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DrawMorseLink

PD Presentation:

X8192 X18,12,19,11 X3,10,4,11 X19,2,20,3 X16,8,17,7 X9,13,10,20 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16

Gauss Code:

{{1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, {8, -9, 10, -5, 7, -2, -4, 6}}

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-22 + 2q-16 + 2q-12 + 2q-10 + 2q-8 + 4q-6 + 2q-4 + 5q-2 + 3 + 2q2 + 2q4

HOMFLY-PT Polynomial:

z-2 + 4 + 2z2 - 2a2z-2 - 6a2 - 6a2z2 - 2a2z4 + a4z-2 + a4 - a4z2 - a4z4 + a6 + a6z2

Kauffman Polynomial:

- z-2 + 7 - 9z2 + 3z4 + 2az-1 - 9az + 7az3 - 3az5 + az7 - 2a2z-2 + 11a2 - 21a2z2 + 14a2z4 - 4a2z6 + a2z8 + 2a3z-1 - 9a3z + 13a3z3 - 8a3z5 + 3a3z7 - a4z-2 + 3a4 - 3a4z2 + 3a4z4 - a4z6 + a4z8 + 3a5z3 - 3a5z5 + 2a5z7 - 2a6 + 7a6z2 - 7a6z4 + 3a6z6 - 3a7z3 + 2a7z5 - 2a8z2 + a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                 2 j = 1               1 1 j = -1             4 1   j = -3           3 3     j = -5         3 2       j = -7       2 3         j = -9     2 3           j = -11     2             j = -13 1 2               j = -15 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[10, NonAlternating, 95]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 95]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 12, 19, 11], X[3, 10, 4, 11], X[19, 2, 20, 3], > X[16, 8, 17, 7], X[9, 13, 10, 20], X[12, 18, 7, 17], X[4, 13, 5, 14], > X[14, 5, 15, 6], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, > {8, -9, 10, -5, 7, -2, -4, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 4 5 6 5 5 -2 + q - -- + -- - -- + -- - -- + - + 2 q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 2 2 4 2 5 2 4 3 + q + --- + --- + --- + -- + -- + -- + -- + 2 q + 2 q 16 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 95]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 6 2 4 - 6 a + a + a + z - ---- + -- + 2 z - 6 a z - a z + a z - 2 2 z z 2 4 4 4 > 2 a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 95]][a, z]
Out[9]=	2 4 3 2 4 6 -2 2 a a 2 a 2 a 3 7 + 11 a + 3 a - 2 a - z - ---- - -- + --- + ---- - 9 a z - 9 a z - 2 2 z z z z 2 2 2 4 2 6 2 8 2 3 3 3 > 9 z - 21 a z - 3 a z + 7 a z - 2 a z + 7 a z + 13 a z + 5 3 7 3 4 2 4 4 4 6 4 8 4 5 > 3 a z - 3 a z + 3 z + 14 a z + 3 a z - 7 a z + a z - 3 a z - 3 5 5 5 7 5 2 6 4 6 6 6 7 3 7 > 8 a z - 3 a z + 2 a z - 4 a z - a z + 3 a z + a z + 3 a z + 5 7 2 8 4 8 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 2 2 2 3 2 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 3 2 3 t 2 3 2 > ----- + ---- + ---- + - + q t + q t + 2 q t 5 2 5 3 q q t q t q t

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