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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 8 + 11q - 12q2 + 12q3 - 9q4 + 7q5 - 3q6 + q7

A2 (sl(3)) Invariant:

q-10 + q-8 + 3q-4 + 2 + 4q2 + 5q6 + q8 + 4q10 + 3q12 + 3q16 - q18 + q22

HOMFLY-PT Polynomial:

a-6 + a-6z2 + a-4z-2 + a-4 - 3a-4z2 - 2a-4z4 - 2a-2z-2 - 4a-2 - a-2z2 + 2a-2z4 + a-2z6 + z-2 - 4z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- a-8z2 + a-8z4 - 2a-7z3 + 3a-7z5 - 3a-6 + 8a-6z2 - 9a-6z4 + 6a-6z6 + a-5z + a-5z3 - 6a-5z5 + 6a-5z7 - a-4z-2 + 2a-4 + 3a-4z2 - 7a-4z4 + 4a-4z8 + 2a-3z-1 - 10a-3z + 24a-3z3 - 27a-3z5 + 10a-3z7 + a-3z9 - 2a-2z-2 + 10a-2 - 18a-2z2 + 15a-2z4 - 15a-2z6 + 7a-2z8 + 2a-1z-1 - 10a-1z + 23a-1z3 - 23a-1z5 + 6a-1z7 + a-1z9 - z-2 + 4 - 7z2 + 8z4 - 8z6 + 3z8 + az + 2az3 - 5az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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 = 15                     1 j = 13                   3 1 j = 11                 4     j = 9               5 3     j = 7             7 4       j = 5           6 6         j = 3         5 6           j = 1       4 7             j = -1     2 4               j = -3     4                 j = -5 1 2                   j = -7 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, 152]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 152]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 14, 15, 13], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[18, 12, 19, 11], X[12, 20, 13, 19], X[14, 18, 5, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {5, -4, 8, -6, 7, -3}, > {9, -1, 4, -5, 10, -2, 6, -7, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 7 -8 + q - -- + - + 11 q - 12 q + 12 q - 9 q + 7 q - 3 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 3 2 6 8 10 12 16 18 22 2 + q + q + -- + 4 q + 5 q + q + 4 q + 3 q + 3 q - q + q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 152]][a, z]
Out[8]=	2 2 2 -6 -4 4 2 -2 1 2 2 z 3 z z 2 2 a + a - -- + 2 a + z + ----- - ----- - 4 z + -- - ---- - -- + a z - 2 4 2 2 2 6 4 2 a a z a z a a a 4 4 6 4 2 z 2 z z > 2 z - ---- + ---- + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[10, Alternating, 152]][a, z]
Out[9]=	3 2 10 2 -2 1 2 2 2 z 10 z 10 z 4 - -- + -- + -- - 2 a - z - ----- - ----- + ---- + --- + -- - ---- - ---- + 6 4 2 4 2 2 2 3 a z 5 3 a a a a a z a z a z a a 2 2 2 2 3 3 3 2 z 8 z 3 z 18 z 2 2 2 z z 24 z > a z - 7 z - -- + ---- + ---- - ----- + 5 a z - ---- + -- + ----- + 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 5 5 23 z 3 4 z 9 z 7 z 15 z 2 4 3 z 6 z > ----- + 2 a z + 8 z + -- - ---- - ---- + ----- - 4 a z + ---- - ---- - a 8 6 4 2 7 5 a a a a a a 5 5 6 6 7 7 27 z 23 z 5 6 6 z 15 z 2 6 6 z 10 z > ----- - ----- - 5 a z - 8 z + ---- - ----- + a z + ---- + ----- + 3 a 6 2 5 3 a a a a a 7 8 8 9 9 6 z 7 8 4 z 7 z z z > ---- + 2 a z + 3 z + ---- + ---- + -- + -- a 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 4 2 4 4 q 3 7 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 6 q t + 7 q t + 4 q t + 5 q t + 3 q t + 4 q t + 13 5 13 6 15 6 > 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a152

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