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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - q-2 + 4q-1 - 4 + 6q - 6q2 + 6q3 - 4q4 + 3q5 - q6

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 2q-6 + 4q-4 + 3q-2 + 4 + 4q2 + q4 + 3q6 + 2q10 + q12 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4z2 - a-4z4 + a-2z-2 + 3a-2 + 5a-2z2 + 4a-2z4 + a-2z6 - 2z-2 - 6 - 7z2 - 2z4 + a2z-2 + 3a2 + a2z2

Kauffman Polynomial:

a-7z3 - 2a-6z2 + 3a-6z4 - 3a-5z3 + 4a-5z5 + a-4 - 4a-4z4 + 4a-4z6 + a-3z3 - 4a-3z5 + 3a-3z7 + a-2z-2 - 3a-2 + 5a-2z2 - 8a-2z4 + 2a-2z6 + a-2z8 - 2a-1z-1 + 6a-1z + 2a-1z3 - 10a-1z5 + 4a-1z7 + 2z-2 - 8 + 11z2 - 6z4 - z6 + z8 - 2az-1 + 6az - 3az3 - 2az5 + az7 + a2z-2 - 5a2 + 8a2z2 - 5a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 2   j = 9               2 1   j = 7             4 2     j = 5           4 4       j = 3         2 2         j = 1       3 5           j = -1     1 1             j = -3     3               j = -5 1 1                 j = -7 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, 48]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 48]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 14, 11, 13], X[8, 16, 9, 15], > X[14, 8, 15, 7], X[16, 10, 17, 9], X[10, 18, 5, 17], X[2, 5, 3, 6], > X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 5, -4, 6, -7}, {9, -2, 3, -5, 4, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 -2 4 2 3 4 5 6 -4 + q - q + - + 6 q - 6 q + 6 q - 4 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 4 3 2 4 6 10 12 16 18 4 + q + -- + -- + -- + -- + 4 q + q + 3 q + 2 q + q + q - q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 48]][a, z]
Out[8]=	2 2 2 4 3 2 2 1 a 2 2 z 5 z 2 2 4 z -6 + -- + 3 a - -- + ----- + -- - 7 z - ---- + ---- + a z - 2 z - -- + 2 2 2 2 2 4 2 4 a z a z z a a a 4 6 4 z z > ---- + -- 2 2 a a
In[9]:=	Kauffman[Link[9, Alternating, 48]][a, z]
Out[9]=	2 -4 3 2 2 1 a 2 2 a 6 z 2 -8 + a - -- - 5 a + -- + ----- + -- - --- - --- + --- + 6 a z + 11 z - 2 2 2 2 2 a z z a a z a z z 2 2 3 3 3 3 4 2 z 5 z 2 2 z 3 z z 2 z 3 4 3 z > ---- + ---- + 8 a z + -- - ---- + -- + ---- - 3 a z - 6 z + ---- - 6 2 7 5 3 a 6 a a a a a a 4 4 5 5 5 6 6 4 z 8 z 2 4 4 z 4 z 10 z 5 6 4 z 2 z > ---- - ---- - 5 a z + ---- - ---- - ----- - 2 a z - z + ---- + ---- + 4 2 5 3 a 4 2 a a a a a a 7 7 8 2 6 3 z 4 z 7 8 z > a z + ---- + ---- + a z + z + -- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 1 3 q 3 5 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 2 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 4 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

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