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DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X14,20,15,19 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 4q-7/2 - 6q-5/2 + 5q-3/2 - 4q-1/2 + 3q1/2 - q3/2

A2 (sl(3)) Invariant:

q-20 + 2q-16 + 3q-14 + q-12 + 3q-10 + q-8 + q-6 - 2q-2 - 2q2 + q6

HOMFLY-PT Polynomial:

- a-1z + az-1 + 3az + 2az3 - 3a3z-1 - 7a3z - 4a3z3 - a3z5 + 2a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

a-1z - a-1z3 + 1 + 3z2 - 3z4 - az-1 + 3az - 5az3 + 2az5 - az7 + 3a2 - 4a2z2 - 2a2z4 + 2a2z6 - a2z8 - 3a3z-1 + 12a3z - 21a3z3 + 14a3z5 - 4a3z7 + 3a4 - 8a4z2 + 6a4z4 - a4z8 - 2a5z-1 + 9a5z - 14a5z3 + 11a5z5 - 3a5z7 - a6z2 + 5a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                 1 j = 2               2   j = 0             2 1   j = -2           4 3     j = -4         2 1       j = -6       2 4         j = -8     2 2           j = -10     2             j = -12 1 2               j = -14 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[10, NonAlternating, 47]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 47]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[6, 7, 1, 8], X[20, 14, 7, 13], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[14, 20, 15, 19], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -6, -7, 5, -3}, > {3, -1, -2, 6, 8, -5, 4, -9, 7, -8, -10, 2, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 4 6 5 4 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 3 Sqrt[q] - q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 3 -12 3 -8 -6 2 2 6 q + --- + --- + q + --- + q + q - -- - 2 q + q 16 14 10 2 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 47]][a, z]
Out[8]=	3 5 a 3 a 2 a z 3 5 3 3 3 5 3 3 5 - - ---- + ---- - - + 3 a z - 7 a z + 2 a z + 2 a z - 4 a z + a z - a z z z z a
In[9]:=	Kauffman[Link[10, NonAlternating, 47]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a z 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + - + 3 a z + 12 a z + 9 a z - a z + z z z a 3 2 2 2 4 2 6 2 z 3 3 3 5 3 > 3 z - 4 a z - 8 a z - a z - -- - 5 a z - 21 a z - 14 a z + a 7 3 4 2 4 4 4 6 4 5 3 5 > 3 a z - 3 z - 2 a z + 6 a z + 5 a z + 2 a z + 14 a z + 5 5 7 5 2 6 6 6 7 3 7 5 7 2 8 > 11 a z - a z + 2 a z - 2 a z - a z - 4 a z - 3 a z - a z - 4 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 2 2 2 2 4 2 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 2 1 4 2 4 2 > ----- + ---- + ---- + t + 2 q t + q t 4 2 4 2 q t q t q t

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