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L11n169 L11n171

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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,21,14,20 X19,13,20,12 X22,16,7,15 X4,22,5,21

Gauss Code:

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

Jones Polynomial:

- q-3/2 + 3q-1/2 - 7q1/2 + 7q3/2 - 10q5/2 + 9q7/2 - 7q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-4 - q-2 + 3 + 2q2 + 4q4 + 6q6 + q8 + 2q10 - 4q12 - 2q14 - q16 - 2q18 + q20 - q22

HOMFLY-PT Polynomial:

2a-5z-1 + 3a-5z + 3a-5z3 + a-5z5 - 5a-3z-1 - 8a-3z - 9a-3z3 - 5a-3z5 - a-3z7 + 3a-1z-1 + 4a-1z + 3a-1z3 + a-1z5

Kauffman Polynomial:

a-8 - 3a-8z2 + 3a-8z4 - a-8z6 - 6a-7z3 + 9a-7z5 - 3a-7z7 - 6a-6z2 + 3a-6z4 + 6a-6z6 - 3a-6z8 + 2a-5z-1 - 3a-5z - 2a-5z3 + 11a-5z5 - 3a-5z7 - a-5z9 - 5a-4 + 2a-4z2 + 3a-4z4 + 5a-4z6 - 4a-4z8 + 5a-3z-1 - 10a-3z + 14a-3z3 - 5a-3z5 - a-3z9 - 5a-2 + 5a-2z2 - 2a-2z6 - a-2z8 + 3a-1z-1 - 7a-1z + 9a-1z3 - 7a-1z5 - 3z4 - 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                 2   j = 12               4 1   j = 10             3 2     j = 8           6 4       j = 6         4 3         j = 4       3 6           j = 2     4 4             j = 0     4               j = -2 1 3                 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, 170]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 170]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[14, 5, 15, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[13, 21, 14, 20], > X[19, 13, 20, 12], X[22, 16, 7, 15], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, -2, 9, -8, -5, 10, -7, 4, 2, -9, 8, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 11/2 -q + ------- - 7 Sqrt[q] + 7 q - 10 q + 9 q - 7 q + 6 q - Sqrt[q] 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 6 8 10 12 14 16 18 3 + q - q + 2 q + 4 q + 6 q + q + 2 q - 4 q - 2 q - q - 2 q + 20 22 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 170]][a, z]
Out[8]=	3 3 3 5 5 5 7 2 5 3 3 z 8 z 4 z 3 z 9 z 3 z z 5 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, 170]][a, z]
Out[9]=	2 2 2 -8 5 5 2 5 3 3 z 10 z 7 z 3 z 6 z 2 z a - -- - -- + ---- + ---- + --- - --- - ---- - --- - ---- - ---- + ---- + 4 2 5 3 a z 5 3 a 8 6 4 a a a z a z a a a a a 2 3 3 3 3 4 4 4 5 z 6 z 2 z 14 z 9 z 3 4 3 z 3 z 3 z > ---- - ---- - ---- + ----- + ---- - a z - 3 z + ---- + ---- + ---- + 2 7 5 3 a 8 6 4 a a a a a a a 5 5 5 5 6 6 6 6 7 7 8 9 z 11 z 5 z 7 z z 6 z 5 z 2 z 3 z 3 z 3 z > ---- + ----- - ---- - ---- - -- + ---- + ---- - ---- - ---- - ---- - ---- - 7 5 3 a 8 6 4 2 7 5 6 a a a a a a a a a a 8 8 9 9 4 z z z z > ---- - -- - -- - -- 4 2 5 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 3 2 4 4 2 6 2 4 + 4 q + ----- + ----- + ---- + 4 q t + 3 q t + 6 q t + 4 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 > 3 q t + 6 q t + 4 q t + 3 q t + 2 q t + 4 q t + q t + 14 6 16 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n170

L11n169 L11n171