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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,12,13,11 X12,14,5,13 X4,17,1,18 X10,20,11,19

Gauss Code:

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

Jones Polynomial:

- q-1 + 3 - 3q + 6q2 - 6q3 + 8q4 - 7q5 + 6q6 - 4q7 + 3q8 - q9

A2 (sl(3)) Invariant:

- q-2 + 1 + q2 + 3q4 + 4q6 + 3q8 + 5q10 + q12 + 4q14 + 2q16 + 2q18 + 2q20 + q24 - q26

HOMFLY-PT Polynomial:

a-6z-2 + a-6 - 3a-6z2 - 4a-6z4 - a-6z6 - 2a-4z-2 - 4a-4 + 5a-4z2 + 11a-4z4 + 6a-4z6 + a-4z8 + a-2z-2 + 3a-2 - 2a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-11z3 - 2a-10z2 + 3a-10z4 - 3a-9z3 + 4a-9z5 + a-8 - 2a-8z2 - 3a-8z4 + 4a-8z6 - 6a-7z5 + 4a-7z7 + a-6z-2 - a-6 - 6a-6z2 + 12a-6z4 - 12a-6z6 + 4a-6z8 - 2a-5z-1 + 4a-5z + 3a-5z3 - 3a-5z5 - 4a-5z7 + 2a-5z9 + 2a-4z-2 - 4a-4 - 13a-4z2 + 39a-4z4 - 31a-4z6 + 7a-4z8 - 2a-3z-1 + 4a-3z + 2a-3z3 + 3a-3z5 - 7a-3z7 + 2a-3z9 + a-2z-2 - 3a-2 - 7a-2z2 + 21a-2z4 - 15a-2z6 + 3a-2z8 + 3a-1z3 - 4a-1z5 + a-1z7

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                     1 j = 17                   2   j = 15                 2 1   j = 13               4 2     j = 11             5 4       j = 9           3 2         j = 7         3 5           j = 5       3 3             j = 3     2 5               j = 1   1 1                 j = -1   2                   j = -3 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 138]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 138]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[18, 10, 19, 9], X[20, 12, 13, 11], X[12, 14, 5, 13], > X[4, 17, 1, 18], X[10, 20, 11, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10, 7, -8}, > {8, -2, 4, -5, 9, -6, 10, -7}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 3 - - - 3 q + 6 q - 6 q + 8 q - 7 q + 6 q - 4 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 16 18 1 - q + q + 3 q + 4 q + 3 q + 5 q + q + 4 q + 2 q + 2 q + 20 24 26 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 138]][a, z]
Out[8]=	2 2 2 4 4 -6 4 3 1 2 1 3 z 5 z 2 z 4 z 11 z a - -- + -- + ----- - ----- + ----- - ---- + ---- - ---- - ---- + ----- - 4 2 6 2 4 2 2 2 6 4 2 6 4 a a a z a z a z a a a a a 4 6 6 6 8 4 z z 6 z z z > ---- - -- + ---- - -- + -- 2 6 4 2 4 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 138]][a, z]
Out[9]=	2 -8 -6 4 3 1 2 1 2 2 4 z 4 z 2 z a - a - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- - ---- - 4 2 6 2 4 2 2 2 5 3 5 3 10 a a a z a z a z a z a z a a a 2 2 2 2 3 3 3 3 3 4 2 z 6 z 13 z 7 z z 3 z 3 z 2 z 3 z 3 z > ---- - ---- - ----- - ---- + --- - ---- + ---- + ---- + ---- + ---- - 8 6 4 2 11 9 5 3 a 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 3 z 12 z 39 z 21 z 4 z 6 z 3 z 3 z 4 z 4 z > ---- + ----- + ----- + ----- + ---- - ---- - ---- + ---- - ---- + ---- - 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 12 z 31 z 15 z 4 z 4 z 7 z z 4 z 7 z 3 z > ----- - ----- - ----- + ---- - ---- - ---- + -- + ---- + ---- + ---- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 9 9 2 z 2 z > ---- + ---- 5 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 2 q q 2 q 5 7 7 2 5 q + 3 q + ----- + ---- + -- + - + ---- + 3 q t + 3 q t + 5 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 3 q t + 2 q t + 5 q t + 4 q t + 4 q t + 2 q t + 2 q t + 15 6 17 6 19 7 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a138

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