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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-7 - q-6 + 2q-5 - 2q-4 + 3q-3 - q-2 + 2q-1

A2 (sl(3)) Invariant:

q-22 + q-20 + 2q-18 + 3q-16 + 2q-14 + 3q-12 + 3q-10 + 4q-8 + 4q-6 + 2q-4 + 2q-2

HOMFLY-PT Polynomial:

a2z-2 + 4a2 + 2a2z2 - 2a4z-2 - 6a4 - 4a4z2 - a4z4 + a6z-2 + 2a6 + a6z2

Kauffman Polynomial:

a2z-2 - 5a2 + 3a2z2 - 2a3z-1 + 6a3z - 3a3z3 + a3z5 + 2a4z-2 - 8a4 + 10a4z2 - 4a4z4 + a4z6 - 2a5z-1 + 6a5z - 5a5z3 + 2a5z5 + a6z-2 - 3a6 + 4a6z2 - 3a6z4 + a6z6 - 2a7z3 + a7z5 + a8 - 3a8z2 + a8z4

Khovanov Homology:

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

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