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L10n103 L10n105

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X15,2,16,3 X7,17,8,16 X14,9,11,10 X20,13,15,14 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-17/2 - q-13/2 - q-11/2 - q-9/2 - 2q-7/2 - q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-28 + 3q-26 + 5q-24 + 7q-22 + 8q-20 + 10q-18 + 11q-16 + 11q-14 + 10q-12 + 7q-10 + 5q-8 + 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z-3 - 4a3z-1 - 4a3z - a3z3 + 3a5z-3 + 9a5z-1 + 6a5z + a5z3 - 3a7z-3 - 6a7z-1 - 2a7z + a9z-3 + a9z-1

Kauffman Polynomial:

a3z-3 - 5a3z-1 + 4a3z - a3z3 - 3a4z-2 + 10a4 - 6a4z2 + a4z4 + 3a5z-3 - 12a5z-1 + 15a5z - 7a5z3 + a5z5 - 6a6z-2 + 19a6 - 20a6z2 + 8a6z4 - a6z6 + 3a7z-3 - 12a7z-1 + 23a7z - 21a7z3 + 8a7z5 - a7z7 - 3a8z-2 + 10a8 - 14a8z2 + 7a8z4 - a8z6 + a9z-3 - 5a9z-1 + 12a9z - 15a9z3 + 7a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                 1 j = -4             1   1 j = -6           1 4     j = -8             3     j = -10       1 3         j = -12         2         j = -14     1             j = -16 1                 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, 104]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 104]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[7, 17, 8, 16], X[14, 9, 11, 10], X[20, 13, 15, 14], 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]=	-(17/2) -(13/2) -(11/2) -(9/2) 2 -(5/2) -(3/2) -q - q - q - q - ---- - q - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 5 7 8 10 11 11 10 7 5 2 -4 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, 104]][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 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - 4 a z + 6 a z - 2 a z - 3 3 3 3 z z z z z z z z 3 3 5 3 > a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 104]][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 4 2 > ----- - ----- - ---- + 4 a z + 15 a z + 23 a z + 12 a z - 6 a z - z z z 6 2 8 2 3 3 5 3 7 3 9 3 4 4 > 20 a z - 14 a z - a z - 7 a z - 21 a z - 15 a z + a z + 6 4 8 4 5 5 7 5 9 5 6 6 8 6 7 7 > 8 a z + 7 a z + a z + 8 a z + 7 a z - a z - a z - a z - 9 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 1 2 3 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 18 8 16 8 14 6 10 5 12 4 10 4 6 3 q t q t q t q t q t q t q t 3 4 1 > ----- + ----- + ----- 8 2 6 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n104

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