© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n86
K11n86 		K11n88
K11n88
K11n87
Knotscape 		This page is passe. Go here instead!

   The Knot K11n87

Visit K11n87's page at Knotilus!

Acknowledgement

K11n87 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X17,22,18,1 X6,19,7,20 X21,16,22,17

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

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

Alexander Polynomial: t-3 - 5t-2 + 12t-1 - 15 + 12t - 5t2 + t3

Conway Polynomial: 1 + z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial: {928, 929, 10163, ...}

Determinant and Signature: {51, -2}

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

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

A2 (sl(3)) Invariant: q-28 - q-24 + q-22 - 2q-20 - 2q-14 + q-12 - q-10 + 3q-8 + 2q-6 + 2q-2 - 1

HOMFLY-PT Polynomial: a2 - a2z2 - a2z4 + 2a4 + 6a4z2 + 4a4z4 + a4z6 - 3a6 - 5a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial: az3 - a2 - 3a2z2 + 4a2z4 + a3z + a3z5 + a3z7 + 2a4 - 10a4z2 + 12a4z4 - 5a4z6 + 2a4z8 - a5z + 5a5z3 - 5a5z5 + a5z7 + a5z9 + 3a6 - 9a6z2 + 14a6z4 - 14a6z6 + 5a6z8 - 4a7z + 14a7z3 - 16a7z5 + 3a7z7 + a7z9 + a8 + 3a8z4 - 8a8z6 + 3a8z8 - 2a9z + 8a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 0}

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 1187. 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         1
j = -1        3 
j = -3       32 
j = -5      52  
j = -7     43   
j = -9    45    
j = -11   34     
j = -13  24      
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, 87]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 87]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], 
 
>   X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 15, 9, 16], 
 
>   X[17, 22, 18, 1], X[6, 19, 7, 20], X[21, 16, 22, 17]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 87]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, 11, -9, -5, 10, 
 
>   -7, -11, 9]
In[5]:=
DTCode[Knot[11, NonAlternating, 87]]
Out[5]=   
DTCode[4, 10, 12, 14, 18, 2, 20, 8, -22, 6, -16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 87]][t]
Out[6]=   
       -3   5    12             2    3
-15 + t   - -- + -- + 12 t - 5 t  + t
             2   t
            t
In[7]:=
Conway[Knot[11, NonAlternating, 87]][z]
Out[7]=   
     2    4    6
1 + z  + z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 87]], KnotSignature[Knot[11, NonAlternating, 87]]}
Out[9]=   
{51, -2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 87]][q]
Out[10]=   
      -9   3    5    7    8    9    8    5    4
-1 + 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, 87]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 87]][q]
Out[12]=   
      -28    -24    -22    2     2     -12    -10   3    2    2
-1 + q    - q    + q    - --- - --- + q    - q    + -- + -- + --
                           20    14                  8    6    2
                          q     q                   q    q    q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 87]][a, z]
Out[13]=   
 2      4      6    8    2  2      4  2      6  2    8  2    2  4      4  4
a  + 2 a  - 3 a  + a  - a  z  + 6 a  z  - 5 a  z  + a  z  - a  z  + 4 a  z  - 
 
       6  4    4  6
>   2 a  z  + a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 87]][a, z]
Out[14]=   
  2      4      6    8    3      5        7        9        2  2       4  2
-a  + 2 a  + 3 a  + a  + a  z - a  z - 4 a  z - 2 a  z - 3 a  z  - 10 a  z  - 
 
       6  2      10  2      3      5  3       7  3      9  3      2  4
>   9 a  z  + 2 a   z  + a z  + 5 a  z  + 14 a  z  + 8 a  z  + 4 a  z  + 
 
        4  4       6  4      8  4      10  4    3  5      5  5       7  5
>   12 a  z  + 14 a  z  + 3 a  z  - 3 a   z  + a  z  - 5 a  z  - 16 a  z  - 
 
        9  5      4  6       6  6      8  6    10  6    3  7    5  7
>   10 a  z  - 5 a  z  - 14 a  z  - 8 a  z  + a   z  + a  z  + a  z  + 
 
       7  7      9  7      4  8      6  8      8  8    5  9    7  9
>   3 a  z  + 3 a  z  + 2 a  z  + 5 a  z  + 3 a  z  + a  z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 87]], Vassiliev[3][Knot[11, NonAlternating, 87]]}
Out[15]=   
{1, 0}
In[16]:=
Kh[Knot[11, NonAlternating, 87]][q, t]
Out[16]=   
2    3     1        2        1        3        2        4        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        4       5       4       3       5      2      3
>   ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 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 K11n87
K11n86
K11n86 		K11n88
K11n88