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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 3q-9/2 + 4q-7/2 - 5q-5/2 + 4q-3/2 - 4q-1/2 + 2q1/2 - q3/2

A2 (sl(3)) Invariant:

q-20 + q-16 + q-14 - q-12 + q-10 + q-6 + q-4 + 2 + q4 + q6

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z + 2az-1 + 4az + 2az3 - 2a3z-1 - 6a3z - 4a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- a-1z-1 + 2a-1z - a-1z3 + 2z2 - 2z4 - 2az-1 + 9az - 11az3 + 4az5 - az7 - a2 + 5a2z2 - 9a2z4 + 4a2z6 - a2z8 - 2a3z-1 + 11a3z - 18a3z3 + 11a3z5 - 3a3z7 - a4z4 + 2a4z6 - a4z8 - a5z-1 + 3a5z - 5a5z3 + 6a5z5 - 2a5z7 - 3a6z2 + 6a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                 1 j = 2               1   j = 0             3 1   j = -2           2 2     j = -4         3 2       j = -6       2 3         j = -8     1 2           j = -10   1 2             j = -12   1               j = -14 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 48]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 48]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[20, 14, 7, 13], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[15, 5, 16, 4], X[11, 16, 12, 17], > X[14, 20, 15, 19], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -6, 7, 5, -3}, > {3, -1, 2, 6, -8, -5, 4, -9, -7, 8, 10, -2, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 3 4 5 4 4 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 -14 -12 -10 -6 -4 4 6 2 + q + q + q - q + q + q + q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 48]][a, z]
Out[8]=	3 5 1 2 a 2 a a z 3 5 3 3 3 -(---) + --- - ---- + -- - - + 4 a z - 6 a z + 2 a z + 2 a z - 4 a z + a z z z z a 5 3 3 5 > a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 48]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 2 z 3 5 7 2 -a - --- - --- - ---- - -- + --- + 9 a z + 11 a z + 3 a z - a z + 2 z + a z z z z a 3 2 2 6 2 z 3 3 3 5 3 7 3 4 > 5 a z - 3 a z - -- - 11 a z - 18 a z - 5 a z + 3 a z - 2 z - a 2 4 4 4 6 4 5 3 5 5 5 7 5 2 6 > 9 a z - a z + 6 a z + 4 a z + 11 a z + 6 a z - a z + 4 a z + 4 6 6 6 7 3 7 5 7 2 8 4 8 > 2 a z - 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 2 2 3 3 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 3 2 2 2 4 2 > ----- + ---- + ---- + t + q t + q t 4 2 4 2 q t q t q t

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L10n47 L10n49