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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,17,5,18 X18,14,19,13 X12,20,13,19 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- 2q-9/2 + 4q-7/2 - 7q-5/2 + 7q-3/2 - 8q-1/2 + 7q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-16 + 3q-14 + 2q-10 + 2q-8 - q-6 + 2q-4 - q-2 + 2 - q4 + q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z + 2a-1z + 2a-1z3 - 2az - 2az3 - az5 - a3z-1 - a3z + a3z3 + a5z-1

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 4a-2z2 + 7a-2z4 - 3a-2z6 - a-1z + 2a-1z3 + 4a-1z5 - 3a-1z7 - 5z2 + 13z4 - 5z6 - z8 + 6az5 - 5az7 - a2z2 + 4a2z4 - 3a2z6 - a2z8 - a3z-1 + 3a3z - 3a3z3 + a3z5 - 2a3z7 + a4 - 2a4z4 - a4z6 - a5z-1 + 3a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                 1 j = 6               2   j = 4             3 1   j = 2           4 2     j = 0         4 3       j = -2       4 5         j = -4     3 3           j = -6   1 4             j = -8 1 3               j = -10 2

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n26

L10n25 L10n27