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L10n70 L10n72

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DrawMorseLink

PD Presentation:

X6172 X3,12,4,13 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X13,4,14,1

Gauss Code:

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

Jones Polynomial:

2q-5 - 4q-4 + 5q-3 - 6q-2 + 7q-1 - 4 + 5q - 2q2 + q3

A2 (sl(3)) Invariant:

q-20 + q-16 - q-14 - q-12 + q-10 - q-8 + 3q-6 + 2q-4 + 5q-2 + 6 + 4q2 + 4q4 + q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 4 - 6z2 - 2z4 + a2z-2 + 3a2 + 5a2z2 + 4a2z4 + a2z6 - 2a4 - 3a4z2 - a4z4 + a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6 - 2a-1z-1 + 4a-1z + 2a-1z3 - 6a-1z5 + 2a-1z7 + 2z-2 - 6 + 17z2 - 16z4 + 2z6 + z8 - 2az-1 + 6az3 - 13az5 + 5az7 + a2z-2 - 3a2 + 14a2z2 - 17a2z4 + 4a2z6 + a2z8 - 6a3z + 7a3z3 - 6a3z5 + 3a3z7 - a4 + 6a4z2 - 5a4z4 + 3a4z6 - 2a5z + 3a5z3 + a5z5 - a6 + 3a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                 1 j = 5               1   j = 3             4 1   j = 1           2 3     j = -1         5 2       j = -3       2 3         j = -5     3 4           j = -7   1 2             j = -9 1 3               j = -11 2

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, 71]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 71]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 12, 4, 13], X[7, 17, 8, 16], X[9, 11, 10, 20], > X[11, 18, 12, 19], X[15, 9, 16, 8], X[19, 5, 20, 10], X[17, 14, 18, 15], > X[2, 5, 3, 6], X[13, 4, 14, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -3, 6, -4, 7}, > {-5, 2, -10, 8, -6, 3, -8, 5, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 5 6 7 2 3 -4 + -- - -- + -- - -- + - + 5 q - 2 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 -14 -12 -10 -8 3 2 5 2 4 6 6 + q + q - q - q + q - q + -- + -- + -- + 4 q + 4 q + q + 6 4 2 q q q 8 10 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 71]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 2 2 4 2 -4 + -- + 3 a - 2 a + a - -- + ----- + -- - 6 z + -- + 5 a z - 3 a z - 2 2 2 2 2 2 a z a z z a 4 2 4 4 4 2 6 > 2 z + 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 71]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 4 z 3 -6 - -- - 3 a - a - a + -- + ----- + -- - --- - --- + --- - 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 5 2 6 z 2 2 4 2 6 2 2 z 3 > 2 a z + 17 z + ---- + 14 a z + 6 a z + 3 a z + ---- + 6 a z + 2 a a 4 5 3 3 5 3 4 4 z 2 4 4 4 6 z 5 > 7 a z + 3 a z - 16 z - ---- - 17 a z - 5 a z - ---- - 13 a z - 2 a a 6 7 3 5 5 5 6 z 2 6 4 6 2 z 7 3 7 > 6 a z + a z + 2 z + -- + 4 a z + 3 a z + ---- + 5 a z + 3 a z + 2 a a 8 2 8 > z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 2 1 3 1 2 3 4 2 2 t -- + - + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 2 q t + 3 q t + 4 q t + q t + q t + q t

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