© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n70
K11n70
K11n72
K11n72
K11n71
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   The Knot K11n71

Visit K11n71's page at Knotilus!

Acknowledgement

K11n71 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X14,6,15,5 X2837 X9,20,10,21 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X19,22,20,1 X21,10,22,11

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

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

Alexander Polynomial: 2t-3 - 7t-2 + 14t-1 - 17 + 14t - 7t2 + 2t3

Conway Polynomial: 1 + 4z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial: {1065, 1077, K11n75, ...}

Determinant and Signature: {63, 2}

Jones Polynomial: - 2 + 5q - 7q2 + 11q3 - 10q4 + 10q5 - 9q6 + 5q7 - 3q8 + q9

Other knots (up to mirrors) with the same Jones Polynomial: {K11n75, ...}

A2 (sl(3)) Invariant: - 2 + q2 - 2q4 + q6 + 4q8 + 2q10 + 6q12 - 3q18 - 5q20 - q24 + q26 + q28

HOMFLY-PT Polynomial: 2a-8 + a-8z2 - 9a-6 - 10a-6z2 - 3a-6z4 + 11a-4 + 18a-4z2 + 10a-4z4 + 2a-4z6 - 3a-2 - 5a-2z2 - 2a-2z4

Kauffman Polynomial: 2a-10z2 - 3a-10z4 + a-10z6 - 4a-9z + 10a-9z3 - 10a-9z5 + 3a-9z7 + 2a-8 + a-8z2 - a-8z4 - 6a-8z6 + 3a-8z8 - 17a-7z + 36a-7z3 - 31a-7z5 + 7a-7z7 + a-7z9 + 9a-6 - 15a-6z2 + 17a-6z4 - 18a-6z6 + 7a-6z8 - 21a-5z + 42a-5z3 - 31a-5z5 + 8a-5z7 + a-5z9 + 11a-4 - 20a-4z2 + 19a-4z4 - 10a-4z6 + 4a-4z8 - 11a-3z + 19a-3z3 - 10a-3z5 + 4a-3z7 + 3a-2 - 6a-2z2 + 4a-2z4 + a-2z6 - 3a-1z + 3a-1z3

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, 5}

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 1171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 19         1
j = 17        2 
j = 15       31 
j = 13      62  
j = 11     43   
j = 9    66    
j = 7   54     
j = 5  26      
j = 3 35       
j = 1 3        
j = -12         


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, 71]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 71]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], 
 
>   X[9, 20, 10, 21], X[16, 12, 17, 11], X[6, 14, 7, 13], X[18, 16, 19, 15], 
 
>   X[12, 18, 13, 17], X[19, 22, 20, 1], X[21, 10, 22, 11]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 71]]
Out[4]=   
GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -8, -10, 
 
>   5, -11, 10]
In[5]:=
DTCode[Knot[11, NonAlternating, 71]]
Out[5]=   
DTCode[4, 8, 14, 2, -20, 16, 6, 18, 12, -22, -10]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 71]][t]
Out[6]=   
      2    7    14             2      3
-17 + -- - -- + -- + 14 t - 7 t  + 2 t
       3    2   t
      t    t
In[7]:=
Conway[Knot[11, NonAlternating, 71]][z]
Out[7]=   
       2      4      6
1 + 4 z  + 5 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], 
 
>   Knot[11, NonAlternating, 75]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 71]], KnotSignature[Knot[11, NonAlternating, 71]]}
Out[9]=   
{63, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 71]][q]
Out[10]=   
              2       3       4       5      6      7      8    9
-2 + 5 q - 7 q  + 11 q  - 10 q  + 10 q  - 9 q  + 5 q  - 3 q  + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 71], Knot[11, NonAlternating, 75]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 71]][q]
Out[12]=   
      2      4    6      8      10      12      18      20    24    26    28
-2 + q  - 2 q  + q  + 4 q  + 2 q   + 6 q   - 3 q   - 5 q   - q   + q   + q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 71]][a, z]
Out[13]=   
                     2       2       2      2      4       4      4      6
2    9    11   3    z    10 z    18 z    5 z    3 z    10 z    2 z    2 z
-- - -- + -- - -- + -- - ----- + ----- - ---- - ---- + ----- - ---- + ----
 8    6    4    2    8     6       4       2      6      4       2      4
a    a    a    a    a     a       a       a      a      a       a      a
In[14]:=
Kauffman[Knot[11, NonAlternating, 71]][a, z]
Out[14]=   
                                                        2    2       2
2    9    11   3    4 z   17 z   21 z   11 z   3 z   2 z    z    15 z
-- + -- + -- + -- - --- - ---- - ---- - ---- - --- + ---- + -- - ----- - 
 8    6    4    2    9      7      5      3     a     10     8     6
a    a    a    a    a      a      a      a           a      a     a
 
        2      2       3       3       3       3      3      4    4       4
    20 z    6 z    10 z    36 z    42 z    19 z    3 z    3 z    z    17 z
>   ----- - ---- + ----- + ----- + ----- + ----- + ---- - ---- - -- + ----- + 
      4       2      9       7       5       3      a      10     8     6
     a       a      a       a       a       a             a      a     a
 
        4      4       5       5       5       5    6       6       6       6
    19 z    4 z    10 z    31 z    31 z    10 z    z     6 z    18 z    10 z
>   ----- + ---- - ----- - ----- - ----- - ----- + --- - ---- - ----- - ----- + 
      4       2      9       7       5       3      10     8      6       4
     a       a      a       a       a       a      a      a      a       a
 
     6      7      7      7      7      8      8      8    9    9
    z    3 z    7 z    8 z    4 z    3 z    7 z    4 z    z    z
>   -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + --
     2     9      7      5      3      8      6      4     7    5
    a     a      a      a      a      a      a      a     a    a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 71]], Vassiliev[3][Knot[11, NonAlternating, 71]]}
Out[15]=   
{4, 5}
In[16]:=
Kh[Knot[11, NonAlternating, 71]][q, t]
Out[16]=   
         3    2       3        5        5  2      7  2      7  3      9  3
3 q + 3 q  + --- + 5 q  t + 2 q  t + 6 q  t  + 5 q  t  + 4 q  t  + 6 q  t  + 
             q t
 
       9  4      11  4      11  5      13  5      13  6      15  6    15  7
>   6 q  t  + 4 q   t  + 3 q   t  + 6 q   t  + 2 q   t  + 3 q   t  + q   t  + 
 
       17  7    19  8
>   2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n71
K11n70
K11n70
K11n72
K11n72