6   7 z    5 z    4 z    z
>   7 z  + ---- - ---- - ----- - ----- + a  z  + ---- + ---- - ---- + -- + 
             8      6      4       2               7      5      3    a
            a      a      a       a               a      a      a
 
                       8       8      8      9      9      9    10    10
         7      8   7 z    12 z    9 z    4 z    7 z    3 z    z     z
>   3 a z  + 4 z  + ---- + ----- + ---- + ---- + ---- + ---- + --- + ---
                      6      4       2      5      3     a      4     2
                     a      a       a      a      a            a     a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 89]], Vassiliev[3][Knot[11, Alternating, 89]]}
Out[15]=   
{-2, -1}
In[16]:=
Kh[Knot[11, Alternating, 89]][q, t]
Out[16]=   
          3     1       2       1       5      2      6    5 q       3
10 q + 7 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q  t + 
               7  4    5  3    3  3    3  2      2   q t    t
              q  t    q  t    q  t    q  t    q t
 
       5        5  2       7  2      7  3      9  3      9  4      11  4
>   9 q  t + 9 q  t  + 10 q  t  + 8 q  t  + 9 q  t  + 5 q  t  + 8 q   t  + 
 
       11  5      13  5    13  6      15  6    17  7
>   3 q   t  + 5 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a89
K11a88
K11a88
K11a90
K11a90