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L10n55 L10n57

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X18,11,19,12 X20,5,9,6 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X8,9,1,10 X4,19,5,20 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - q-9/2 - q-5/2 - q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-18 + q-10 + q-8 + 2q-6 + q-4 + q-2 + 1 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + az-1 + 5az + 5az3 + az5 - a3z - a5z

Kauffman Polynomial:

- a-1z-1 + 6a-1z - 10a-1z3 + 6a-1z5 - a-1z7 + 1 + 3z2 - 9z4 + 6z6 - z8 - az-1 + 10az - 22az3 + 13az5 - 2az7 + a2z2 - 8a2z4 + 6a2z6 - a2z8 + 2a3z - 8a3z3 + 6a3z5 - a3z7 + a4z2 - 2a5z + 4a5z3 - a5z5 + 3a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                   1 j = 4                     j = 2               1 1   j = 0           2 1       j = -2         1 2 1       j = -4       1 1 1         j = -6       1             j = -8   1 1 1             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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 56]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 56]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[20, 5, 9, 6], X[7, 15, 8, 14], > X[12, 4, 13, 3], X[13, 16, 14, 17], X[15, 7, 16, 6], X[8, 9, 1, 10], > X[4, 19, 5, 20], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -9, 3, 7, -4, -8}, > {8, -1, 2, -5, -6, 4, -7, 6, 10, -2, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) -(5/2) 3/2 5/2 q - q - q - Sqrt[q] + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -10 -8 2 -4 -2 4 6 8 1 - q + q + q + -- + q + q + q + q + q 6 q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 56]][a, z]
Out[8]=	3 1 a 3 z 3 5 z 3 5 -(---) + - - --- + 5 a z - a z - a z - -- + 5 a z + a z a z z a a
In[9]:=	Kauffman[Link[10, NonAlternating, 56]][a, z]
Out[9]=	1 a 6 z 3 5 2 2 2 4 2 6 2 1 - --- - - + --- + 10 a z + 2 a z - 2 a z + 3 z + a z + a z + 3 a z - a z z a 3 5 10 z 3 3 3 5 3 4 2 4 6 4 6 z > ----- - 22 a z - 8 a z + 4 a z - 9 z - 8 a z - a z + ---- + a a 7 5 3 5 5 5 6 2 6 z 7 3 7 8 > 13 a z + 6 a z - a z + 6 z + 6 a z - -- - 2 a z - a z - z - a 2 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 1 1 1 2 + q + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 12 5 8 4 8 3 8 2 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t t 2 2 2 3 6 4 > t + -- + q t + q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n56

L10n55 L10n57