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L10n56 L10n58

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X20,5,9,6 X3,15,4,14 X15,5,16,4 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X6,19,7,20

Gauss Code:

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

Jones Polynomial:

q-9/2 - q-7/2 - q-3/2 - q-1/2 + q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-14 - q-12 + 2q-8 + 2q-6 + 2q-4 + 2q-2 + 1 + q2 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 - a-1z + az-1 + 5az + 5az3 + az5 - 3a3z - a3z3

Kauffman Polynomial:

- 2a-3z + 4a-3z3 - a-3z5 - 5a-2z2 + 5a-2z4 - a-2z6 - a-1z-1 + 2a-1z + 1 - 7z2 + 6z4 - z6 - az-1 + 10az - 14az3 + 7az5 - az7 - 7a2z2 + 6a2z4 - a2z6 + 6a3z - 10a3z3 + 6a3z5 - a3z7 - 5a4z2 + 5a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   1 j = 6                     j = 4             1 1 1   j = 2           1 1       j = 0         1 3 1       j = -2       1 1 2         j = -4       1             j = -6   1 1 1             j = -8                     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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 57]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 57]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 5, 9, 6], X[3, 15, 4, 14], X[15, 5, 16, 4], > X[7, 17, 8, 16], X[11, 18, 12, 19], X[17, 12, 18, 13], X[2, 9, 3, 10], > X[13, 1, 14, 8], X[6, 19, 7, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, -3, 4, 2, -10, -5, 9}, > {8, -1, -6, 7, -9, 3, -4, 5, -7, 6, 10, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) -(3/2) 1 5/2 7/2 q - q - q - ------- + q - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 2 2 2 2 12 1 - q - q + -- + -- + -- + -- + q + q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 57]][a, z]
Out[8]=	1 a z z 3 3 3 3 5 -(---) + - - -- - - + 5 a z - 3 a z + 5 a z - a z + a z a z z 3 a a
In[9]:=	Kauffman[Link[10, NonAlternating, 57]][a, z]
Out[9]=	2 1 a 2 z 2 z 3 2 5 z 2 2 4 2 1 - --- - - - --- + --- + 10 a z + 6 a z - 7 z - ---- - 7 a z - 5 a z + a z z 3 a 2 a a 3 4 5 4 z 3 3 3 4 5 z 2 4 4 4 z 5 > ---- - 14 a z - 10 a z + 6 z + ---- + 6 a z + 5 a z - -- + 7 a z + 3 2 3 a a a 6 3 5 6 z 2 6 4 6 7 3 7 > 6 a z - z - -- - a z - a z - a z - a z 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 1 1 1 3 + -- + q + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + t + 2 10 5 6 4 6 3 6 2 4 2 2 2 t 2 q q t q t q t q t q t q t q t 2 4 4 2 4 3 8 4 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n57

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