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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X11,13,12,16 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,9,16,12

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

3 + 3q2 + 5q4 + 8q6 + 10q8 + 13q10 + 12q12 + 11q14 + 8q16 + 4q18 + 3q20 + q22

HOMFLY-PT Polynomial:

- a-7z-3 - a-7z-1 + 3a-5z-3 + 5a-5z-1 + 3a-5z - 3a-3z-3 - 7a-3z-1 - 6a-3z - 2a-3z3 + a-1z-3 + 3a-1z-1 + 3a-1z

Kauffman Polynomial:

- a-7z-3 + 4a-7z-1 - 6a-7z + 4a-7z3 - a-7z5 + 3a-6z-2 - 8a-6 + 6a-6z2 + a-6z4 - a-6z6 - 3a-5z-3 + 9a-5z-1 - 14a-5z + 16a-5z3 - 5a-5z5 + 6a-4z-2 - 15a-4 + 12a-4z2 - 2a-4z4 - a-4z6 - 3a-3z-3 + 9a-3z-1 - 14a-3z + 12a-3z3 - 4a-3z5 + 3a-2z-2 - 8a-2 + 6a-2z2 - 3a-2z4 - a-1z-3 + 4a-1z-1 - 6a-1z

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14             1 j = 12               j = 10         4 1   j = 8       1 4     j = 6     3         j = 4 1   1         j = 2 4 3           j = 0 3

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8n7

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