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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X16,8,17,7 X18,10,19,9 X20,12,5,11 X8,18,9,17 X10,20,11,19 X12,16,13,15 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 4q-1/2 + 4q1/2 - 6q3/2 + 7q5/2 - 7q7/2 + 6q9/2 - 4q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 2q-4 + 4q-2 + 3 + 3q2 + q4 - 2q6 - 3q10 - q14 - q16 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z - 3a-5z3 - a-5z5 + 2a-3z-1 + 6a-3z + 8a-3z3 + 5a-3z5 + a-3z7 - 5a-1z-1 - 11a-1z - 9a-1z3 - 2a-1z5 + 3az-1 + 4az + az3

Kauffman Polynomial:

- a-9z3 + 2a-8z2 - 3a-8z4 + 3a-7z3 - 4a-7z5 - a-6 + 2a-6z2 + 3a-6z4 - 4a-6z6 + a-5z - 3a-5z3 + 7a-5z5 - 4a-5z7 + a-4z2 - 4a-4z4 + 7a-4z6 - 3a-4z8 - 2a-3z-1 + 5a-3z - 11a-3z3 + 11a-3z5 - a-3z7 - a-3z9 + 5a-2 - 8a-2z2 - 8a-2z4 + 14a-2z6 - 4a-2z8 - 5a-1z-1 + 14a-1z - 16a-1z3 + 6a-1z5 + 2a-1z7 - a-1z9 + 5 - 9z2 + 2z4 + 3z6 - z8 - 3az-1 + 10az - 12az3 + 6az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                     1 j = 14                   2   j = 12                 2 1   j = 10               4 2     j = 8             3 2       j = 6           4 4         j = 4         2 3           j = 2       3 5             j = 0     1 1               j = -2     3                 j = -4 1 1                   j = -6 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, Alternating, 44]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 44]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[20, 12, 5, 11], X[8, 18, 9, 17], X[10, 20, 11, 19], X[12, 16, 13, 15], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7, 5, -8, 10, -2, 8, -3, 6, -4, > 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 4 3/2 5/2 7/2 9/2 -q + q - ------- + 4 Sqrt[q] - 6 q + 7 q - 7 q + 6 q - Sqrt[q] 11/2 13/2 15/2 > 4 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 2 4 2 4 6 10 14 16 20 22 3 + q + -- + -- + -- + 3 q + q - 2 q - 3 q - q - q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 44]][a, z]
Out[8]=	3 3 3 5 2 5 3 a z 6 z 11 z 3 z 8 z 9 z 3 z ---- - --- + --- - -- + --- - ---- + 4 a z - ---- + ---- - ---- + a z - -- + 3 a z z 5 3 a 5 3 a 5 a z a a a a a 5 5 7 5 z 2 z z > ---- - ---- + -- 3 a 3 a a
In[9]:=	Kauffman[Link[10, Alternating, 44]][a, z]
Out[9]=	2 -6 5 2 5 3 a z 5 z 14 z 2 2 z 5 - a + -- - ---- - --- - --- + -- + --- + ---- + 10 a z - 9 z + ---- + 2 3 a z z 5 3 a 8 a a z a a a 2 2 2 3 3 3 3 3 2 z z 8 z z 3 z 3 z 11 z 16 z 3 4 > ---- + -- - ---- - -- + ---- - ---- - ----- - ----- - 12 a z + 2 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 3 z 3 z 4 z 8 z 4 z 7 z 11 z 6 z 5 6 > ---- + ---- - ---- - ---- - ---- + ---- + ----- + ---- + 6 a z + 3 z - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 8 8 9 9 4 z 7 z 14 z 4 z z 2 z 7 8 3 z 4 z z z > ---- + ---- + ----- - ---- - -- + ---- - a z - z - ---- - ---- - -- - -- 6 4 2 5 3 a 4 2 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 3 1 3 q 4 5 q + 2 q + ----- + ----- + ----- + t + ----- + - + ---- + 3 q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 4 q t + 4 q t + 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + 12 5 14 5 16 6 > q t + 2 q t + q t

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