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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X22,12,7,11 X16,9,17,10 X14,22,15,21 X10,15,11,16 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 8q-1/2 + 13q1/2 - 18q3/2 + 21q5/2 - 22q7/2 + 19q9/2 - 15q11/2 + 9q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 4q-2 - 2 + 2q2 + q4 - 4q6 + 4q8 - 3q10 + 4q12 + q14 - q16 + 4q18 - 3q20 + q22 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 - a-5z-1 - 2a-5z - 4a-5z3 - 2a-5z5 + 2a-3z-1 + 4a-3z + 4a-3z3 + 3a-3z5 + a-3z7 - 2a-1z-1 - 4a-1z - 5a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- a-10z4 + a-9z3 - 4a-9z5 - 2a-8z2 + 7a-8z4 - 9a-8z6 + 3a-7z - 12a-7z3 + 20a-7z5 - 14a-7z7 + a-6z2 - 8a-6z4 + 21a-6z6 - 14a-6z8 - a-5z-1 + 9a-5z - 29a-5z3 + 33a-5z5 - a-5z7 - 8a-5z9 + 8a-4z2 - 39a-4z4 + 55a-4z6 - 18a-4z8 - 2a-4z10 - 2a-3z-1 + 13a-3z - 28a-3z3 + 10a-3z5 + 22a-3z7 - 12a-3z9 - a-2 + 9a-2z2 - 34a-2z4 + 35a-2z6 - 7a-2z8 - 2a-2z10 - 2a-1z-1 + 11a-1z - 18a-1z3 + 5a-1z5 + 8a-1z7 - 4a-1z9 + 4z2 - 11z4 + 10z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     3   j = 14                   6 1   j = 12                 9 3     j = 10               10 6       j = 8             12 9         j = 6           10 11           j = 4         8 11             j = 2       6 11               j = 0     2 7                 j = -2   1 6                   j = -4   2                     j = -6 1

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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 238]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 238]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 12, 7, 11], X[16, 9, 17, 10], > X[14, 22, 15, 21], X[10, 15, 11, 16], X[18, 6, 19, 5], X[20, 18, 21, 17], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 7, -11}, > {9, -1, 4, -6, 3, -2, 10, -5, 6, -4, 8, -7, 11, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 8 3/2 5/2 7/2 -q + ---- - ------- + 13 Sqrt[q] - 18 q + 21 q - 22 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 19 q - 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 4 2 4 6 8 10 12 14 16 18 -2 + q + -- + 2 q + q - 4 q + 4 q - 3 q + 4 q + q - q + 4 q - 2 q 20 22 24 26 > 3 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 238]][a, z]
Out[8]=	3 3 3 1 2 2 a z 2 z 4 z 4 z z 4 z 4 z -(----) + ---- - --- + - + -- - --- + --- - --- + 2 a z + -- - ---- + ---- - 5 3 a z z 7 5 3 a 7 5 3 a z a z a a a a a a 3 5 5 5 7 5 z 3 2 z 3 z 2 z z > ---- + a z - ---- + ---- - ---- + -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 238]][a, z]
Out[9]=	2 -2 1 2 2 a 3 z 9 z 13 z 11 z 2 2 z -a - ---- - ---- - --- - - + --- + --- + ---- + ---- + 4 a z + 4 z - ---- + 5 3 a z z 7 5 3 a 8 a z a z a a a a 2 2 2 3 3 3 3 3 z 8 z 9 z z 12 z 29 z 28 z 18 z 3 4 > -- + ---- + ---- + -- - ----- - ----- - ----- - ----- - 6 a z - 11 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 4 5 5 5 5 5 z 7 z 8 z 39 z 34 z 4 z 20 z 33 z 10 z 5 z > --- + ---- - ---- - ----- - ----- - ---- + ----- + ----- + ----- + ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 7 7 7 7 5 6 9 z 21 z 55 z 35 z 14 z z 22 z 8 z > 4 a z + 10 z - ---- + ----- + ----- + ----- - ----- - -- + ----- + ---- - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 14 z 18 z 7 z 8 z 12 z 4 z 2 z 2 z > a z - 3 z - ----- - ----- - ---- - ---- - ----- - ---- - ----- - ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 6 7 6 q 4 11 q + 8 q + ----- + ----- + ----- + -- + ----- + - + ---- + 11 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 10 q t + 11 q t + 12 q t + 9 q t + 10 q t + 6 q t + 9 q t + 12 5 14 5 14 6 16 6 18 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a238

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