© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n74
K11n74
K11n76
K11n76
K11n75
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   The Knot K11n75

Visit K11n75's page at Knotilus!

Acknowledgement

K11n75 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 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, K11n71, ...}

Determinant and Signature: {63, -2}

Jones Polynomial: q-9 - 3q-8 + 5q-7 - 9q-6 + 10q-5 - 10q-4 + 11q-3 - 7q-2 + 5q-1 - 2

Other knots (up to mirrors) with the same Jones Polynomial: {K11n71, ...}

A2 (sl(3)) Invariant: q-28 + q-26 - q-24 - 5q-20 - 3q-18 + 6q-12 + 2q-10 + 4q-8 + q-6 - 2q-4 + q-2 - 2

HOMFLY-PT Polynomial: - 3a2 - 5a2z2 - 2a2z4 + 11a4 + 18a4z2 + 10a4z4 + 2a4z6 - 9a6 - 10a6z2 - 3a6z4 + 2a8 + a8z2

Kauffman Polynomial: - 3az + 3az3 + 3a2 - 6a2z2 + 4a2z4 + a2z6 - 11a3z + 19a3z3 - 10a3z5 + 4a3z7 + 11a4 - 20a4z2 + 19a4z4 - 10a4z6 + 4a4z8 - 21a5z + 42a5z3 - 31a5z5 + 8a5z7 + a5z9 + 9a6 - 15a6z2 + 17a6z4 - 18a6z6 + 7a6z8 - 17a7z + 36a7z3 - 31a7z5 + 7a7z7 + a7z9 + 2a8 + a8z2 - a8z4 - 6a8z6 + 3a8z8 - 4a9z + 10a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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 1175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 1         2
j = -1        3 
j = -3       53 
j = -5      62  
j = -7     45   
j = -9    66    
j = -11   34     
j = -13  26      
j = -15 13       
j = -17 2        
j = -191         


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


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n75
K11n74
K11n74
K11n76
K11n76