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L10n106 L10n108

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

q-9/2 - 2q-7/2 + q-5/2 - 2q-3/2 - 2q-1/2 - 2q1/2 - 2q3/2 + q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-14 + 3q-8 + 6q-6 + 10q-4 + 15q-2 + 15 + 15q2 + 10q4 + 6q6 + 3q8 - q14

HOMFLY-PT Polynomial:

- a-3z-3 + 2a-3z + a-3z3 + 3a-1z-3 - 6a-1z - 5a-1z3 - a-1z5 - 3az-3 + 6az + 5az3 + az5 + a3z-3 - 2a3z - a3z3

Kauffman Polynomial:

- 2a-4z2 + 4a-4z4 - a-4z6 - a-3z-3 + 8a-3z - 12a-3z3 + 10a-3z5 - 2a-3z7 + 3a-2z-2 - 8a-2z2 + 2a-2z4 + 4a-2z6 - a-2z8 - 3a-1z-3 - 3a-1z-1 + 24a-1z - 44a-1z3 + 26a-1z5 - 4a-1z7 + 6z-2 + 1 - 12z2 - 4z4 + 10z6 - 2z8 - 3az-3 - 3az-1 + 24az - 44az3 + 26az5 - 4az7 + 3a2z-2 - 8a2z2 + 2a2z4 + 4a2z6 - a2z8 - a3z-3 + 8a3z - 12a3z3 + 10a3z5 - 2a3z7 - 2a4z2 + 4a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                     1 j = 8                   1   j = 6               1 1 1   j = 4               2 1     j = 2           5 2 1       j = 0         2 8 2         j = -2       1 2 5           j = -4     1 2               j = -6   1 1 1               j = -8   1                   j = -10 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, 107]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 107]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 11, 10, 14], X[13, 15, 14, 20], X[19, 5, 20, 10], > X[11, 18, 12, 19], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, -3, -6, 8}, > {4, -5, 10, 9, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 -(5/2) 2 2 3/2 5/2 q - ---- + q - ---- - ------- - 2 Sqrt[q] - 2 q + q - 7/2 3/2 Sqrt[q] q q 7/2 9/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 6 10 15 2 4 6 8 14 15 - q + -- + -- + -- + -- + 15 q + 10 q + 6 q + 3 q - q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 107]][a, z]
Out[8]=	3 3 3 1 3 3 a a 2 z 6 z 3 z 5 z 3 -(-----) + ---- - --- + -- + --- - --- + 6 a z - 2 a z + -- - ---- + 5 a z - 3 3 3 3 3 3 a 3 a a z a z z z a a 5 3 3 z 5 > a z - -- + a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 107]][a, z]
Out[9]=	3 2 1 3 3 a a 6 3 3 a 3 3 a 8 z 24 z 1 - ----- - ---- - --- - -- + -- + ----- + ---- - --- - --- + --- + ---- + 3 3 3 3 3 2 2 2 2 a z z 3 a a z a z z z z a z z a 2 2 3 3 3 2 2 z 8 z 2 2 4 2 12 z 44 z > 24 a z + 8 a z - 12 z - ---- - ---- - 8 a z - 2 a z - ----- - ----- - 4 2 3 a a a a 4 4 5 3 3 3 4 4 z 2 z 2 4 4 4 10 z > 44 a z - 12 a z - 4 z + ---- + ---- + 2 a z + 4 a z + ----- + 4 2 3 a a a 5 6 6 7 26 z 5 3 5 6 z 4 z 2 6 4 6 2 z > ----- + 26 a z + 10 a z + 10 z - -- + ---- + 4 a z - a z - ---- - a 4 2 3 a a a 7 8 4 z 7 3 7 8 z 2 8 > ---- - 4 a z - 2 a z - 2 z - -- - a z a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	5 2 1 1 1 1 1 1 2 8 + -- + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 2 10 5 8 4 6 4 6 3 4 3 6 2 4 2 q q t q t q t q t q t q t q t 1 2 2 2 2 2 4 2 6 2 4 3 6 3 > ----- + - + ---- + 2 t + 2 q t + q t + 2 q t + q t + q t + q t + 2 2 t 2 q t q t 6 4 8 4 10 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n107

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