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DrawMorseLink

PD Presentation:

X8192 X18,10,19,9 X17,6,18,1 X7,17,8,16 X3,10,4,11 X5,14,6,15 X13,4,14,5 X11,13,12,20 X15,7,16,12 X2,19,3,20

Gauss Code:

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

Jones Polynomial:

q-6 - q-5 + 2q-4 - 2q-3 + 3q-2 - 2q-1 + 3 - q + q2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-14 + 2q-12 + 2q-10 + 3q-8 + 2q-6 + 3q-4 + 3q-2 + 3 + 3q2 + q4 + q6

HOMFLY-PT Polynomial:

z-2 + 5 + 4z2 + z4 - 2a2z-2 - 9a2 - 12a2z2 - 6a2z4 - a2z6 + a4z-2 + 4a4 + 4a4z2 + a4z4

Kauffman Polynomial:

- a-2 + a-2z2 + a-1z3 - z-2 + 5 - 4z2 + 2z4 + 2az-1 - 9az + 11az3 - 5az5 + az7 - 2a2z-2 + 11a2 - 21a2z2 + 16a2z4 - 6a2z6 + a2z8 + 2a3z-1 - 9a3z + 13a3z3 - 9a3z5 + 2a3z7 - a4z-2 + 5a4 - 10a4z2 + 9a4z4 - 5a4z6 + a4z8 + 3a5z3 - 4a5z5 + a5z7 - a6 + 6a6z2 - 5a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                 1 j = 3               1 1 j = 1             2     j = -1           2 3     j = -3         1         j = -5       1 2         j = -7     1 1           j = -9     1             j = -11 1 1               j = -13 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, 92]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 92]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 10, 19, 9], X[17, 6, 18, 1], X[7, 17, 8, 16], > X[3, 10, 4, 11], X[5, 14, 6, 15], X[13, 4, 14, 5], X[11, 13, 12, 20], > X[15, 7, 16, 12], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -5, 7, -6, 3}, {-4, -1, 2, 5, -8, 9}, > {-7, 6, -9, 4, -3, -2, 10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 -5 2 2 3 2 2 3 + q - q + -- - -- + -- - - - q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 2 3 2 3 3 2 4 6 3 + q + q + --- + --- + --- + -- + -- + -- + -- + 3 q + q + q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 92]][a, z]
Out[8]=	2 4 2 4 -2 2 a a 2 2 2 4 2 4 2 4 5 - 9 a + 4 a + z - ---- + -- + 4 z - 12 a z + 4 a z + z - 6 a z + 2 2 z z 4 4 2 6 > a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 92]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a 3 5 - a + 11 a + 5 a - a - z - ---- - -- + --- + ---- - 9 a z - 9 a z - 2 2 z z z z 2 3 2 z 2 2 4 2 6 2 z 3 3 3 > 4 z + -- - 21 a z - 10 a z + 6 a z + -- + 11 a z + 13 a z + 2 a a 5 3 4 2 4 4 4 6 4 5 3 5 > 3 a z + 2 z + 16 a z + 9 a z - 5 a z - 5 a z - 9 a z - 5 5 2 6 4 6 6 6 7 3 7 5 7 2 8 4 8 > 4 a z - 6 a z - 5 a z + a z + a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 1 1 2 - + 2 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 1 2 3 3 2 5 2 > ----- + --- + q t + q t + q t 3 2 q t q t

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