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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - q-3 + 4q-2 - 4q-1 + 7 - 6q + 5q2 - 4q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-14 + 2q-12 + 2q-10 + 5q-8 + 5q-6 + 3q-4 + 5q-2 + 2 + 2q2 - q6 + q8 - q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z2 + a-2z4 + z-2 + 2 + z2 + z4 - 2a2z-2 - 3a2 - 2a2z2 + a4z-2 + a4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 8a-4z4 + 3a-4z6 + 5a-3z3 - 8a-3z5 + 3a-3z7 + 2a-2z2 - 5a-2z4 + a-2z6 + a-2z8 + 7a-1z3 - 9a-1z5 + 4a-1z7 - z-2 + 3 - 4z2 + 4z4 - z6 + z8 + 2az-1 - 3az + az5 + az7 - 2a2z-2 + 5a2 - 6a2z2 + 2a2z4 + a2z6 + 2a3z-1 - 3a3z + a3z5 - a4z-2 + 3a4 - 3a4z2 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                   1 j = 9                 2   j = 7               2 1   j = 5             3 2     j = 3           3 2       j = 1         4 3         j = -1       3 6           j = -3     1 1             j = -5     3               j = -7 1 1                 j = -9 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, 45]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 45]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 9, 11], X[16, 14, 17, 13], > X[8, 16, 5, 15], X[14, 8, 15, 7], X[12, 18, 13, 17], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -3 4 4 2 3 4 5 7 + q - q + -- - - - 6 q + 5 q - 4 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 5 5 3 5 2 6 8 10 12 14 2 + q + --- + --- + -- + -- + -- + -- + 2 q - q + q - q + q + q - 12 10 8 6 4 2 q q q q q q 16 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 45]][a, z]
Out[8]=	2 4 2 2 4 2 4 -2 2 a a 2 z z 2 2 4 z 2 - 3 a + a + z - ---- + -- + z - -- + -- - 2 a z + z + -- 2 2 4 2 2 z z a a a
In[9]:=	Kauffman[Link[9, Alternating, 45]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 3 z 3 + 5 a + 3 a - z - ---- - -- + --- + ---- - 3 a z - 3 a z - 4 z + ---- + 2 2 z z 4 z z a 2 3 3 3 4 4 2 z 2 2 4 2 2 z 5 z 7 z 4 8 z 5 z > ---- - 6 a z - 3 a z - ---- + ---- + ---- + 4 z - ---- - ---- + 2 5 3 a 4 2 a a a a a 5 5 5 6 6 2 4 4 4 z 8 z 9 z 5 3 5 6 3 z z > 2 a z + a z + -- - ---- - ---- + a z + a z - z + ---- + -- + 5 3 a 4 2 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]=	6 1 1 1 3 1 1 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 3 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

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