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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 7q-7/2 + 11q-5/2 - 16q-3/2 + 18q-1/2 - 19q1/2 + 17q3/2 - 13q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 2q-12 + q-10 - q-8 + 4q-6 - 2q-4 + q-2 + 1 - 2q2 + 4q4 - 3q6 + 3q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 5a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + 2az-1 + 8az + 9az3 + 3az5 - 2a3z-1 - 6a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 8a-4z6 + 2a-3z - 7a-3z3 + 14a-3z5 - 11a-3z7 - 2a-2z2 + a-2z4 + 11a-2z6 - 10a-2z8 - a-1z-1 + 10a-1z - 33a-1z3 + 39a-1z5 - 8a-1z7 - 5a-1z9 + 7z2 - 34z4 + 48z6 - 17z8 - z10 - 2az-1 + 18az - 44az3 + 30az5 + 8az7 - 8az9 - a2 + 12a2z2 - 40a2z4 + 40a2z6 - 10a2z8 - a2z10 - 2a3z-1 + 14a3z - 26a3z3 + 13a3z5 + 4a3z7 - 3a3z9 + 5a4z2 - 13a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   5 1   j = 6                 8 3     j = 4               9 5       j = 2             10 8         j = 0           9 10           j = -2         7 9             j = -4       5 10               j = -6     2 6                 j = -8   1 5                   j = -10   2                     j = -12 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, 158]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 158]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[16, 12, 17, 11], X[14, 5, 15, 6], X[4, 15, 5, 16], X[20, 18, 21, 17], > X[18, 13, 19, 14], X[12, 19, 13, 20], X[6, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -11}, > {4, -1, 3, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 11 16 18 3/2 -q + ---- - ---- + ---- - ---- + ------- - 19 Sqrt[q] + 17 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 2 -10 -8 4 2 -2 2 4 1 + q + q - q + --- + q - q + -- - -- + q - 2 q + 4 q - 12 6 4 q q q 6 8 10 12 14 16 > 3 q + 3 q + q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 158]][a, z]
Out[8]=	3 5 3 3 1 2 a 2 a a z 5 z 3 5 2 z 7 z -(---) + --- - ---- + -- + -- - --- + 8 a z - 6 a z + a z + ---- - ---- + a z z z z 3 a 3 a a a 5 5 7 3 3 3 z 4 z 5 z > 9 a z - 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 158]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 2 z 10 z 3 5 2 -a - --- - --- - ---- - -- + --- + ---- + 18 a z + 14 a z + 4 a z + 7 z - a z z z z 3 a a 2 2 3 3 3 2 z 2 z 2 2 4 2 2 z 7 z 33 z 3 > ---- - ---- + 12 a z + 5 a z + ---- - ---- - ----- - 44 a z - 4 2 5 3 a a a a a 4 4 4 5 3 3 5 3 4 z 7 z z 2 4 4 4 4 z > 26 a z - 6 a z - 34 z - -- + ---- + -- - 40 a z - 13 a z - ---- + 6 4 2 5 a a a a 5 5 6 6 14 z 39 z 5 3 5 5 5 6 8 z 11 z > ----- + ----- + 30 a z + 13 a z + 4 a z + 48 z - ---- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 11 z 8 z 7 3 7 5 7 8 > 40 a z + 11 a z - ----- - ---- + 8 a z + 4 a z - a z - 17 z - 3 a a 8 9 10 z 2 8 4 8 5 z 9 3 9 10 2 10 > ----- - 10 a z - 3 a z - ---- - 8 a z - 3 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 6 5 10 10 + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 7 9 9 2 4 4 2 6 2 6 3 > ----- + - + ---- + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a158

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