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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

- q-3/2 + q-1/2 - 3q1/2 + 3q3/2 - 4q5/2 + 3q7/2 - 3q9/2 + 2q11/2 - q13/2 + q15/2

A2 (sl(3)) Invariant:

q-4 + q-2 + 3 + 2q2 + 2q4 + 2q6 + q8 + 2q10 - q16 - 2q18 - q20 - q22

HOMFLY-PT Polynomial:

2a-5z-1 + 7a-5z + 5a-5z3 + a-5z5 - 5a-3z-1 - 16a-3z - 17a-3z3 - 7a-3z5 - a-3z7 + 3a-1z-1 + 8a-1z + 5a-1z3 + a-1z5

Kauffman Polynomial:

a-8 - 6a-8z2 + 5a-8z4 - a-8z6 - 3a-7z3 + 4a-7z5 - a-7z7 + 2a-6z2 - 4a-6z4 + 4a-6z6 - a-6z8 + 2a-5z-1 - 9a-5z + 19a-5z3 - 15a-5z5 + 6a-5z7 - a-5z9 - 5a-4 + 20a-4z2 - 23a-4z4 + 11a-4z6 - 2a-4z8 + 5a-3z-1 - 20a-3z + 29a-3z3 - 21a-3z5 + 7a-3z7 - a-3z9 - 5a-2 + 13a-2z2 - 15a-2z4 + 6a-2z6 - a-2z8 + 3a-1z-1 - 9a-1z + 6a-1z3 - 2a-1z5 + z2 - z4 + 2az - az3

Khovanov Homology:

trqj 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                     j = 12               2 1   j = 10             1       j = 8           2 2       j = 6         2 1         j = 4       1 2           j = 2     2 2             j = 0     2               j = -2 1 1                 j = -4 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, NonAlternating, 135]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 135]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 14, 21, 13], > X[22, 16, 7, 15], X[4, 20, 5, 19], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 5, -6}, > {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, -4, 2, 10, -8, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 1 3/2 5/2 7/2 9/2 11/2 -q + ------- - 3 Sqrt[q] + 3 q - 4 q + 3 q - 3 q + 2 q - Sqrt[q] 13/2 15/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 6 8 10 16 18 20 22 3 + q + q + 2 q + 2 q + 2 q + q + 2 q - q - 2 q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 135]][a, z]
Out[8]=	3 3 3 5 5 5 7 2 5 3 7 z 16 z 8 z 5 z 17 z 5 z z 7 z z z ---- - ---- + --- + --- - ---- + --- + ---- - ----- + ---- + -- - ---- + -- - -- 5 3 a z 5 3 a 5 3 a 5 3 a 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 135]][a, z]
Out[9]=	2 -8 5 5 2 5 3 9 z 20 z 9 z 2 6 z a - -- - -- + ---- + ---- + --- - --- - ---- - --- + 2 a z + z - ---- + 4 2 5 3 a z 5 3 a 8 a a a z a z a a a 2 2 2 3 3 3 3 4 2 z 20 z 13 z 3 z 19 z 29 z 6 z 3 4 5 z > ---- + ----- + ----- - ---- + ----- + ----- + ---- - a z - z + ---- - 6 4 2 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 5 6 6 6 4 z 23 z 15 z 4 z 15 z 21 z 2 z z 4 z 11 z > ---- - ----- - ----- + ---- - ----- - ----- - ---- - -- + ---- + ----- + 6 4 2 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 8 8 8 9 9 6 z z 6 z 7 z z 2 z z z z > ---- - -- + ---- + ---- - -- - ---- - -- - -- - -- 2 7 5 3 6 4 2 5 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 4 4 2 6 2 6 3 2 + 2 q + ----- + ----- + ---- + 2 q t + q t + 2 q t + 2 q t + q t + 4 2 2 2 2 q t q t q t 8 3 8 4 10 4 12 5 12 6 16 7 > 2 q t + 2 q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n135

L11n134 L11n136