© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n101
K11n101 		K11n103
K11n103
K11n102
Knotscape 		This page is passe. Go here instead!

   The Knot K11n102

Visit K11n102's page at Knotilus!

Acknowledgement

K11n102 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,9,20,8 X21,16,22,17

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

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

Alexander Polynomial: - t-2 + t-1 + 1 + t - t2

Conway Polynomial: 1 - 3z2 - z4

Other knots with the same Alexander/Conway Polynomial: {K11n38, ...}

Determinant and Signature: {3, -2}

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

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

A2 (sl(3)) Invariant: q-26 + q-24 - q-18 - q-14 - q-12 - q-8 + q-6 + q-2 + 1 + q2 + q4

HOMFLY-PT Polynomial: 2 + z2 - a2 - 3a2z2 - a2z4 - a6 - a6z2 + a8

Kauffman Polynomial: 2 - 3z2 + z4 - az - 2az3 + az5 + a2 - 5a2z2 + 2a2z4 - 6a4z2 + 13a4z4 - 7a4z6 + a4z8 + 6a5z - 11a5z3 + 14a5z5 - 7a5z7 + a5z9 + a6 - 15a6z2 + 27a6z4 - 14a6z6 + 2a6z8 + 5a7z - 13a7z3 + 15a7z5 - 7a7z7 + a7z9 + a8 - 11a8z2 + 15a8z4 - 7a8z6 + a8z8

V2 and V3, the type 2 and 3 Vassiliev invariants: {-3, 6}

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 11102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 3          1
j = 1           
j = -1        21 
j = -3      111  
j = -5      11   
j = -7    121    
j = -9   1 1     
j = -11   11      
j = -13 11        
j = -15           
j = -171          

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, 102]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 102]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 12, 8, 13], 
 
>   X[9, 19, 10, 18], X[2, 11, 3, 12], X[13, 6, 14, 7], X[15, 22, 16, 1], 
 
>   X[17, 20, 18, 21], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 102]]
Out[4]=   
GaussCode[1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 
 
>   9, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 102]]
Out[5]=   
DTCode[4, 10, -14, -12, -18, 2, -6, -22, -20, -8, -16]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 102]][t]
Out[6]=   
     -2   1        2
1 - t   + - + t - t
          t
In[7]:=
Conway[Knot[11, NonAlternating, 102]][z]
Out[7]=   
       2    4
1 - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 38], Knot[11, NonAlternating, 102]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 102]], KnotSignature[Knot[11, NonAlternating, 102]]}
Out[9]=   
{3, -2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 102]][q]
Out[10]=   
      -8    -7    -6    -5    -4    -3    -2   2
-1 + q   - 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, 102]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 102]][q]
Out[12]=   
     -26    -24    -18    -14    -12    -8    -6    -2    2    4
1 + q    + q    - q    - q    - q    - q   + q   + q   + q  + q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 102]][a, z]
Out[13]=   
     2    6    8    2      2  2    6  2    2  4
2 - a  - a  + a  + z  - 3 a  z  - a  z  - a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 102]][a, z]
Out[14]=   
     2    6    8            5        7        2      2  2      4  2
2 + a  + a  + a  - a z + 6 a  z + 5 a  z - 3 z  - 5 a  z  - 6 a  z  - 
 
        6  2       8  2        3       5  3       7  3    4      2  4
>   15 a  z  - 11 a  z  - 2 a z  - 11 a  z  - 13 a  z  + z  + 2 a  z  + 
 
        4  4       6  4       8  4      5       5  5       7  5      4  6
>   13 a  z  + 27 a  z  + 15 a  z  + a z  + 14 a  z  + 15 a  z  - 7 a  z  - 
 
        6  6      8  6      5  7      7  7    4  8      6  8    8  8    5  9
>   14 a  z  - 7 a  z  - 7 a  z  - 7 a  z  + a  z  + 2 a  z  + a  z  + a  z  + 
 
     7  9
>   a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 102]], Vassiliev[3][Knot[11, NonAlternating, 102]]}
Out[15]=   
{-3, 6}
In[16]:=
Kh[Knot[11, NonAlternating, 102]][q, t]
Out[16]=   
 -3   2     1        1        1        1        1       1        1       1
q   + - + ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 
      q    17  8    13  7    13  6    11  5    9  5    11  4    7  4    9  3
          q   t    q   t    q   t    q   t    q  t    q   t    q  t    q  t
 
      2       1       1       1      1      1     t    3  2
>   ----- + ----- + ----- + ----- + ---- + ---- + - + q  t
     7  3    7  2    5  2    3  2    5      3     q
    q  t    q  t    q  t    q  t    q  t   q  t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n102
K11n101
K11n101 		K11n103
K11n103