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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,11,17,12 X14,8,15,7 X8,14,9,13 X18,15,13,16 X12,17,5,18 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 5q-6 + 8q-5 - 7q-4 + 9q-3 - 7q-2 + 5q-1 - 2 + q

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 + 2q-18 + 5q-16 + 3q-14 + 6q-12 + 3q-10 + 3q-8 + 2q-6 - q-4 + 3q-2 + q4

HOMFLY-PT Polynomial:

1 + z2 + a2z-2 + a2 - a2z4 - 2a4z-2 - 4a4 - 4a4z2 - 2a4z4 + a6z-2 + 3a6 + 3a6z2 - a8

Kauffman Polynomial:

1 - 2z2 + z4 - 2az3 + 2az5 + a2z-2 - 3a2 + 3a2z2 - 3a2z4 + 3a2z6 - 2a3z-1 + 5a3z - a3z3 - 2a3z5 + 3a3z7 + 2a4z-2 - 10a4 + 20a4z2 - 19a4z4 + 7a4z6 + a4z8 - 2a5z-1 + 7a5z - 2a5z3 - 8a5z5 + 6a5z7 + a6z-2 - 9a6 + 20a6z2 - 22a6z4 + 7a6z6 + a6z8 + 3a7z - 5a7z3 - 3a7z5 + 3a7z7 - 2a8 + 5a8z2 - 7a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   1 j = 1                 1   j = -1               4 1   j = -3             5 3     j = -5           4 2       j = -7         3 5         j = -9       5 4           j = -11     2 5             j = -13   1 3               j = -15   2                 j = -17 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[9, Alternating, 52]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 52]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[14, 8, 15, 7], > X[8, 14, 9, 13], X[18, 15, 13, 16], X[12, 17, 5, 18], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 5 8 7 9 7 5 -2 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 2 5 3 6 3 3 2 -4 3 4 -q - q + --- + --- + --- + --- + --- + --- + -- + -- - q + -- + q 22 18 16 14 12 10 8 6 2 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 52]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 4 2 6 2 2 4 1 + a - 4 a + 3 a - a + -- - ---- + -- + z - 4 a z + 3 a z - a z - 2 2 2 z z z 4 4 > 2 a z
In[9]:=	Kauffman[Link[9, Alternating, 52]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 1 - 3 a - 10 a - 9 a - 2 a + -- + ---- + -- - ---- - ---- + 5 a z + 2 2 2 z z z z z 5 7 9 2 2 2 4 2 6 2 8 2 > 7 a z + 3 a z + a z - 2 z + 3 a z + 20 a z + 20 a z + 5 a z - 3 3 3 5 3 7 3 9 3 4 2 4 4 4 > 2 a z - a z - 2 a z - 5 a z - 2 a z + z - 3 a z - 19 a z - 6 4 8 4 5 3 5 5 5 7 5 9 5 > 22 a z - 7 a z + 2 a z - 2 a z - 8 a z - 3 a z + a z + 2 6 4 6 6 6 8 6 3 7 5 7 7 7 > 3 a z + 7 a z + 7 a z + 3 a z + 3 a z + 6 a z + 3 a z + 4 8 6 8 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 3 2 5 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 3 5 4 2 5 t 3 2 > ----- + ----- + ----- + ---- + ---- + - + q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

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