© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11a110
K11a110
K11a112
K11a112
K11a111
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   The Knot K11a111

Visit K11a111's page at Knotilus!

Acknowledgement

K11a111 as Morse Link
DrawMorseLink

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

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

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

Alexander Polynomial: 2t-3 - 10t-2 + 24t-1 - 31 + 24t - 10t2 + 2t3

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

Other knots with the same Alexander/Conway Polynomial: {10117, K11a23, ...}

Determinant and Signature: {103, 2}

Jones Polynomial: - q-4 + 3q-3 - 6q-2 + 10q-1 - 13 + 16q - 16q2 + 15q3 - 11q4 + 7q5 - 4q6 + q7

Other knots (up to mirrors) with the same Jones Polynomial: {K11a68, ...}

A2 (sl(3)) Invariant: - q-12 + q-10 - q-8 - q-6 + 3q-4 - 2q-2 + 2 + 2q2 - q4 + 3q6 - 2q8 + 2q10 - 2q14 + 2q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial: a-6z2 - a-4 - 3a-4z2 - 2a-4z4 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 + 2 + 4z2 + 3z4 + z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial: a-8z4 - 3a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 7a-6z6 + a-5z - a-5z3 - 7a-5z5 + 8a-5z7 - a-4 + 5a-4z2 - 6a-4z4 - 5a-4z6 + 7a-4z8 - a-3z + 12a-3z3 - 19a-3z5 + 3a-3z7 + 4a-3z9 - a-2 - 4a-2z2 + 24a-2z4 - 33a-2z6 + 11a-2z8 + a-2z10 - 5a-1z + 17a-1z3 - 7a-1z5 - 13a-1z7 + 7a-1z9 + 2 - 14z2 + 37z4 - 33z6 + 7z8 + z10 - 5az + 12az3 - 3az5 - 7az7 + 3az9 + a2 - 7a2z2 + 15a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 1}

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 11111. 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         41 
j = 9        73  
j = 7       84   
j = 5      87    
j = 3     88     
j = 1    69      
j = -1   47       
j = -3  26        
j = -5 14         
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, 111]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, Alternating, 111]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], 
 
>   X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 13, 21, 14], X[18, 16, 19, 15], 
 
>   X[8, 18, 9, 17], X[6, 19, 7, 20], X[12, 21, 13, 22]]
In[4]:=
GaussCode[Knot[11, Alternating, 111]]
Out[4]=   
GaussCode[1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, 
 
>   -7, 11, -6]
In[5]:=
DTCode[Knot[11, Alternating, 111]]
Out[5]=   
DTCode[4, 10, 14, 16, 2, 22, 20, 18, 8, 6, 12]
In[6]:=
alex = Alexander[Knot[11, Alternating, 111]][t]
Out[6]=   
      2    10   24              2      3
-31 + -- - -- + -- + 24 t - 10 t  + 2 t
       3    2   t
      t    t
In[7]:=
Conway[Knot[11, Alternating, 111]][z]
Out[7]=   
       2      4      6
1 + 2 z  + 2 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[10, 117], Knot[11, Alternating, 23], Knot[11, Alternating, 111]}
In[9]:=
{KnotDet[Knot[11, Alternating, 111]], KnotSignature[Knot[11, Alternating, 111]]}
Out[9]=   
{103, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 111]][q]
Out[10]=   
       -4   3    6    10              2       3       4      5      6    7
-13 - q   + -- - -- + -- + 16 q - 16 q  + 15 q  - 11 q  + 7 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, 68], Knot[11, Alternating, 111]}
In[12]:=
A2Invariant[Knot[11, Alternating, 111]][q]
Out[12]=   
     -12    -10    -8    -6   3    2       2    4      6      8      10
2 - q    + q    - q   - q   + -- - -- + 2 q  - q  + 3 q  - 2 q  + 2 q   - 
                               4    2
                              q    q
 
       14      16      18    20    22
>   2 q   + 2 q   - 2 q   - q   + q
In[13]:=
HOMFLYPT[Knot[11, Alternating, 111]][a, z]
Out[13]=   
                             2      2      2                       4      4
     -4    -2    2      2   z    3 z    2 z       2  2      4   2 z    2 z
2 - a   + a   - a  + 4 z  + -- - ---- + ---- - 2 a  z  + 3 z  - ---- + ---- - 
                             6     4      2                       4      2
                            a     a      a                       a      a
 
                  6
     2  4    6   z
>   a  z  + z  + --
                  2
                 a
In[14]:=
Kauffman[Knot[11, Alternating, 111]][a, z]
Out[14]=   
                                                                 2      2
     -4    -2    2   z    z    5 z              3         2   2 z    5 z
2 - a   - a   + a  + -- - -- - --- - 5 a z - 2 a  z - 14 z  + ---- + ---- - 
                      5    3    a                               6      4
                     a    a                                    a      a
 
       2                3    3       3       3
    4 z       2  2   3 z    z    12 z    17 z          3      3  3       4
>   ---- - 7 a  z  - ---- - -- + ----- + ----- + 12 a z  + 5 a  z  + 37 z  + 
      2                7     5     3       a
     a                a     a     a
 
     4      4      4       4                 5      5       5      5
    z    7 z    6 z    24 z        2  4   4 z    7 z    19 z    7 z         5
>   -- - ---- - ---- + ----- + 15 a  z  + ---- - ---- - ----- - ---- - 3 a z  - 
     8     6      4      2                  7      5      3      a
    a     a      a      a                  a      a      a
 
                         6      6       6                 7      7       7
       3  5       6   7 z    5 z    33 z        2  6   8 z    3 z    13 z
>   4 a  z  - 33 z  + ---- - ---- - ----- - 12 a  z  + ---- + ---- - ----- - 
                        6      4      2                  5      3      a
                       a      a      a                  a      a
 
                               8       8                9      9
         7    3  7      8   7 z    11 z       2  8   4 z    7 z         9
>   7 a z  + a  z  + 7 z  + ---- + ----- + 3 a  z  + ---- + ---- + 3 a z  + 
                              4      2                 3     a
                             a      a                 a
 
           10
     10   z
>   z   + ---
           2
          a
In[15]:=
{Vassiliev[2][Knot[11, Alternating, 111]], Vassiliev[3][Knot[11, Alternating, 111]]}
Out[15]=   
{2, 1}
In[16]:=
Kh[Knot[11, Alternating, 111]][q, t]
Out[16]=   
         3     1       2       1       4       2       6      4      7    6 q
9 q + 8 q  + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 
              9  5    7  4    5  4    5  3    3  3    3  2      2   q t    t
             q  t    q  t    q  t    q  t    q  t    q  t    q t
 
       3        5        5  2      7  2      7  3      9  3      9  4
>   8 q  t + 8 q  t + 7 q  t  + 8 q  t  + 4 q  t  + 7 q  t  + 3 q  t  + 
 
       11  4    11  5      13  5    15  6
>   4 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a111
K11a110
K11a110
K11a112
K11a112