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L10n92 L10n94

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DrawMorseLink

PD Presentation:

X8192 X9,18,10,19 X17,6,18,1 X16,7,17,8 X3,10,4,11 X5,14,6,15 X13,4,14,5 X20,11,13,12 X12,15,7,16 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-11 + q-9 + q-6 + q-4

A2 (sl(3)) Invariant:

q-40 + q-36 + 2q-34 + 3q-32 + 4q-30 + 3q-28 + 3q-26 + 2q-24 + 2q-22 + 2q-20 + 2q-18 + q-16 + q-14

HOMFLY-PT Polynomial:

a8z-2 + 7a8 + 21a8z2 + 21a8z4 + 8a8z6 + a8z8 - 2a10z-2 - 9a10 - 14a10z2 - 7a10z4 - a10z6 + a12z-2 + 2a12 + a12z2

Kauffman Polynomial:

- a8z-2 + 7a8 - 21a8z2 + 21a8z4 - 8a8z6 + a8z8 + 2a9z-1 - 9a9z + 14a9z3 - 7a9z5 + a9z7 - 2a10z-2 + 11a10 - 23a10z2 + 21a10z4 - 8a10z6 + a10z8 + 2a11z-1 - 9a11z + 14a11z3 - 7a11z5 + a11z7 - a12z-2 + 3a12 - a12z2 - 2a14 + a14z2

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -7                 1 j = -9                 1 j = -11             1     j = -13         1         j = -15         1 1       j = -17     2 1           j = -19     2 1           j = -21 1                 j = -23 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, 93]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 93]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 18, 10, 19], X[17, 6, 18, 1], X[16, 7, 17, 8], > X[3, 10, 4, 11], X[5, 14, 6, 15], X[13, 4, 14, 5], X[20, 11, 13, 12], > X[12, 15, 7, 16], 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]=	-11 -9 -6 -4 q + q + q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -36 2 3 4 3 3 2 2 2 2 -16 -14 q + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + q + q 34 32 30 28 26 24 22 20 18 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 93]][a, z]
Out[8]=	8 10 12 8 10 12 a 2 a a 8 2 10 2 12 2 7 a - 9 a + 2 a + -- - ----- + --- + 21 a z - 14 a z + a z + 2 2 2 z z z 8 4 10 4 8 6 10 6 8 8 > 21 a z - 7 a z + 8 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 93]][a, z]
Out[9]=	8 10 12 9 11 8 10 12 14 a 2 a a 2 a 2 a 9 7 a + 11 a + 3 a - 2 a - -- - ----- - --- + ---- + ----- - 9 a z - 2 2 2 z z z z z 11 8 2 10 2 12 2 14 2 9 3 11 3 > 9 a z - 21 a z - 23 a z - a z + a z + 14 a z + 14 a z + 8 4 10 4 9 5 11 5 8 6 10 6 9 7 > 21 a z + 21 a z - 7 a z - 7 a z - 8 a z - 8 a z + a z + 11 7 8 8 10 8 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-9 -7 1 1 2 2 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 8 21 8 19 6 17 6 19 5 17 5 15 4 q t q t q t q t q t q t q t 1 1 1 > ------ + ------ + ------ 13 4 15 3 11 2 q t q t q t

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