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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,8,17,7 X17,20,18,5 X11,19,12,18 X19,11,20,10 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

3 + 2q2 + 2q4 + 3q6 + q8 + q10 - q12 - q14 - q18 + q20 - q24

HOMFLY-PT Polynomial:

a-7z + a-5z-1 - a-5z3 - 3a-3z-1 - 4a-3z - 2a-3z3 + 2a-1z-1 + 3a-1z

Kauffman Polynomial:

- 4a-8z2 + 4a-8z4 - a-8z6 + 2a-7z - 8a-7z3 + 8a-7z5 - 2a-7z7 - a-6 + a-6z2 + a-6z4 + 2a-6z6 - a-6z8 + a-5z-1 - 5a-5z3 + 9a-5z5 - 3a-5z7 - 3a-4 + 9a-4z2 - 6a-4z4 + 3a-4z6 - a-4z8 + 3a-3z-1 - 7a-3z + 3a-3z3 + a-3z5 - a-3z7 - 3a-2 + 4a-2z2 - 3a-2z4 + 2a-1z-1 - 5a-1z

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16               1 j = 14             1   j = 12           2 1   j = 10         2 1     j = 8       2 2       j = 6     2 2         j = 4   1 2           j = 2 2 2             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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 34]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 34]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[17, 20, 18, 5], > X[11, 19, 12, 18], X[19, 11, 20, 10], X[14, 10, 15, 9], X[8, 16, 9, 15], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 7, 6, -5, -2, 10, -7, 8, -3, -4, 5, > -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -3 Sqrt[q] + 3 q - 4 q + 4 q - 4 q + 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 18 20 24 3 + 2 q + 2 q + 3 q + q + q - q - q - q + q - q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 34]][a, z]
Out[8]=	3 3 1 3 2 z 4 z 3 z z 2 z ---- - ---- + --- + -- - --- + --- - -- - ---- 5 3 a z 7 3 a 5 3 a z a z a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 34]][a, z]
Out[9]=	2 2 2 -6 3 3 1 3 2 2 z 7 z 5 z 4 z z 9 z -a - -- - -- + ---- + ---- + --- + --- - --- - --- - ---- + -- + ---- + 4 2 5 3 a z 7 3 a 8 6 4 a a a z a z a a a a a 2 3 3 3 4 4 4 4 5 5 5 4 z 8 z 5 z 3 z 4 z z 6 z 3 z 8 z 9 z z > ---- - ---- - ---- + ---- + ---- + -- - ---- - ---- + ---- + ---- + -- - 2 7 5 3 8 6 4 2 7 5 3 a a a a a a a a a a a 6 6 6 7 7 7 8 8 z 2 z 3 z 2 z 3 z z z z > -- + ---- + ---- - ---- - ---- - -- - -- - -- 8 6 4 7 5 3 6 4 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 4 2 6 2 6 3 8 3 8 4 3 + 2 q + 2 q t + q t + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 10 4 10 5 12 5 12 6 14 6 16 7 > 2 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n34

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