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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 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 + 2q-1/2 - 5q1/2 + 5q3/2 - 7q5/2 + 6q7/2 - 5q9/2 + 4q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

2a-5z-1 + 5a-5z + 4a-5z3 + a-5z5 - 5a-3z-1 - 12a-3z - 13a-3z3 - 6a-3z5 - a-3z7 + 3a-1z-1 + 6a-1z + 4a-1z3 + a-1z5

Kauffman Polynomial:

a-8 - 4a-8z2 + 4a-8z4 - a-8z6 - 5a-7z3 + 7a-7z5 - 2a-7z7 - 3a-6z4 + 6a-6z6 - 2a-6z8 + 2a-5z-1 - 6a-5z + 7a-5z3 - 2a-5z5 + 2a-5z7 - a-5z9 - 5a-4 + 14a-4z2 - 15a-4z4 + 10a-4z6 - 3a-4z8 + 5a-3z-1 - 15a-3z + 20a-3z3 - 13a-3z5 + 4a-3z7 - a-3z9 - 5a-2 + 11a-2z2 - 10a-2z4 + 3a-2z6 - a-2z8 + 3a-1z-1 - 8a-1z + 7a-1z3 - 4a-1z5 + z2 - 2z4 + az - 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                 1   j = 12               3 1   j = 10             2 1     j = 8           4 3       j = 6         3 2         j = 4       2 4           j = 2     3 3             j = 0     3               j = -2 1 2                 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, 131]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 131]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[5, 13, 6, 12], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 14, 21, 13], > X[22, 16, 7, 15], X[19, 4, 20, 5], 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) 2 3/2 5/2 7/2 9/2 11/2 -q + ------- - 5 Sqrt[q] + 5 q - 7 q + 6 q - 5 q + 4 q - Sqrt[q] 13/2 15/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 4 6 8 10 12 14 16 18 22 3 + q + 2 q + 3 q + 4 q + q + 2 q - 2 q - q - q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 131]][a, z]
Out[8]=	3 3 3 5 5 5 7 2 5 3 5 z 12 z 6 z 4 z 13 z 4 z z 6 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, 131]][a, z]
Out[9]=	2 -8 5 5 2 5 3 6 z 15 z 8 z 2 4 z a - -- - -- + ---- + ---- + --- - --- - ---- - --- + a z + z - ---- + 4 2 5 3 a z 5 3 a 8 a a a z a z a a a 2 2 3 3 3 3 4 4 14 z 11 z 5 z 7 z 20 z 7 z 3 4 4 z 3 z > ----- + ----- - ---- + ---- + ----- + ---- - a z - 2 z + ---- - ---- - 4 2 7 5 3 a 8 6 a a a a a a a 4 4 5 5 5 5 6 6 6 6 15 z 10 z 7 z 2 z 13 z 4 z z 6 z 10 z 3 z > ----- - ----- + ---- - ---- - ----- - ---- - -- + ---- + ----- + ---- - 4 2 7 5 3 a 8 6 4 2 a a a a a a a a a 7 7 7 8 8 8 9 9 2 z 2 z 4 z 2 z 3 z z z z > ---- + ---- + ---- - ---- - ---- - -- - -- - -- 7 5 3 6 4 2 5 3 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 2 4 4 2 6 2 3 + 3 q + ----- + ----- + ---- + 3 q t + 2 q t + 4 q t + 3 q t + 4 2 2 2 2 q t q t q t 6 3 8 3 8 4 10 4 10 5 12 5 12 6 > 2 q t + 4 q t + 3 q t + 2 q t + q t + 3 q t + q t + 14 6 16 7 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n131

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