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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 6q5/2 - 9q7/2 + 8q9/2 - 7q11/2 + 6q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 + 4q8 + 2q10 + 2q12 + 2q14 - 2q16 - 3q20 - q22 + q24 - q26

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-10z2 - a-10z4 + 3a-9z3 - 3a-9z5 + 2a-8 - 5a-8z2 + 8a-8z4 - 5a-8z6 - a-7z-1 + 2a-7z - 4a-7z3 + 5a-7z5 - 4a-7z7 + 5a-6 - 13a-6z2 + 16a-6z4 - 7a-6z6 - a-6z8 - 2a-5z-1 + 3a-5z - 6a-5z3 + 10a-5z5 - 6a-5z7 + 3a-4 - 8a-4z2 + 11a-4z4 - 4a-4z6 - a-4z8 - 2a-3z + 4a-3z3 + a-3z5 - 2a-3z7 - a-2 - a-2z2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 3a-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               4 1   j = 12             3 2     j = 10           5 4       j = 8         4 3         j = 6       2 5           j = 4     3 4             j = 2   1 4               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, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[18, 14, 5, 13], X[14, 18, 15, 17], X[8, 16, 9, 15], X[2, 5, 3, 6], > X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 4, -2, 9, -3, 5, -6, 7, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 6 q - 9 q + 8 q - 7 q + Sqrt[q] 13/2 15/2 17/2 > 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 8 10 12 14 16 20 22 24 26 q + q + 3 q + 4 q + 2 q + 2 q + 2 q - 2 q - 3 q - q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 4]][a, z]
Out[8]=	3 3 3 3 5 5 1 2 1 z 2 z z 2 z z 2 z 2 z z z z ---- - ---- + --- + -- - --- - -- + --- + -- - ---- - ---- + -- - -- - -- 7 5 a z 7 5 3 a 7 5 3 a 5 3 a z a z a a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 4]][a, z]
Out[9]=	2 2 2 5 3 -2 1 2 1 2 z 3 z 2 z 3 z z 5 z -- + -- + -- - a - ---- - ---- + --- + --- + --- - --- - --- + --- - ---- - 8 6 4 7 5 a z 7 5 3 a 10 8 a a a a z a z a a a a a 2 2 2 3 3 3 3 3 4 4 4 13 z 8 z z 3 z 4 z 6 z 4 z 3 z z 8 z 16 z > ----- - ---- - -- + ---- - ---- - ---- + ---- + ---- - --- + ---- + ----- + 6 4 2 9 7 5 3 a 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 6 11 z 4 z 3 z 5 z 10 z z z 5 z 7 z 4 z 2 z > ----- + ---- - ---- + ---- + ----- + -- - -- - ---- - ---- - ---- - ---- - 4 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a a 7 7 7 8 8 4 z 6 z 2 z z z > ---- - ---- - ---- - -- - -- 7 5 3 6 4 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 4 q + 3 q + ----- + - + -- + 4 q t + 2 q t + 5 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 > 5 q t + 4 q t + 3 q t + 2 q t + 4 q t + q t + 2 q t + 18 7 > q t

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