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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X18,7,19,8 X20,10,21,9 X22,17,7,18 X14,6,15,5 X16,12,17,11 X12,16,13,15 X4,14,5,13 X6,20,1,19 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 9q-3/2 + 15q-1/2 - 21q1/2 + 23q3/2 - 24q5/2 + 20q7/2 - 15q9/2 + 9q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 - 3q-2 + 5 - 3q2 + 4q4 + 2q6 + 5q10 - 4q12 + 3q14 - 2q18 + 2q20 - q22

HOMFLY-PT Polynomial:

a-5z + 2a-5z3 + a-5z5 - a-3z-1 - a-3z - 3a-3z3 - 3a-3z5 - a-3z7 + a-1z-1 - 3a-1z3 - 3a-1z5 - a-1z7 + az + 2az3 + az5

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + a-7z - 6a-7z3 + 9a-7z5 - 4a-7z7 + a-6z2 - 9a-6z4 + 15a-6z6 - 7a-6z8 + a-5z - 8a-5z3 + 9a-5z5 + 5a-5z7 - 6a-5z9 + 3a-4z2 - 19a-4z4 + 33a-4z6 - 13a-4z8 - 2a-4z10 + a-3z-1 - 9a-3z3 + 10a-3z5 + 12a-3z7 - 12a-3z9 - a-2 + a-2z2 - 13a-2z4 + 30a-2z6 - 15a-2z8 - 2a-2z10 + a-1z-1 + 2a-1z - 16a-1z3 + 23a-1z5 - 5a-1z7 - 6a-1z9 - 2z2 + 9z6 - 9z8 + 2az - 8az3 + 12az5 - 8az7 - 2a2z2 + 5a2z4 - 4a2z6 + a3z3 - a3z5

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 = 16                       1 j = 14                     3   j = 12                   6 1   j = 10                 9 3     j = 8               11 6       j = 6             13 9         j = 4           11 12           j = 2         10 12             j = 0       6 12               j = -2     3 9                 j = -4   1 6                   j = -6   3                     j = -8 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, 183]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 183]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[18, 7, 19, 8], X[20, 10, 21, 9], > X[22, 17, 7, 18], X[14, 6, 15, 5], X[16, 12, 17, 11], X[12, 16, 13, 15], > X[4, 14, 5, 13], X[6, 20, 1, 19], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 6, -10}, > {3, -1, 4, -2, 7, -8, 9, -6, 8, -7, 5, -3, 10, -4, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 9 15 3/2 5/2 7/2 -q + ---- - ---- + ------- - 21 Sqrt[q] + 23 q - 24 q + 20 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 3 2 4 6 10 12 14 18 5 + q - -- + -- - -- - 3 q + 4 q + 2 q + 5 q - 4 q + 3 q - 2 q + 8 6 2 q q q 20 22 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 183]][a, z]
Out[8]=	3 3 3 5 5 1 1 z z 2 z 3 z 3 z 3 z 3 z -(----) + --- + -- - -- + a z + ---- - ---- - ---- + 2 a z + -- - ---- - 3 a z 5 3 5 3 a 5 3 a z a a a a a a 5 7 7 3 z 5 z z > ---- + a z - -- - -- a 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 183]][a, z]
Out[9]=	2 2 2 2 -2 1 1 z z 2 z 2 z z 3 z z -a + ---- + --- + -- + -- + --- + 2 a z - 2 z - -- + -- + ---- + -- - 3 a z 7 5 a 8 6 4 2 a z a a a a a a 3 3 3 3 4 4 2 2 6 z 8 z 9 z 16 z 3 3 3 2 z 9 z > 2 a z - ---- - ---- - ---- - ----- - 8 a z + a z + ---- - ---- - 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 19 z 13 z 2 4 9 z 9 z 10 z 23 z 5 3 5 > ----- - ----- + 5 a z + ---- + ---- + ----- + ----- + 12 a z - a z + 4 2 7 5 3 a a a a a a 6 6 6 6 7 7 7 7 6 z 15 z 33 z 30 z 2 6 4 z 5 z 12 z 5 z > 9 z - -- + ----- + ----- + ----- - 4 a 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 7 z 13 z 15 z 6 z 12 z 6 z 2 z 2 z > 8 a z - 9 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 1 3 1 6 3 6 9 2 12 + 10 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 12 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 11 q t + 12 q t + 13 q t + 9 q t + 11 q t + 6 q t + 9 q t + 10 5 12 5 12 6 14 6 16 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 L11a183

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