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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X18,16,7,15 X14,5,15,6 X4,13,5,14 X12,18,13,17 X16,12,17,11 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 6q-7/2 - 8q-5/2 + 8q-3/2 - 7q-1/2 + 5q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

q-20 + 2q-16 + 4q-14 + q-12 + 3q-10 - q-6 - 2q-2 + 2 - q2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 + az-1 - az - 2az3 - az5 - 3a3z-1 - 5a3z - 3a3z3 - a3z5 + 2a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

a-2z2 - a-2z4 - a-1z + 4a-1z3 - 3a-1z5 + 1 - 2z2 + 5z4 - 4z6 - az-1 + 2az3 + az5 - 3az7 + 3a2 - 13a2z2 + 15a2z4 - 6a2z6 - a2z8 - 3a3z-1 + 11a3z - 15a3z3 + 13a3z5 - 6a3z7 + 3a4 - 11a4z2 + 13a4z4 - 4a4z6 - a4z8 - 2a5z-1 + 8a5z - 10a5z3 + 8a5z5 - 3a5z7 - a6z2 + 4a6z4 - 2a6z6 - 2a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   1 j = 4                 2   j = 2               3 1   j = 0             4 2     j = -2           5 4       j = -4         3 3         j = -6       3 5           j = -8     2 3             j = -10     3               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[9, Alternating, 23]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 23]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[18, 16, 7, 15], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[12, 18, 13, 17], X[16, 12, 17, 11], X[2, 7, 3, 8], > X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 6 8 8 7 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 5 Sqrt[q] - 3 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 4 -12 3 -6 2 2 6 8 2 + q + --- + --- + q + --- - q - -- - q + q - q 16 14 10 2 q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 23]][a, z]
Out[8]=	3 5 3 a 3 a 2 a z 3 5 z 3 3 3 5 3 - - ---- + ---- + - - a z - 5 a z + 2 a z + -- - 2 a z - 3 a z + a z - z z z a a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[9, Alternating, 23]][a, z]
Out[9]=	3 5 2 2 4 a 3 a 2 a z 3 5 7 2 z 1 + 3 a + 3 a - - - ---- - ---- - - + 11 a z + 8 a z - 2 a z - 2 z + -- - z z z a 2 a 3 2 2 4 2 6 2 4 z 3 3 3 5 3 > 13 a z - 11 a z - a z + ---- + 2 a z - 15 a z - 10 a z + a 4 5 7 3 4 z 2 4 4 4 6 4 3 z 5 > 3 a z + 5 z - -- + 15 a z + 13 a z + 4 a z - ---- + a z + 2 a a 3 5 5 5 7 5 6 2 6 4 6 6 6 7 > 13 a z + 8 a z - a z - 4 z - 6 a z - 4 a z - 2 a z - 3 a z - 3 7 5 7 2 8 4 8 > 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 2 3 2 3 3 5 4 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 3 3 5 2 2 2 4 2 6 3 > ----- + ---- + ---- + 2 t + 3 q t + q t + 2 q t + q t 4 2 4 2 q t q t q t

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