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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X18,16,5,15 X16,12,17,11 X12,18,13,17 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-16 + 3q-14 + q-12 + 2q-10 + q-8 - q-6 + q-4 - q-2 + 2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 - 3az - 3az3 - az5 - a3z-1 + a3z3 + a5z-1

Kauffman Polynomial:

a-2z2 - a-2z4 - a-1z + 5a-1z3 - 3a-1z5 + 4z4 - 3z6 - 2az + 6az3 - 3az5 - az7 - 2a2z2 + 5a2z4 - 4a2z6 - a3z-1 + 3a3z - 2a3z3 - a3z7 + a4 - a4z2 - a4z6 - a5z-1 + 4a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6               1 j = 4             2   j = 2           2 1   j = 0         3 2     j = -2       3 3       j = -4     2 2         j = -6   1 3           j = -8 1 2             j = -10 2

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

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