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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X16,8,5,7 X14,10,15,9 X10,14,11,13 X8,16,9,15 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-9/2 + q-7/2 - 3q-5/2 + 3q-3/2 - 4q-1/2 + 3q1/2 - 2q3/2 + 2q5/2 - q7/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 + q-12 + 2q-10 + 2q-8 + q-4 + 1 - q4 - q8 + q12

HOMFLY-PT Polynomial:

- a-3z + a-1z + a-1z3 + az3 - a3z-1 - 2a3z + a5z-1

Kauffman Polynomial:

- a-3z + 3a-3z3 - a-3z5 - 5a-2z2 + 7a-2z4 - 2a-2z6 + 2a-1z5 - a-1z7 - 5z2 + 8z4 - 3z6 + az - 3az3 + 2az5 - az7 - a2z6 - a3z-1 + 2a3z - a3z3 - a3z5 + a4 - a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

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

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