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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X18,12,9,11 X2,9,3,10 X4,18,5,17 X16,8,17,7 X14,6,15,5 X6,16,7,15 X8,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 4q3/2 + 6q5/2 - 8q7/2 + 7q9/2 - 7q11/2 + 5q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-2 + q4 - 2q6 + 2q8 + q10 + 2q12 + 3q14 + 2q18 - q20 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 - a-5z-1 - a-5z - 2a-5z3 - a-5z5 + a-3z-1 - 2a-3z3 - a-3z5 + 2a-1z + a-1z3

Kauffman Polynomial:

a-10z2 - a-10z4 - a-9z + 4a-9z3 - 3a-9z5 - a-8z2 + 5a-8z4 - 4a-8z6 + 2a-7z5 - 3a-7z7 - 3a-6z2 + 7a-6z4 - 4a-6z6 - a-6z8 - a-5z-1 + 5a-5z - 8a-5z3 + 9a-5z5 - 5a-5z7 + a-4 - 4a-4z2 + 6a-4z4 - 2a-4z6 - a-4z8 - a-3z-1 + 2a-3z - a-3z3 + 3a-3z5 - 2a-3z7 - 3a-2z2 + 5a-2z4 - 2a-2z6 - 2a-1z + 3a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                   1 j = 16                 2   j = 14               3 1   j = 12             4 2     j = 10           4 4       j = 8         4 3         j = 6       2 4           j = 4     2 4             j = 2   1 3               j = 0   1                 j = -2 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, 37]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 37]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[2, 9, 3, 10], > X[4, 18, 5, 17], X[16, 8, 17, 7], X[14, 6, 15, 5], X[6, 16, 7, 15], > X[8, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 4 q + 6 q - 8 q + 7 q - 7 q + Sqrt[q] 13/2 15/2 17/2 > 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 12 14 18 20 24 26 q + q - 2 q + 2 q + q + 2 q + 3 q + 2 q - q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 37]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z z 2 z z 2 z 2 z z z z -(----) + ---- + -- - -- + --- + -- - ---- - ---- + -- - -- - -- 5 3 7 5 a 7 5 3 a 5 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 37]][a, z]
Out[9]=	2 2 2 2 2 -4 1 1 z 5 z 2 z 2 z z z 3 z 4 z 3 z a - ---- - ---- - -- + --- + --- - --- + --- - -- - ---- - ---- - ---- + 5 3 9 5 3 a 10 8 6 4 2 a z a z a a a a a a a a 3 3 3 3 4 4 4 4 4 5 5 4 z 8 z z 3 z z 5 z 7 z 6 z 5 z 3 z 2 z > ---- - ---- - -- + ---- - --- + ---- + ---- + ---- + ---- - ---- + ---- + 9 5 3 a 10 8 6 4 2 9 7 a a a a a a a a a a 5 5 5 6 6 6 6 7 7 7 8 8 9 z 3 z z 4 z 4 z 2 z 2 z 3 z 5 z 2 z z z > ---- + ---- - -- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - -- - -- 5 3 a 8 6 4 2 7 5 3 6 4 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 q 4 6 6 2 8 2 8 3 3 q + 2 q + ----- + - + -- + 4 q t + 2 q t + 4 q t + 4 q t + 3 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 16 6 > 4 q t + 4 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 18 7 > q t

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