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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,5,17,6 X20,10,13,9 X18,8,19,7 X8,14,9,13 X12,15,5,16 X10,20,11,19 X2,11,3,12 X4,18,1,17

Gauss Code:

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

Jones Polynomial:

q-4 - 4q-3 + 8q-2 - 11q-1 + 14 - 13q + 14q2 - 9q3 + 6q4 - 3q5 + q6

A2 (sl(3)) Invariant:

q-12 - 2q-10 + q-8 + q-6 - 2q-4 + 4q-2 + 5q2 + 5q4 + 4q6 + 7q8 + 2q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4z-2 + a-4 + 2a-4z2 + a-4z4 - 2a-2z-2 - 3a-2 - 4a-2z2 - 3a-2z4 - a-2z6 + z-2 + 2 - 2z4 - z6 + a2z2 + a2z4

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 + 7a-5z3 - 9a-5z5 + 3a-5z7 - a-4z-2 + 3a-4 - 7a-4z2 + 11a-4z4 - 11a-4z6 + 4a-4z8 + 2a-3z-1 - 3a-3z + 10a-3z3 - 13a-3z5 + 2a-3z7 + 2a-3z9 - 2a-2z-2 + 5a-2 - 13a-2z2 + 20a-2z4 - 23a-2z6 + 10a-2z8 + 2a-1z-1 - 3a-1z + 8a-1z3 - 19a-1z5 + 8a-1z7 + 2a-1z9 - z-2 + 3 - z2 - 4z4 - 3z6 + 6z8 + 3az3 - 11az5 + 9az7 + 3a2z2 - 9a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                     1 j = 11                   2   j = 9                 4 1   j = 7               6 3     j = 5             8 3       j = 3           5 6         j = 1         9 8           j = -1       5 8             j = -3     3 6               j = -5   1 5                 j = -7   3                   j = -9 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, 147]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 147]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[20, 10, 13, 9], > X[18, 8, 19, 7], X[8, 14, 9, 13], X[12, 15, 5, 16], X[10, 20, 11, 19], > X[2, 11, 3, 12], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {3, -1, 5, -6, 4, -8, 9, -7}, > {6, -2, 7, -3, 10, -5, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 8 11 2 3 4 5 6 14 + q - -- + -- - -- - 13 q + 14 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 -8 -6 2 4 2 4 6 8 12 14 q - --- + q + q - -- + -- + 5 q + 5 q + 4 q + 7 q + 2 q + q - 10 4 2 q q q 16 18 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 147]][a, z]
Out[8]=	2 2 4 4 -4 3 -2 1 2 2 z 4 z 2 2 4 z 3 z 2 + a - -- + z + ----- - ----- + ---- - ---- + a z - 2 z + -- - ---- + 2 4 2 2 2 4 2 4 2 a a z a z a a a a 6 2 4 6 z > a z - z - -- 2 a
In[9]:=	Kauffman[Link[10, Alternating, 147]][a, z]
Out[9]=	2 2 3 5 -2 1 2 2 2 3 z 3 z 2 2 z 7 z 3 + -- + -- - z - ----- - ----- + ---- + --- - --- - --- - z + ---- - ---- - 4 2 4 2 2 2 3 a z 3 a 6 4 a a a z a z a z a a a 2 3 3 3 4 13 z 2 2 7 z 10 z 8 z 3 3 3 4 3 z > ----- + 3 a z + ---- + ----- + ---- + 3 a z - 2 a z - 4 z - ---- + 2 5 3 a 6 a a a a 4 4 5 5 5 11 z 20 z 2 4 4 4 9 z 13 z 19 z 5 > ----- + ----- - 9 a z + a z - ---- - ----- - ----- - 11 a z + 4 2 5 3 a a a a a 6 6 6 7 7 7 3 5 6 z 11 z 23 z 2 6 3 z 2 z 8 z > 4 a z - 3 z + -- - ----- - ----- + 8 a z + ---- + ---- + ---- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 4 z 10 z 2 z 2 z > 9 a z + 6 z + ---- + ----- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 5 3 6 5 3 - + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 8 q t + 5 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 8 q t + 3 q t + 6 q t + 3 q t + 4 q t + q t + 11 5 13 6 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a147

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