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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,8,17,7 X18,10,5,9 X8,18,9,17 X14,12,15,11 X10,16,11,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-1/2 + q1/2 - 4q3/2 + 5q5/2 - 7q7/2 + 7q9/2 - 6q11/2 + 5q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-2 + 1 + 2q2 + 4q4 + q6 + 3q8 - q12 - 2q16 + q18 - q20 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 + a-5z-1 - 2a-5z3 - a-5z5 - 3a-3z-1 - 4a-3z - 3a-3z3 - a-3z5 + 2a-1z-1 + 3a-1z + a-1z3

Kauffman Polynomial:

a-10z2 - a-10z4 + 4a-9z3 - 3a-9z5 - 2a-8z2 + 6a-8z4 - 4a-8z6 + 2a-7z - 4a-7z3 + 4a-7z5 - 3a-7z7 - a-6 + a-6z2 + 2a-6z4 - 2a-6z6 - a-6z8 + a-5z-1 - 5a-5z3 + 7a-5z5 - 4a-5z7 - 3a-4 + 7a-4z2 - 4a-4z4 + a-4z6 - a-4z8 + 3a-3z-1 - 7a-3z + 7a-3z3 - a-3z5 - a-3z7 - 3a-2 + 3a-2z2 + a-2z4 - a-2z6 + 2a-1z-1 - 5a-1z + 4a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                   1 j = 16                 2   j = 14               3 1   j = 12             3 2     j = 10           4 3       j = 8         3 3         j = 6       2 4           j = 4     2 3             j = 2   1 4               j = 0                     j = -2 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[9, Alternating, 17]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 17]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[18, 10, 5, 9], > X[8, 18, 9, 17], X[14, 12, 15, 11], X[10, 16, 11, 15], X[2, 5, 3, 6], > X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -5, 4, -7, 6, -2, 9, -6, 7, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 13/2 -(-------) + Sqrt[q] - 4 q + 5 q - 7 q + 7 q - 6 q + 5 q - Sqrt[q] 15/2 17/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 12 16 18 20 24 26 1 + q + 2 q + 4 q + q + 3 q - q - 2 q + q - q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 17]][a, z]
Out[8]=	3 3 3 3 5 5 1 3 2 z 4 z 3 z z 2 z 3 z z z z ---- - ---- + --- + -- - --- + --- + -- - ---- - ---- + -- - -- - -- 5 3 a z 7 3 a 7 5 3 a 5 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 17]][a, z]
Out[9]=	2 2 2 2 -6 3 3 1 3 2 2 z 7 z 5 z z 2 z z 7 z -a - -- - -- + ---- + ---- + --- + --- - --- - --- + --- - ---- + -- + ---- + 4 2 5 3 a z 7 3 a 10 8 6 4 a a a z a z a a a a a a 2 3 3 3 3 3 4 4 4 4 4 3 z 4 z 4 z 5 z 7 z 4 z z 6 z 2 z 4 z z > ---- + ---- - ---- - ---- + ---- + ---- - --- + ---- + ---- - ---- + -- - 2 9 7 5 3 a 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 7 7 7 3 z 4 z 7 z z z 4 z 2 z z z 3 z 4 z z > ---- + ---- + ---- - -- - -- - ---- - ---- + -- - -- - ---- - ---- - -- - 9 7 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a a a 8 8 z z > -- - -- 6 4 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 q 4 6 6 2 8 2 8 3 4 q + 2 q + ----- + -- + 3 q t + 2 q t + 4 q t + 3 q t + 3 q t + 2 2 t q t 10 3 10 4 12 4 12 5 14 5 14 6 16 6 > 4 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 18 7 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a17

L9a16 L9a18