© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n37
K11n37 		K11n39
K11n39
K11n38
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   The Knot K11n38

Visit K11n38's page at Knotilus!

Acknowledgement

K11n38 as Morse Link
DrawMorseLink

PD Presentation: X4251 X8493 X5,12,6,13 X2837 X9,19,10,18 X11,6,12,7 X13,22,14,1 X15,20,16,21 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, 8, -11, 7}

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

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

Conway Polynomial: 1 - 3z2 - z4

Other knots with the same Alexander/Conway Polynomial: {K11n102, ...}

Determinant and Signature: {3, 2}

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

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

A2 (sl(3)) Invariant: q-16 + q-14 + q-12 - q-8 - q-6 - 2q-4 + q2 + q4 + q6 + q8

HOMFLY-PT Polynomial: a-2 + 1 - 3a2 - 4a2z2 - a2z4 + 2a4 + a4z2

Kauffman Polynomial: - a-2 + a-1z - a-1z3 + 1 - 9z2 + 14z4 - 7z6 + z8 + 5az - 13az3 + 15az5 - 7az7 + az9 + 3a2 - 20a2z2 + 29a2z4 - 14a2z6 + 2a2z8 + 4a3z - 12a3z3 + 15a3z5 - 7a3z7 + a3z9 + 2a4 - 11a4z2 + 15a4z4 - 7a4z6 + a4z8

V2 and V3, the type 2 and 3 Vassiliev invariants: {-3, 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 1138. 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      11 
j = -1    121  
j = -3   1     
j = -5   11    
j = -7 11      
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, 38]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 38]]
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, 22, 14, 1], X[15, 20, 16, 21], 
 
>   X[17, 11, 18, 10], X[19, 16, 20, 17], X[21, 14, 22, 15]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 38]]
Out[4]=   
GaussCode[1, -4, 2, -1, -3, 6, 4, -2, -5, 9, -6, 3, -7, 11, -8, 10, -9, 5, -10, 
 
>   8, -11, 7]
In[5]:=
DTCode[Knot[11, NonAlternating, 38]]
Out[5]=   
DTCode[4, 8, -12, 2, -18, -6, -22, -20, -10, -16, -14]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 38]][t]
Out[6]=   
     -2   1        2
1 - t   + - + t - t
          t
In[7]:=
Conway[Knot[11, NonAlternating, 38]][z]
Out[7]=   
       2    4
1 - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[11, NonAlternating, 38], Knot[11, NonAlternating, 102]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 38]], KnotSignature[Knot[11, NonAlternating, 38]]}
Out[9]=   
{3, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 38]][q]
Out[10]=   
 -5    -4    -3    -2    2
q   - q   + q   - q   + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[11, NonAlternating, 38]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 38]][q]
Out[12]=   
 -16    -14    -12    -8    -6   2     2    4    6    8
q    + q    + q    - q   - q   - -- + q  + q  + q  + q
                                  4
                                 q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 38]][a, z]
Out[13]=   
     -2      2      4      2  2    4  2    2  4
1 + a   - 3 a  + 2 a  - 4 a  z  + a  z  - a  z
In[14]:=
Kauffman[Knot[11, NonAlternating, 38]][a, z]
Out[14]=   
                                                                           3
     -2      2      4   z              3        2       2  2       4  2   z
1 - a   + 3 a  + 2 a  + - + 5 a z + 4 a  z - 9 z  - 20 a  z  - 11 a  z  - -- - 
                        a                                                 a
 
          3       3  3       4       2  4       4  4         5       3  5
>   13 a z  - 12 a  z  + 14 z  + 29 a  z  + 15 a  z  + 15 a z  + 15 a  z  - 
 
       6       2  6      4  6        7      3  7    8      2  8    4  8
>   7 z  - 14 a  z  - 7 a  z  - 7 a z  - 7 a  z  + z  + 2 a  z  + a  z  + 
 
       9    3  9
>   a z  + a  z
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 38]], Vassiliev[3][Knot[11, NonAlternating, 38]]}
Out[15]=   
{-3, 2}
In[16]:=
Kh[Knot[11, NonAlternating, 38]][q, t]
Out[16]=   
1        3     1        1       1       1       1       1      1      2
- + q + q  + ------ + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
q             11  6    7  5    7  4    5  3    3  3    5  2      2   q t
             q   t    q  t    q  t    q  t    q  t    q  t    q t
 
           5  2
>   q t + q  t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n38
K11n37
K11n37 		K11n39
K11n39