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L10n107 L10n109

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,11,20,14 X13,15,14,20 X9,18,10,19 X11,10,12,5 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-17/2 + q-15/2 - 3q-13/2 + q-11/2 - 4q-9/2 + q-7/2 - 3q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

q-30 + 2q-26 + 4q-24 + 7q-22 + 10q-20 + 11q-18 + 13q-16 + 10q-14 + 9q-12 + 6q-10 + 4q-8 + 2q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- a3z-3 - 4a3z-1 - 7a3z - 5a3z3 - a3z5 + 3a5z-3 + 9a5z-1 + 13a5z + 12a5z3 + 6a5z5 + a5z7 - 3a7z-3 - 6a7z-1 - 7a7z - 5a7z3 - a7z5 + a9z-3 + a9z-1 + a9z

Kauffman Polynomial:

a3z-3 - 5a3z-1 + 11a3z - 12a3z3 + 6a3z5 - a3z7 - 3a4z-2 + 10a4 - 11a4z2 + 4a4z6 - a4z8 + 3a5z-3 - 12a5z-1 + 27a5z - 37a5z3 + 22a5z5 - 4a5z7 - 6a6z-2 + 19a6 - 23a6z2 + 9a6z4 + 2a6z6 - a6z8 + 3a7z-3 - 12a7z-1 + 25a7z - 28a7z3 + 16a7z5 - 3a7z7 - 3a8z-2 + 10a8 - 13a8z2 + 9a8z4 - 2a8z6 + a9z-3 - 5a9z-1 + 8a9z - 3a9z3 - a10z2 - a11z

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                 1 j = -2                   j = -4             3 1   j = -6         1 1 2     j = -8         4 1       j = -10     2 1 2         j = -12     4 2           j = -14 1   1             j = -16 2 2               j = -18 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, 108]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 108]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[19, 11, 20, 14], X[13, 15, 14, 20], X[9, 18, 10, 19], > X[11, 10, 12, 5], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, > {-4, -5, 10, 8, -6, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) 3 -(11/2) 4 -(7/2) 3 -(3/2) -q + q - ----- + q - ---- + q - ---- + q - 13/2 9/2 5/2 q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 4 7 10 11 13 10 9 6 4 2 -4 -2 q + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + -- + q + q 26 24 22 20 18 16 14 12 10 8 6 q q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 108]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 5 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - 7 a z + 13 a z - 3 3 3 3 z z z z z z z z 7 9 3 3 5 3 7 3 3 5 5 5 7 5 > 7 a z + a z - 5 a z + 12 a z - 5 a z - a z + 6 a z - a z + 5 7 > a z
In[9]:=	Kauffman[Link[10, NonAlternating, 108]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 11 > ----- - ----- - ---- + 11 a z + 27 a z + 25 a z + 8 a z - a z - z z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 11 a z - 23 a z - 13 a z - a z - 12 a z - 37 a z - 28 a z - 9 3 6 4 8 4 3 5 5 5 7 5 4 6 > 3 a z + 9 a z + 9 a z + 6 a z + 22 a z + 16 a z + 4 a z + 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 2 a z - 2 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 2 1 2 1 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 18 6 16 6 14 6 16 5 14 4 12 4 10 4 q q q t q t q t q t q t q t q t 2 1 2 4 1 1 1 t 2 > ------ + ------ + ------ + ----- + ----- + ---- + ---- + -- + t 12 3 10 3 10 2 8 2 6 2 8 6 4 q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n108

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