© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n18
K11n18 		K11n20
K11n20
K11n19
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   The Knot K11n19

Visit K11n19's page at Knotilus!

Acknowledgement

K11n19 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8394 X10,6,11,5 X7,17,8,16 X2,9,3,10 X11,18,12,19 X13,20,14,21 X15,22,16,1 X17,7,18,6 X19,12,20,13 X21,14,22,15

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

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

Alexander Polynomial: - t-3 + 2t-2 - 1 + 2t2 - t3

Conway Polynomial: 1 - z2 - 4z4 - z6

Other knots with the same Alexander/Conway Polynomial: {K11n135, ...}

Determinant and Signature: {5, -4}

Jones Polynomial: q-2 - q-1 + 1 - q + q2

Other knots (up to mirrors) with the same Jones Polynomial: {41, ...}

A2 (sl(3)) Invariant: - q-20 + q-16 + q-14 + q-12 - q-6 - q-4 + q2 + q4 + q6

HOMFLY-PT Polynomial: 3 + 4z2 + z4 - 5a2 - 10a2z2 - 6a2z4 - a2z6 + 4a4 + 5a4z2 + a4z4 - a6

Kauffman Polynomial: 3 - 11z2 + 15z4 - 7z6 + z8 + 2az - 11az3 + 15az5 - 7az7 + az9 + 5a2 - 23a2z2 + 30a2z4 - 14a2z6 + 2a2z8 + 3a3z - 12a3z3 + 15a3z5 - 7a3z7 + a3z9 + 4a4 - 13a4z2 + 15a4z4 - 7a4z6 + a4z8 + a5z - a5z3 + a6 - a6z2

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=-4 is the signature of 1119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 5       1
j = 3        
j = 1     11 
j = -1   11   
j = -3   11   
j = -5 111    
j = -7        
j = -911      

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, 19]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 19]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 17, 8, 16], 
 
>   X[2, 9, 3, 10], X[11, 18, 12, 19], X[13, 20, 14, 21], X[15, 22, 16, 1], 
 
>   X[17, 7, 18, 6], X[19, 12, 20, 13], X[21, 14, 22, 15]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 19]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 
 
>   7, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 19]]
Out[5]=   
DTCode[4, 8, 10, -16, 2, -18, -20, -22, -6, -12, -14]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 19]][t]
Out[6]=   
      -3   2       2    3
-1 - t   + -- + 2 t  - t
            2
           t
In[7]:=
Conway[Knot[11, NonAlternating, 19]][z]
Out[7]=   
     2      4    6
1 - z  - 4 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 19], Knot[11, NonAlternating, 135]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 19]], KnotSignature[Knot[11, NonAlternating, 19]]}
Out[9]=   
{5, -4}
In[10]:=
J=Jones[Knot[11, NonAlternating, 19]][q]
Out[10]=   
     -2   1        2
1 + q   - - - q + q
          q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[4, 1], Knot[11, NonAlternating, 19]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 19]][q]
Out[12]=   
  -20    -16    -14    -12    -6    -4    2    4    6
-q    + q    + q    + q    - q   - q   + q  + q  + q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 19]][a, z]
Out[13]=   
       2      4    6      2       2  2      4  2    4      2  4    4  4    2  6
3 - 5 a  + 4 a  - a  + 4 z  - 10 a  z  + 5 a  z  + z  - 6 a  z  + a  z  - a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 19]][a, z]
Out[14]=   
       2      4    6              3      5         2       2  2       4  2
3 + 5 a  + 4 a  + a  + 2 a z + 3 a  z + a  z - 11 z  - 23 a  z  - 13 a  z  - 
 
     6  2         3       3  3    5  3       4       2  4       4  4
>   a  z  - 11 a z  - 12 a  z  - a  z  + 15 z  + 30 a  z  + 15 a  z  + 
 
          5       3  5      6       2  6      4  6        7      3  7    8
>   15 a z  + 15 a  z  - 7 z  - 14 a  z  - 7 a  z  - 7 a z  - 7 a  z  + z  + 
 
       2  8    4  8      9    3  9
>   2 a  z  + a  z  + a z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 19]], Vassiliev[3][Knot[11, NonAlternating, 19]]}
Out[15]=   
{-1, 0}
In[16]:=
Kh[Knot[11, NonAlternating, 19]][q, t]
Out[16]=   
 -5    -3   1     1       1       1      1     t    t      2      3    5  4
q   + q   + - + ----- + ----- + ----- + ---- + -- + - + q t  + q t  + q  t
            q    9  3    9  2    5  2    5      3   q
                q  t    q  t    q  t    q  t   q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n19
K11n18
K11n18 		K11n20
K11n20