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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 3q-3 + 5q-2 - 5q-1 + 8 - 5q + 5q2 - 3q3 + q4

A2 (sl(3)) Invariant:

q-12 - q-10 + q-8 + q-6 + 2q-4 + 7q-2 + 5 + 7q2 + 2q4 + q6 + q8 - q10 + q12

HOMFLY-PT Polynomial:

a-2z-2 + a-2 + 2a-2z2 + a-2z4 - 2z-2 - 2 - 5z2 - 4z4 - z6 + a2z-2 + a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- a-4z2 + a-4z4 - 4a-3z3 + 3a-3z5 + a-2z-2 - 2a-2 + 5a-2z2 - 7a-2z4 + 4a-2z6 - 2a-1z-1 + 2a-1z - 4a-1z3 + a-1z5 + 2a-1z7 + 2z-2 - 3 + 12z2 - 16z4 + 8z6 - 2az-1 + 2az - 4az3 + az5 + 2az7 + a2z-2 - 2a2 + 5a2z2 - 7a2z4 + 4a2z6 - 4a3z3 + 3a3z5 - a4z2 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                 1 j = 7               2   j = 5             3 1   j = 3           3 3     j = 1         5 2       j = -1       2 5         j = -3     3 3           j = -5   1 3             j = -7   2               j = -9 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, Alternating, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 19]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[8, 12, 9, 11], > X[16, 8, 11, 7], X[10, 13, 5, 14], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 5 5 2 3 4 8 + q - -- + -- - - - 5 q + 5 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 -6 2 7 2 4 6 8 10 12 5 + q - q + q + q + -- + -- + 7 q + 2 q + q + q - q + q 4 2 q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 19]][a, z]
Out[8]=	2 2 4 -2 2 2 1 a 2 2 z 2 2 4 z 2 4 6 -2 + a + a - -- + ----- + -- - 5 z + ---- + 2 a z - 4 z + -- + a z - z 2 2 2 2 2 2 z a z z a a
In[9]:=	Kauffman[Link[8, Alternating, 19]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z + 12 z - -- + 2 2 2 2 2 a z z a 4 a z a z z a 2 3 3 4 5 z 2 2 4 2 4 z 4 z 3 3 3 4 z > ---- + 5 a z - a z - ---- - ---- - 4 a z - 4 a z - 16 z + -- - 2 3 a 4 a a a 4 5 5 6 7 z 2 4 4 4 3 z z 5 3 5 6 4 z > ---- - 7 a z + a z + ---- + -- + a z + 3 a z + 8 z + ---- + 2 3 a 2 a a a 7 2 6 2 z 7 > 4 a z + ---- + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 3 3 3 2 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + 3 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 9 4 > 3 q t + 3 q t + q t + 2 q t + q t

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