© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a137
K11a137
K11a139
K11a139
K11a138
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   The Knot K11a138

Visit K11a138's page at Knotilus!

Acknowledgement

K11a138 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X16,5,17,6 X14,8,15,7 X18,10,19,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X12,18,13,17 X8,20,9,19 X6,21,7,22

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

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

Alexander Polynomial: - 2t-3 + 14t-2 - 38t-1 + 53 - 38t + 14t2 - 2t3

Conway Polynomial: 1 + 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial: {K11a285, ...}

Determinant and Signature: {161, 0}

Jones Polynomial: - q-5 + 4q-4 - 9q-3 + 16q-2 - 22q-1 + 26 - 26q + 23q2 - 17q3 + 11q4 - 5q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial: {K11a285, ...}

A2 (sl(3)) Invariant: - q-16 + q-14 + 2q-12 - 4q-10 + 3q-8 + q-6 - 4q-4 + 5q-2 - 3 + 3q2 - 2q6 + 5q8 - 5q10 + 2q12 + 2q14 - 3q16 + q18

HOMFLY-PT Polynomial: a-4z4 - a-2z2 - a-2z4 - a-2z6 + 1 + z2 - z6 + a2z2 + 2a2z4 - a4z2

Kauffman Polynomial: - a-6z4 + a-6z6 + 3a-5z3 - 9a-5z5 + 5a-5z7 - a-4z2 + 12a-4z4 - 22a-4z6 + 10a-4z8 - a-3z + 8a-3z3 - 6a-3z5 - 12a-3z7 + 9a-3z9 - 7a-2z2 + 35a-2z4 - 49a-2z6 + 16a-2z8 + 3a-2z10 - 3a-1z + 10a-1z3 + 2a-1z5 - 27a-1z7 + 17a-1z9 + 1 - 9z2 + 32z4 - 42z6 + 16z8 + 3z10 - 3az + 12az3 - 13az5 - 2az7 + 8az9 - a2z2 + 5a2z4 - 12a2z6 + 10a2z8 - a3z + 6a3z3 - 11a3z5 + 8a3z7 + 2a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {0, 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=0 is the signature of 11138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 13           1
j = 11          4 
j = 9         71 
j = 7        104  
j = 5       137   
j = 3      1310    
j = 1     1313     
j = -1    1014      
j = -3   612       
j = -5  310        
j = -7 16         
j = -9 3          
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, Alternating, 138]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 138]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], 
 
>   X[18, 10, 19, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 15, 1, 16], 
 
>   X[12, 18, 13, 17], X[8, 20, 9, 19], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 138]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -11, 4, -10, 5, -2, 6, -9, 7, -4, 8, -3, 9, -5, 10, 
 
>   -7, 11, -8]
In[5]:=
DTCode[Knot[11, Alternating, 138]]
Out[5]=   
DTCode[4, 10, 16, 14, 18, 2, 20, 22, 12, 8, 6]
In[6]:=
alex = Alexander[Knot[11, Alternating, 138]][t]
Out[6]=   
     2    14   38              2      3
53 - -- + -- - -- - 38 t + 14 t  - 2 t
      3    2   t
     t    t
In[7]:=
Conway[Knot[11, Alternating, 138]][z]
Out[7]=   
       4      6
1 + 2 z  - 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 138], Knot[11, Alternating, 285]}
In[9]:=
{KnotDet[Knot[11, Alternating, 138]], KnotSignature[Knot[11, Alternating, 138]]}
Out[9]=   
{161, 0}
In[10]:=
J=Jones[Knot[11, Alternating, 138]][q]
Out[10]=   
      -5   4    9    16   22              2       3       4      5    6
26 - q   + -- - -- + -- - -- - 26 q + 23 q  - 17 q  + 11 q  - 5 q  + q
            4    3    2   q
           q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 138], Knot[11, Alternating, 285]}
In[12]:=
A2Invariant[Knot[11, Alternating, 138]][q]
Out[12]=   
      -16    -14    2     4    3     -6   4    5       2      6      8
-3 - q    + q    + --- - --- + -- + q   - -- + -- + 3 q  - 2 q  + 5 q  - 
                    12    10    8          4    2
                   q     q     q          q    q
 
       10      12      14      16    18
>   5 q   + 2 q   + 2 q   - 3 q   + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 138]][a, z]
Out[13]=   
          2                    4    4                   6
     2   z     2  2    4  2   z    z       2  4    6   z
1 + z  - -- + a  z  - a  z  + -- - -- + 2 a  z  - z  - --
          2                    4    2                   2
         a                    a    a                   a
In[14]:=
Kauffman[Knot[11, Alternating, 138]][a, z]
Out[14]=   
                                      2      2                        3
    z    3 z            3        2   z    7 z     2  2      4  2   3 z
1 - -- - --- - 3 a z - a  z - 9 z  - -- - ---- - a  z  + 2 a  z  + ---- + 
     3    a                           4     2                        5
    a                                a     a                        a
 
       3       3                                        4       4       4
    8 z    10 z          3      3  3    5  3       4   z    12 z    35 z
>   ---- + ----- + 12 a z  + 6 a  z  - a  z  + 32 z  - -- + ----- + ----- + 
      3      a                                          6     4       2
     a                                                 a     a       a
 
                           5      5      5
       2  4      4  4   9 z    6 z    2 z          5       3  5    5  5
>   5 a  z  - 5 a  z  - ---- - ---- + ---- - 13 a z  - 11 a  z  + a  z  - 
                          5      3     a
                         a      a
 
             6       6       6                           7       7       7
        6   z    22 z    49 z        2  6      4  6   5 z    12 z    27 z
>   42 z  + -- - ----- - ----- - 12 a  z  + 4 a  z  + ---- - ----- - ----- - 
             6     4       2                            5      3       a
            a     a       a                            a      a
 
                                   8       8                 9       9
         7      3  7       8   10 z    16 z        2  8   9 z    17 z
>   2 a z  + 8 a  z  + 16 z  + ----- + ----- + 10 a  z  + ---- + ----- + 
                                 4       2                  3      a
                                a       a                  a
 
                        10
         9      10   3 z
>   8 a z  + 3 z   + -----
                       2
                      a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 138]], Vassiliev[3][Knot[11, Alternating, 138]]}
Out[15]=   
{0, 0}
In[16]:=
Kh[Knot[11, Alternating, 138]][q, t]
Out[16]=   
14            1        3       1       6       3      10       6      12
-- + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 
q            11  5    9  4    7  4    7  3    5  3    5  2    3  2    3
            q   t    q  t    q  t    q  t    q  t    q  t    q  t    q  t
 
    10                 3         3  2       5  2      5  3       7  3
>   --- + 13 q t + 13 q  t + 10 q  t  + 13 q  t  + 7 q  t  + 10 q  t  + 
    q t
 
       7  4      9  4    9  5      11  5    13  6
>   4 q  t  + 7 q  t  + q  t  + 4 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a138
K11a137
K11a137
K11a139
K11a139