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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X14,8,5,7 X12,10,13,9 X8,14,9,13 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + q2 + q4 + q10 + q14 - q18

HOMFLY-PT Polynomial:

a-5z - a-3z - a-3z3 - a-1z-1 - a-1z - a-1z3 + az-1 + az

Kauffman Polynomial:

2a-6z2 - a-6z4 - 2a-5z + 5a-5z3 - 2a-5z5 + a-4z4 - a-4z6 - 2a-3z + 5a-3z3 - 3a-3z5 - 2a-2z2 + a-2z4 - a-2z6 - a-1z-1 + 2a-1z - a-1z3 - a-1z5 + 1 - z4 - az-1 + 2az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12               1 j = 10             1   j = 8           1 1   j = 6         2 1     j = 4       1 1       j = 2     2 2         j = 0   1 3           j = -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[7, Alternating, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[7, Alternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 5, 7], X[12, 10, 13, 9], > X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 2, -7}, {6, -1, 3, -5, 4, -2, 7, -4, 5, -3}]
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 - 3 q + 2 q - 2 q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 10 14 18 3 + q + q + q + q + q + q + q - q
In[8]:=	HOMFLYPT[Link[7, Alternating, 4]][a, z]
Out[8]=	3 3 1 a z z z z z -(---) + - + -- - -- - - + a z - -- - -- a z z 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[7, Alternating, 4]][a, z]
Out[9]=	2 2 3 3 3 1 a 2 z 2 z 2 z 2 z 2 z 5 z 5 z z 3 1 - --- - - - --- - --- + --- + 2 a z + ---- - ---- + ---- + ---- - -- - a z - a z z 5 3 a 6 2 5 3 a a a a a a a 4 4 4 5 5 5 6 6 4 z z z 2 z 3 z z z z > z - -- + -- + -- - ---- - ---- - -- - -- - -- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 4 2 6 2 6 3 8 3 3 + 2 q + ----- + - + 2 q t + q t + q t + 2 q t + q t + q t + 4 2 t q t 8 4 10 4 12 5 > q t + q t + q t

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