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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X11,20,12,13 X7,18,8,19 X9,16,10,17 X17,8,18,9 X13,10,14,11 X19,12,20,5 X2536 X4,16,1,15

Gauss Code:

{{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}}

Jones Polynomial:

q-8 + q-6 + q-5 - 2q-4 + 3q-3 - 2q-2 + 3q-1 - 2 + q

A2 (sl(3)) Invariant:

q-26 + 2q-24 + 3q-22 + 4q-20 + 4q-18 + 5q-16 + 2q-14 + q-12 + q-10 + 2q-6 + q-2 + q4

HOMFLY-PT Polynomial:

1 + z2 - 2a2z2 - a2z4 + a4z-2 + a4 + a4z2 - 2a6z-2 - 3a6 - a6z2 + a8z-2 + a8

Kauffman Polynomial:

1 - 2z2 + z4 - 4az3 + 2az5 - 3a2z2 + a2z6 - 2a3z3 + 2a3z5 - a4z-2 + 5a4 - 6a4z2 + 3a4z4 + 2a5z-1 - 11a5z + 19a5z3 - 8a5z5 + a5z7 - 2a6z-2 + 13a6 - 27a6z2 + 25a6z4 - 9a6z6 + a6z8 + 2a7z-1 - 11a7z + 17a7z3 - 8a7z5 + a7z7 - a8z-2 + 8a8 - 22a8z2 + 21a8z4 - 8a8z6 + a8z8

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                     1 j = 1                   1   j = -1                 2 1   j = -3               2 3     j = -5           1 2         j = -7           1 2         j = -9       1 2 2           j = -11         2             j = -13     1                 j = -15 1                     j = -17 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[10, NonAlternating, 82]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 82]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[11, 20, 12, 13], X[7, 18, 8, 19], > X[9, 16, 10, 17], X[17, 8, 18, 9], X[13, 10, 14, 11], X[19, 12, 20, 5], > X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, > {-7, -2, 10, 5, -6, 4, -8, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -6 -5 2 3 2 3 -2 + q + q + q - -- + -- - -- + - + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 3 4 4 5 2 -12 -10 2 -2 4 q + --- + --- + --- + --- + --- + --- + q + q + -- + q + q 24 22 20 18 16 14 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 82]][a, z]
Out[8]=	4 6 8 4 6 8 a 2 a a 2 2 2 4 2 6 2 2 4 1 + a - 3 a + a + -- - ---- + -- + z - 2 a z + a z - a z - a z 2 2 2 z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 82]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 1 + 5 a + 13 a + 8 a - -- - ---- - -- + ---- + ---- - 11 a z - 11 a z - 2 2 2 z z z z z 2 2 2 4 2 6 2 8 2 3 3 3 > 2 z - 3 a z - 6 a z - 27 a z - 22 a z - 4 a z - 2 a z + 5 3 7 3 4 4 4 6 4 8 4 5 > 19 a z + 17 a z + z + 3 a z + 25 a z + 21 a z + 2 a z + 3 5 5 5 7 5 2 6 6 6 8 6 5 7 7 7 > 2 a z - 8 a z - 8 a z + a z - 9 a z - 8 a z + a z + a z + 6 8 8 8 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 1 1 2 2 2 1 -- + - + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 3 q 17 8 15 8 13 6 9 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t q t 1 2 2 2 t 3 2 > ----- + ----- + ----- + ---- + - + q t + q t 5 3 7 2 5 2 3 q q t q t q t q t

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