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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 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-9 + q-7 + q-5 + q-3

A2 (sl(3)) Invariant:

q-32 + q-30 + 2q-28 + 3q-26 + 4q-24 + 4q-22 + 3q-20 + 3q-18 + 2q-16 + 2q-14 + q-12 + q-10

HOMFLY-PT Polynomial:

a6z-2 + 5a6 + 10a6z2 + 6a6z4 + a6z6 - 2a8z-2 - 6a8 - 5a8z2 - a8z4 + a10z-2 + a10

Kauffman Polynomial:

a6z-2 - 5a6 + 10a6z2 - 6a6z4 + a6z6 - 2a7z-1 + 6a7z - 5a7z3 + a7z5 + 2a8z-2 - 8a8 + 11a8z2 - 6a8z4 + a8z6 - 2a9z-1 + 6a9z - 5a9z3 + a9z5 + a10z-2 - 3a10 + a10z2 + a12

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5             1 j = -7             1 j = -9         1     j = -11     1         j = -13     2 1       j = -15 1             j = -17 2 1           j = -19 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, 3]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, NonAlternating, 3]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > 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]=	-9 -7 -5 -3 q + q + q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 3 4 4 3 3 2 2 -12 -10 q + q + --- + --- + --- + --- + --- + --- + --- + --- + q + q 28 26 24 22 20 18 16 14 q q q q q q q q
In[8]:=	HOMFLYPT[Link[8, NonAlternating, 3]][a, z]
Out[8]=	6 8 10 6 8 10 a 2 a a 6 2 8 2 6 4 8 4 5 a - 6 a + a + -- - ---- + --- + 10 a z - 5 a z + 6 a z - a z + 2 2 2 z z z 6 6 > a z
In[9]:=	Kauffman[Link[8, NonAlternating, 3]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 a 2 a a 2 a 2 a 7 9 -5 a - 8 a - 3 a + a + -- + ---- + --- - ---- - ---- + 6 a z + 6 a z + 2 2 2 z z z z z 6 2 8 2 10 2 7 3 9 3 6 4 8 4 > 10 a z + 11 a z + a z - 5 a z - 5 a z - 6 a z - 6 a z + 7 5 9 5 6 6 8 6 > a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 1 1 2 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- 19 6 17 6 15 6 17 5 13 4 11 4 13 3 9 2 q t q t q t q t q t q t q t q t

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