© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a162
K11a162
K11a164
K11a164
K11a163
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   The Knot K11a163

Visit K11a163's page at Knotilus!

Acknowledgement

K11a163 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: - t-4 + 6t-3 - 15t-2 + 24t-1 - 27 + 24t - 15t2 + 6t3 - t4

Conway Polynomial: 1 + 2z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial: {K11a66, ...}

Determinant and Signature: {119, 2}

Jones Polynomial: - q-4 + 3q-3 - 7q-2 + 12q-1 - 15 + 19q - 19q2 + 17q3 - 13q4 + 8q5 - 4q6 + q7

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

A2 (sl(3)) Invariant: - q-12 - 2q-6 + 3q-4 - q-2 + 3 + 3q2 - 2q4 + 4q6 - 4q8 + 2q10 - q12 - 2q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial: 2a-4z2 + 3a-4z4 + a-4z6 - 2a-2 - 7a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + 5 + 10z2 + 8z4 + 2z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial: a-8z4 - 2a-7z3 + 4a-7z5 + a-6z2 - 7a-6z4 + 8a-6z6 + 5a-5z3 - 14a-5z5 + 11a-5z7 - 2a-4z2 + 8a-4z4 - 17a-4z6 + 11a-4z8 - 2a-3z + 12a-3z3 - 13a-3z5 - 5a-3z7 + 7a-3z9 + 2a-2 - 14a-2z2 + 40a-2z4 - 44a-2z6 + 12a-2z8 + 2a-2z10 - 4a-1z + 6a-1z3 + 13a-1z5 - 28a-1z7 + 11a-1z9 + 5 - 18z2 + 37z4 - 30z6 + 4z8 + 2z10 - 4az + 6az3 + 4az5 - 11az7 + 4az9 + 2a2 - 7a2z2 + 13a2z4 - 11a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 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=2 is the signature of 11163. 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 = 15           1
j = 13          3 
j = 11         51 
j = 9        83  
j = 7       95   
j = 5      108    
j = 3     99     
j = 1    711      
j = -1   58       
j = -3  27        
j = -5 15         
j = -7 2          
j = -91           


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, 163]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 163]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], 
 
>   X[16, 10, 17, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 16, 9, 15], 
 
>   X[22, 17, 1, 18], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 163]]
Out[4]=   
GaussCode[1, -6, 2, -1, 3, -11, 4, -8, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, 
 
>   -7, 11, -9]
In[5]:=
DTCode[Knot[11, Alternating, 163]]
Out[5]=   
DTCode[4, 10, 18, 14, 16, 2, 20, 8, 22, 12, 6]
In[6]:=
alex = Alexander[Knot[11, Alternating, 163]][t]
Out[6]=   
       -4   6    15   24              2      3    4
-27 - t   + -- - -- + -- + 24 t - 15 t  + 6 t  - t
             3    2   t
            t    t
In[7]:=
Conway[Knot[11, Alternating, 163]][z]
Out[7]=   
       2    4      6    8
1 + 2 z  + z  - 2 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, Alternating, 66], Knot[11, Alternating, 163]}
In[9]:=
{KnotDet[Knot[11, Alternating, 163]], KnotSignature[Knot[11, Alternating, 163]]}
Out[9]=   
{119, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 163]][q]
Out[10]=   
       -4   3    7    12              2       3       4      5      6    7
-15 - q   + -- - -- + -- + 19 q - 19 q  + 17 q  - 13 q  + 8 q  - 4 q  + q
             3    2   q
            q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, Alternating, 163]}
In[12]:=
A2Invariant[Knot[11, Alternating, 163]][q]
Out[12]=   
     -12   2    3     -2      2      4      6      8      10    12      14
3 - q    - -- + -- - q   + 3 q  - 2 q  + 4 q  - 4 q  + 2 q   - q   - 2 q   + 
            6    4
           q    q
 
       16      18    20
>   2 q   - 2 q   + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 163]][a, z]
Out[13]=   
                           2      2                       4      4
    2       2       2   2 z    7 z       2  2      4   3 z    9 z     2  4
5 - -- - 2 a  + 10 z  + ---- - ---- - 3 a  z  + 8 z  + ---- - ---- - a  z  + 
     2                    4      2                       4      2
    a                    a      a                       a      a
 
            6      6    8
       6   z    5 z    z
>   2 z  + -- - ---- - --
            4     2     2
           a     a     a
In[14]:=
Kauffman[Knot[11, Alternating, 163]][a, z]
Out[14]=   
                                                      2      2       2
    2       2   2 z   4 z              3         2   z    2 z    14 z
5 + -- + 2 a  - --- - --- - 4 a z - 2 a  z - 18 z  + -- - ---- - ----- - 
     2           3     a                              6     4      2
    a           a                                    a     a      a
 
                 3      3       3      3                               4
       2  2   2 z    5 z    12 z    6 z         3      3  3       4   z
>   7 a  z  - ---- + ---- + ----- + ---- + 6 a z  + 5 a  z  + 37 z  + -- - 
                7      5      3      a                                 8
               a      a      a                                        a
 
       4      4       4                 5       5       5       5
    7 z    8 z    40 z        2  4   4 z    14 z    13 z    13 z         5
>   ---- + ---- + ----- + 13 a  z  + ---- - ----- - ----- + ----- + 4 a z  - 
      6      4      2                  7      5       3       a
     a      a      a                  a      a       a
 
                         6       6       6                  7      7       7
       3  5       6   8 z    17 z    44 z        2  6   11 z    5 z    28 z
>   4 a  z  - 30 z  + ---- - ----- - ----- - 11 a  z  + ----- - ---- - ----- - 
                        6      4       2                  5       3      a
                       a      a       a                  a       a
 
                                 8       8                9       9
          7    3  7      8   11 z    12 z       2  8   7 z    11 z         9
>   11 a z  + a  z  + 4 z  + ----- + ----- + 3 a  z  + ---- + ----- + 4 a z  + 
                               4       2                 3      a
                              a       a                 a
 
               10
       10   2 z
>   2 z   + -----
              2
             a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 163]], Vassiliev[3][Knot[11, Alternating, 163]]}
Out[15]=   
{2, 0}
In[16]:=
Kh[Knot[11, Alternating, 163]][q, t]
Out[16]=   
          3     1       2       1       5       2       7      5      8
11 q + 9 q  + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
               9  5    7  4    5  4    5  3    3  3    3  2      2   q t
              q  t    q  t    q  t    q  t    q  t    q  t    q t
 
    7 q      3         5        5  2      7  2      7  3      9  3      9  4
>   --- + 9 q  t + 10 q  t + 8 q  t  + 9 q  t  + 5 q  t  + 8 q  t  + 3 q  t  + 
     t
 
       11  4    11  5      13  5    15  6
>   5 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a163
K11a162
K11a162
K11a164
K11a164