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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X15,20,16,5 X19,14,20,15 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 3q-1/2 + 2q1/2 - 3q3/2 + 2q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-8 + q-4 + q-2 + 1 + 2q2 + q4 + 2q6 + q8 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - a-1z-1 - 4a-1z - 4a-1z3 - a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 + 4a-3z - 10a-3z3 + 9a-3z5 - 2a-3z7 - 3a-2z2 + a-2z4 + 3a-2z6 - a-2z8 - a-1z-1 + 8a-1z - 18a-1z3 + 14a-1z5 - 3a-1z7 + 1 - z2 - 3z4 + 4z6 - z8 - az-1 + 4az - 8az3 + 5az5 - az7 - a2z2

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n24

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