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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-22 + q-18 + 3q-16 + 3q-14 + 5q-12 + 4q-10 + 4q-8 + 3q-6 + q-4 + 2q-2

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 2a2z2 - 2a4z-2 - 4a4 - 3a4z2 - a4z4 + a6z-2 + a6 + a6z2

Kauffman Polynomial:

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

Khovanov Homology:

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

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