© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n137
K11n137 		K11n139
K11n139
K11n138
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   The Knot K11n138

Visit K11n138's page at Knotilus!

Acknowledgement

K11n138 as Morse Link
DrawMorseLink

PD Presentation: X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X9,19,10,18 X11,21,12,20 X2,13,3,14 X15,6,16,7 X17,22,18,1 X19,11,20,10 X21,9,22,8

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

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

Alexander Polynomial: - 2t-2 + 4t-1 - 3 + 4t - 2t2

Conway Polynomial: 1 - 4z2 - 2z4

Other knots with the same Alexander/Conway Polynomial: {K11n79, ...}

Determinant and Signature: {15, 2}

Jones Polynomial: q-5 - q-4 + 2q-3 - 3q-2 + 2q-1 - 2 + 2q - q2 + q3

Other knots (up to mirrors) with the same Jones Polynomial: {K11n79, ...}

A2 (sl(3)) Invariant: q-16 + q-14 + q-12 + q-10 - q-8 - q-6 - 2q-4 - q-2 + q4 + q6 + q8 + q10

HOMFLY-PT Polynomial: 2a-2 + a-2z2 - 1 - 3z2 - z4 - 2a2 - 3a2z2 - a2z4 + 2a4 + a4z2

Kauffman Polynomial: - a-3z - 2a-2 + 5a-2z2 - 5a-2z4 + a-2z6 - 3a-1z + 11a-1z3 - 10a-1z5 + 2a-1z7 - 1 - z2 + 11z4 - 10z6 + 2z8 + az + 3az3 + az5 - 4az7 + az9 + 2a2 - 19a2z2 + 32a2z4 - 18a2z6 + 3a2z8 + 3a3z - 8a3z3 + 11a3z5 - 6a3z7 + a3z9 + 2a4 - 13a4z2 + 16a4z4 - 7a4z6 + a4z8

V2 and V3, the type 2 and 3 Vassiliev invariants: {-4, 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=2 is the signature of 11138. 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 = 7        1
j = 5         
j = 3      21 
j = 1     11  
j = -1    121  
j = -3   21    
j = -5   1     
j = -7 12      
j = -9         
j = -111        

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, 138]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 138]]
Out[3]=   
PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[5, 16, 6, 17], X[7, 14, 8, 15], 
 
>   X[9, 19, 10, 18], X[11, 21, 12, 20], X[2, 13, 3, 14], X[15, 6, 16, 7], 
 
>   X[17, 22, 18, 1], X[19, 11, 20, 10], X[21, 9, 22, 8]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 138]]
Out[4]=   
GaussCode[1, -7, 2, -1, -3, 8, -4, 11, -5, 10, -6, -2, 7, 4, -8, 3, -9, 5, -10, 
 
>   6, -11, 9]
In[5]:=
DTCode[Knot[11, NonAlternating, 138]]
Out[5]=   
DTCode[4, 12, -16, -14, -18, -20, 2, -6, -22, -10, -8]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 138]][t]
Out[6]=   
     2    4            2
-3 - -- + - + 4 t - 2 t
      2   t
     t
In[7]:=
Conway[Knot[11, NonAlternating, 138]][z]
Out[7]=   
       2      4
1 - 4 z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 79], Knot[11, NonAlternating, 138]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 138]], KnotSignature[Knot[11, NonAlternating, 138]]}
Out[9]=   
{15, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 138]][q]
Out[10]=   
      -5    -4   2    3    2          2    3
-2 + q   - q   + -- - -- + - + 2 q - q  + q
                  3    2   q
                 q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 79], Knot[11, NonAlternating, 138]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 138]][q]
Out[12]=   
 -16    -14    -12    -10    -8    -6   2     -2    4    6    8    10
q    + q    + q    + q    - q   - q   - -- - q   + q  + q  + q  + q
                                         4
                                        q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 138]][a, z]
Out[13]=   
                                2
     2       2      4      2   z       2  2    4  2    4    2  4
-1 + -- - 2 a  + 2 a  - 3 z  + -- - 3 a  z  + a  z  - z  - a  z
      2                         2
     a                         a
In[14]:=
Kauffman[Knot[11, NonAlternating, 138]][a, z]
Out[14]=   
                                                          2
     2       2      4   z    3 z            3      2   5 z        2  2
-1 - -- + 2 a  + 2 a  - -- - --- + a z + 3 a  z - z  + ---- - 19 a  z  - 
      2                  3    a                          2
     a                  a                               a
 
                   3                                 4
        4  2   11 z         3      3  3       4   5 z        2  4       4  4
>   13 a  z  + ----- + 3 a z  - 8 a  z  + 11 z  - ---- + 32 a  z  + 16 a  z  - 
                 a                                  2
                                                   a
 
        5                              6                           7
    10 z       5       3  5       6   z        2  6      4  6   2 z         7
>   ----- + a z  + 11 a  z  - 10 z  + -- - 18 a  z  - 7 a  z  + ---- - 4 a z  - 
      a                                2                         a
                                      a
 
       3  7      8      2  8    4  8      9    3  9
>   6 a  z  + 2 z  + 3 a  z  + a  z  + a z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 138]], Vassiliev[3][Knot[11, NonAlternating, 138]]}
Out[15]=   
{-4, 2}
In[16]:=
Kh[Knot[11, NonAlternating, 138]][q, t]
Out[16]=   
1          3     1        1       2       1       2       1      1      2
- + q + 2 q  + ------ + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
q               11  6    7  5    7  4    5  3    3  3    3  2      2   q t
               q   t    q  t    q  t    q  t    q  t    q  t    q t
 
    q    3      7  2
>   - + q  t + q  t
    t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n138
K11n137
K11n137 		K11n139
K11n139