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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X14,9,15,10 X2,14,3,13 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 4q-5/2 - 7q-3/2 + 8q-1/2 - 8q1/2 + 7q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 + q-8 + 4q-6 - 1 - 2q2 + 2q4 + 3q8 + q10 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - a-3z3 + 2a-1z-1 + 2a-1z + 2a-1z3 + a-1z5 - 2az-1 + 2az3 + az5 + a3z-1 - a3z - a3z3

Kauffman Polynomial:

- a-5z3 - 3a-4z4 + a-3z-1 - 3a-3z + 6a-3z3 - 6a-3z5 - 4a-2z2 + 10a-2z4 - 7a-2z6 + 2a-1z-1 - 6a-1z + 5a-1z3 + 5a-1z5 - 5a-1z7 + 1 - 6z2 + 14z4 - 3z6 - 2z8 + 2az-1 - 2az - 13az3 + 22az5 - 8az7 - 4a2z2 + 4a2z4 + 3a2z6 - 2a2z8 + a3z-1 + a3z - 11a3z3 + 11a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:

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

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