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L10n102 L10n104

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-15/2 + q-13/2 - 6q-11/2 + 4q-9/2 - 7q-7/2 + 4q-5/2 - 5q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-26 + 4q-24 + 7q-22 + 9q-20 + 14q-18 + 13q-16 + 12q-14 + 10q-12 + 5q-10 + 5q-8 + q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-3 - 3a3z-1 - a3z + 2a3z3 + a3z5 + 3a5z-3 + 6a5z-1 + 2a5z - a5z3 - 3a7z-3 - 3a7z-1 + a9z-3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 7a2z4 - 3a2z6 + a3z-3 - 3a3z-1 + 4a3z - 3a3z3 + 7a3z5 - 3a3z7 - 3a4z-2 + 6a4 - 8a4z2 + 7a4z4 - a4z6 - a4z8 + 3a5z-3 - 6a5z-1 + 11a5z - 16a5z3 + 12a5z5 - 4a5z7 - 6a6z-2 + 11a6 - 9a6z2 - a6z4 + 2a6z6 - a6z8 + 3a7z-3 - 6a7z-1 + 9a7z - 12a7z3 + 4a7z5 - a7z7 - 3a8z-2 + 6a8 - 2a8z2 - a8z4 + a9z-3 - 3a9z-1 + 3a9z - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                 1 j = 0               2   j = -2             3 1   j = -4           3 4     j = -6         4 1       j = -8     1 1 3         j = -10     6 4           j = -12   1 6             j = -14                   j = -16 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, 103]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 103]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 18, 12, 19], X[20, 16, 17, 15], X[16, 20, 9, 19], X[17, 12, 18, 13], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, > {10, -2, -5, 8, 4, -3, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) -(13/2) 6 4 7 4 5 3 -q + q - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 4 7 9 14 13 12 10 5 5 -4 2 -1 + q + --- + --- + --- + --- + --- + --- + --- + --- + -- + q + q 24 22 20 18 16 14 12 10 8 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 103]][a, z]
Out[8]=	3 5 7 9 3 5 7 a 3 a 3 a a 3 a 6 a 3 a 3 5 3 -(--) + ---- - ---- + -- - ---- + ---- - ---- - a z - a z + 2 a z - a z + 3 3 3 3 z z z z z z z 3 3 5 3 3 5 > 2 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 103]][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 3 a 6 a + 11 a + 6 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 6 a 6 a 3 a 3 5 7 9 2 2 > ---- - ---- - ---- - a z + 4 a z + 11 a z + 9 a z + 3 a z - a z - z z z 4 2 6 2 8 2 3 3 3 5 3 7 3 > 8 a z - 9 a z - 2 a z + 2 a z - 3 a z - 16 a z - 12 a z - 9 3 2 4 4 4 6 4 8 4 5 3 5 5 5 > a z + 7 a z + 7 a z - a z - a z - a z + 7 a z + 12 a z + 7 5 2 6 4 6 6 6 3 7 5 7 7 7 4 8 > 4 a z - 3 a z - a z + 2 a z - 3 a z - 4 a z - a z - a z - 6 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 6 6 1 4 1 3 -- + -- + ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 4 2 16 6 12 5 12 4 10 4 8 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 1 3 t 2 2 > ----- + ---- + ---- + 2 t + -- + q t 6 2 6 4 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n103

L10n102 L10n104