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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,12,9,11 X18,14,19,13 X14,18,15,17 X12,20,13,19 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-7 - q-6 + 4q-5 - 4q-4 + 7q-3 - 7q-2 + 7q-1 - 5 + 4q - 3q2 + q3

A2 (sl(3)) Invariant:

q-24 + 3q-22 + 3q-20 + 4q-18 + 6q-16 + 3q-14 + 3q-12 + 3q-10 + q-6 - q-4 + q-2 + 1 - q2 + q4 - q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 - z2 - z4 - a2z4 + a4z-2 + 3a4 + 3a4z2 - 2a6z-2 - 3a6 + a8z-2

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 8a-1z3 - 11a-1z5 + 3a-1z7 - 2z2 + 10z4 - 11z6 + 3z8 + 5az3 - 7az5 + az9 - 3a2z2 + 13a2z4 - 13a2z6 + 4a2z8 - 3a3z3 + 4a3z5 - 2a3z7 + a3z9 - a4z-2 + 3a4 - 3a4z2 + a4z4 + a4z8 + 2a5z-1 - 3a5z + a5z5 + a5z7 - 2a6z-2 + 5a6 - 6a6z2 + 2a6z4 + a6z6 + 2a7z-1 - 3a7z + a7z5 - a8z-2 + 3a8 - 3a8z2 + a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                     1 j = 5                   2   j = 3                 2 1   j = 1               3 2     j = -1             4 2       j = -3           4 4         j = -5         3 3           j = -7       1 4             j = -9     3 3               j = -11   1 4                 j = -13                       j = -15 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, 126]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 126]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[20, 12, 9, 11], X[18, 14, 19, 13], X[14, 18, 15, 17], 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, 3, -4}, > {10, -2, 5, -8, 6, -7, 4, -3, 7, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 4 4 7 7 7 2 3 -5 + q - q + -- - -- + -- - -- + - + 4 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 4 6 3 3 3 -6 -4 -2 2 1 + q + --- + --- + --- + --- + --- + --- + --- + q - q + q - q + 22 20 18 16 14 12 10 q q q q q q q 4 6 8 10 > q - q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 126]][a, z]
Out[8]=	4 6 8 2 4 6 a 2 a a 2 z 4 2 4 2 4 3 a - 3 a + -- - ---- + -- - z + -- + 3 a z - z - a z 2 2 2 2 z z z a
In[9]:=	Kauffman[Link[10, Alternating, 126]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 2 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z - 2 z + 2 2 2 z z z z z 2 3 z 2 2 4 2 6 2 8 2 8 z 3 3 3 > -- - 3 a z - 3 a z - 6 a z - 3 a z + ---- + 5 a z - 3 a z + 2 a a 4 5 4 3 z 2 4 4 4 6 4 8 4 11 z 5 > 10 z - ---- + 13 a z + a z + 2 a z + a z - ----- - 7 a z + 2 a a 6 7 3 5 5 5 7 5 6 z 2 6 6 6 3 z 3 7 > 4 a z + a z + a z - 11 z + -- - 13 a z + a z + ---- - 2 a z + 2 a a 5 7 8 2 8 4 8 9 3 9 > a z + 3 z + 4 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 1 1 4 3 3 1 4 3 -- + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 15 6 11 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t q t 3 4 2 t 2 3 2 3 3 5 3 7 4 > ---- + ---- + --- + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t 5 3 q q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a126

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