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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 5q-1 + 6 - 5q + 5q2 - 2q3 + q4

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + 2q-6 + 2q-4 + 5q-2 + 3 + 5q2 + 2q4 + 2q6 + 2q8 - q10 + q12 + q14

HOMFLY-PT Polynomial:

a-4 + a-2z-2 - 2a-2z2 - 2z-2 - 2 + z4 + a2z-2 - 2a2z2 + a4

Kauffman Polynomial:

a-4 - 2a-4z2 + a-4z4 - 2a-3z3 + 2a-3z5 + a-2z-2 - 4a-2 + 5a-2z2 - 5a-2z4 + 3a-2z6 - 2a-1z-1 + 8a-1z - 12a-1z3 + 5a-1z5 + a-1z7 + 2z-2 - 9 + 14z2 - 12z4 + 6z6 - 2az-1 + 8az - 12az3 + 5az5 + az7 + a2z-2 - 4a2 + 5a2z2 - 5a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

Khovanov Homology:

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

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