© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n36
K11n36 		K11n38
K11n38
K11n37
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   The Knot K11n37

Visit K11n37's page at Knotilus!

Acknowledgement

K11n37 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X5,12,6,13 X2837 X9,19,10,18 X11,6,12,7 X13,20,14,21 X15,22,16,1 X17,11,18,10 X19,16,20,17 X21,14,22,15

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

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

Alexander Polynomial: - t-3 + 3t-2 - 5t-1 + 7 - 5t + 3t2 - t3

Conway Polynomial: 1 - 2z2 - 3z4 - z6

Other knots with the same Alexander/Conway Polynomial: {89, 10155, ...}

Determinant and Signature: {25, 0}

Jones Polynomial: q-6 - 2q-5 + 3q-4 - 4q-3 + 4q-2 - 4q-1 + 4 - 2q + q2

Other knots (up to mirrors) with the same Jones Polynomial: {10137, 10155, ...}

A2 (sl(3)) Invariant: q-18 + q-14 - q-10 - 2q-6 + 1 + 2q2 + q6

HOMFLY-PT Polynomial: 3 + 3z2 + z4 - 4a2 - 8a2z2 - 5a2z4 - a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial: a-1z3 + 3 - 9z2 + 13z4 - 6z6 + z8 - 3az3 + 8az5 - 5az7 + az9 + 4a2 - 17a2z2 + 25a2z4 - 15a2z6 + 3a2z8 - 2a3z + 3a3z3 - 3a3z7 + a3z9 + 2a4 - 5a4z2 + 8a4z4 - 8a4z6 + 2a4z8 - 2a5z + 7a5z3 - 8a5z5 + 2a5z7 + 3a6z2 - 4a6z4 + a6z6

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

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=0 is the signature of 1137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 5        1
j = 3       1 
j = 1      31 
j = -1     22  
j = -3    22   
j = -5   22    
j = -7  12     
j = -9 12      
j = -11 1       
j = -131        

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, 37]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 37]]
Out[3]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[2, 8, 3, 7], 
 
>   X[9, 19, 10, 18], X[11, 6, 12, 7], X[13, 20, 14, 21], X[15, 22, 16, 1], 
 
>   X[17, 11, 18, 10], X[19, 16, 20, 17], X[21, 14, 22, 15]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 37]]
Out[4]=   
GaussCode[1, -4, 2, -1, -3, 6, 4, -2, -5, 9, -6, 3, -7, 11, -8, 10, -9, 5, -10, 
 
>   7, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 37]]
Out[5]=   
DTCode[4, 8, -12, 2, -18, -6, -20, -22, -10, -16, -14]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 37]][t]
Out[6]=   
     -3   3    5            2    3
7 - t   + -- - - - 5 t + 3 t  - t
           2   t
          t
In[7]:=
Conway[Knot[11, NonAlternating, 37]][z]
Out[7]=   
       2      4    6
1 - 2 z  - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[8, 9], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 37]], KnotSignature[Knot[11, NonAlternating, 37]]}
Out[9]=   
{25, 0}
In[10]:=
J=Jones[Knot[11, NonAlternating, 37]][q]
Out[10]=   
     -6   2    3    4    4    4          2
4 + q   - -- + -- - -- + -- - - - 2 q + q
           5    4    3    2   q
          q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[10, 137], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 37]][q]
Out[12]=   
     -18    -14    -10   2       2    6
1 + q    + q    - q    - -- + 2 q  + q
                          6
                         q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 37]][a, z]
Out[13]=   
       2      4      2      2  2      4  2    4      2  4    4  4    2  6
3 - 4 a  + 2 a  + 3 z  - 8 a  z  + 3 a  z  + z  - 5 a  z  + a  z  - a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 37]][a, z]
Out[14]=   
                                                                           3
       2      4      3        5        2       2  2      4  2      6  2   z
3 + 4 a  + 2 a  - 2 a  z - 2 a  z - 9 z  - 17 a  z  - 5 a  z  + 3 a  z  + -- - 
                                                                          a
 
         3      3  3      5  3       4       2  4      4  4      6  4
>   3 a z  + 3 a  z  + 7 a  z  + 13 z  + 25 a  z  + 8 a  z  - 4 a  z  + 
 
         5      5  5      6       2  6      4  6    6  6        7      3  7
>   8 a z  - 8 a  z  - 6 z  - 15 a  z  - 8 a  z  + a  z  - 5 a z  - 3 a  z  + 
 
       5  7    8      2  8      4  8      9    3  9
>   2 a  z  + z  + 3 a  z  + 2 a  z  + a z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 37]], Vassiliev[3][Knot[11, NonAlternating, 37]]}
Out[15]=   
{-2, 2}
In[16]:=
Kh[Knot[11, NonAlternating, 37]][q, t]
Out[16]=   
2           1        1        1       2       1       2       2       2
- + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          13  6    11  5    9  5    9  4    7  4    7  3    5  3    5  2
          q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
      2      2      2           3      5  2
>   ----- + ---- + --- + q t + q  t + q  t
     3  2    3     q t
    q  t    q  t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n37
K11n36
K11n36 		K11n38
K11n38