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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-7/2 + q-5/2 - q-3/2 + q-1/2 - 2q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

q-12 + q-10 + q-8 + q-6 + q4 + q6 + 2q8 + q10

HOMFLY-PT Polynomial:

- a-3z-1 + 2a-1z-1 + 2a-1z - 2az-1 - 3az - az3 + a3z-1 + a3z

Kauffman Polynomial:

a-3z-1 - a-3z - a-2z2 + 2a-1z-1 - 6a-1z + 4a-1z3 - a-1z5 + 1 - 4z2 + 4z4 - z6 + 2az-1 - 8az + 8az3 - 2az5 - 3a2z2 + 4a2z4 - a2z6 + a3z-1 - 3a3z + 4a3z3 - a3z5

Khovanov Homology:

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

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