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L10n97 L10n99

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DrawMorseLink

PD Presentation:

X6172 X2536 X13,15,14,20 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X11,19,12,18 X19,13,20,12 X17,9,18,14

Gauss Code:

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

Jones Polynomial:

- q-11/2 + q-9/2 - 4q-7/2 + 2q-5/2 - 5q-3/2 + 3q-1/2 - 4q1/2 + 2q3/2 - 2q5/2

A2 (sl(3)) Invariant:

q-18 + 3q-16 + 4q-14 + 7q-12 + 8q-10 + 10q-8 + 11q-6 + 9q-4 + 9q-2 + 6 + 5q2 + 4q4 + 2q6 + 2q8

HOMFLY-PT Polynomial:

- a-1z-3 - 4a-1z-1 - 6a-1z - 2a-1z3 + 3az-3 + 11az-1 + 16az + 10az3 + 2az5 - 3a3z-3 - 10a3z-1 - 11a3z - 3a3z3 + a5z-3 + 3a5z-1 + a5z

Kauffman Polynomial:

a-3z-1 - 3a-3z - a-2 - a-2z4 + a-1z-3 - 3a-1z-1 + 3a-1z + 2a-1z3 - 2a-1z5 - 3z-2 + 11 - 16z2 + 14z4 - 4z6 + 3az-3 - 12az-1 + 21az - 24az3 + 17az5 - 4az7 - 6a2z-2 + 24a2 - 33a2z2 + 19a2z4 - a2z6 - a2z8 + 3a3z-3 - 14a3z-1 + 28a3z - 39a3z3 + 25a3z5 - 5a3z7 - 3a4z-2 + 13a4 - 17a4z2 + 4a4z4 + 3a4z6 - a4z8 + a5z-3 - 6a5z-1 + 13a5z - 13a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 6                 2 j = 4               3 3 j = 2             1   1 j = 0           3 4     j = -2         2 1 1     j = -4       1 4         j = -6     3 1           j = -8   1 4             j = -10                   j = -12 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 98]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 98]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[13, 15, 14, 20], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[16, 7, 17, 8], X[8, 15, 5, 16], X[11, 19, 12, 18], > X[19, 13, 20, 12], X[17, 9, 18, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, > {7, -6, -10, 8, -9, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) 4 2 5 3 3/2 5/2 -q + q - ---- + ---- - ---- + ------- - 4 Sqrt[q] + 2 q - 2 q 7/2 5/2 3/2 Sqrt[q] q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 4 7 8 10 11 9 9 2 4 6 8 6 + q + --- + --- + --- + --- + -- + -- + -- + -- + 5 q + 4 q + 2 q + 2 q 16 14 12 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 98]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 4 11 a 10 a 3 a 6 z -(----) + --- - ---- + -- - --- + ---- - ----- + ---- - --- + 16 a z - 3 3 3 3 a z z z z a a z z z z 3 3 5 2 z 3 3 3 5 > 11 a z + a z - ---- + 10 a z - 3 a z + 2 a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 98]][a, z]
Out[9]=	3 5 2 4 -2 2 4 1 3 a 3 a a 3 6 a 3 a 1 11 - a + 24 a + 13 a + ---- + --- + ---- + -- - -- - ---- - ---- + ---- - 3 3 3 3 2 2 2 3 a z z z z z z z a z 3 5 3 12 a 14 a 6 a 3 z 3 z 3 5 > --- - ---- - ----- - ---- - --- + --- + 21 a z + 28 a z + 13 a z - a z z z z 3 a a 3 2 2 2 4 2 2 z 3 3 3 5 3 > 16 z - 33 a z - 17 a z + ---- - 24 a z - 39 a z - 13 a z + a 4 5 4 z 2 4 4 4 2 z 5 3 5 5 5 > 14 z - -- + 19 a z + 4 a z - ---- + 17 a z + 25 a z + 6 a z - 2 a a 6 2 6 4 6 7 3 7 5 7 2 8 4 8 > 4 z - a z + 3 a z - 4 a z - 5 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 4 3 1 1 4 2 4 + q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 8 5 8 4 6 4 6 3 4 3 4 2 2 2 q t q t q t q t q t q t q t q t 3 1 4 2 2 4 2 6 2 > - + ---- + 3 q t + q t + 3 q t + 2 q t t 2 q t

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L10n97 L10n99