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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 5q-1/2 + 5q1/2 - 9q3/2 + 8q5/2 - 10q7/2 + 7q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 3q-4 + 7q-2 + 7 + 10q2 + 10q4 + 8q6 + 11q8 + 6q10 + 8q12 + 4q14 + 2q16 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-3 - 3a-5z-1 - 3a-5z - 3a-5z3 - a-5z5 + 3a-3z-3 + 10a-3z-1 + 13a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 3a-1z-3 - 11a-1z-1 - 14a-1z - 9a-1z3 - 2a-1z5 + az-3 + 4az-1 + 4az + az3

Kauffman Polynomial:

- a-9z3 - 3a-8z4 + a-7z-1 - 3a-7z + 6a-7z3 - 6a-7z5 - a-6 + 8a-6z4 - 7a-6z6 + a-5z-3 - 3a-5z-1 + 3a-5z - 6a-5z3 + 13a-5z5 - 7a-5z7 - 3a-4z-2 + 11a-4 - 16a-4z2 + 11a-4z4 + 4a-4z6 - 4a-4z8 + 3a-3z-3 - 12a-3z-1 + 21a-3z - 30a-3z3 + 27a-3z5 - 6a-3z7 - a-3z9 - 6a-2z-2 + 24a-2 - 33a-2z2 + 6a-2z4 + 13a-2z6 - 5a-2z8 + 3a-1z-3 - 14a-1z-1 + 28a-1z - 30a-1z3 + 14a-1z5 - a-1z9 - 3z-2 + 13 - 17z2 + 6z4 + 2z6 - z8 + az-3 - 6az-1 + 13az - 13az3 + 6az5 - az7

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 = 16                     1 j = 14                   3 1 j = 12                 3     j = 10               4 3     j = 8             6 3       j = 6           5 7         j = 4         4 3           j = 2       4 8             j = 0     1 1               j = -2     4                 j = -4 1 1                   j = -6 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 167]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 167]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 12, 19, 11], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[8, 18, 5, 17], X[16, 8, 17, 7], X[20, 14, 15, 13], > X[14, 16, 9, 15], X[12, 20, 13, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -10, 8, -9}, > {9, -7, 6, -3, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 5 3/2 5/2 7/2 9/2 -q + q - ------- + 5 Sqrt[q] - 9 q + 8 q - 10 q + 7 q - Sqrt[q] 11/2 13/2 15/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 3 7 2 4 6 8 10 12 14 7 + q + -- + -- + -- + 10 q + 10 q + 8 q + 11 q + 6 q + 8 q + 4 q + 6 4 2 q q q 16 18 20 22 > 2 q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 167]][a, z]
Out[8]=	1 3 3 a 3 10 11 4 a 3 z 13 z 14 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - --- + ---- - ---- + 5 3 3 3 3 3 5 3 a z z 5 3 a a z a z a z z a z a z a a 3 3 3 5 5 5 7 3 z 10 z 9 z 3 z 5 z 2 z z > 4 a z - ---- + ----- - ---- + a z - -- + ---- - ---- + -- 5 3 a 5 3 a 3 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 167]][a, z]
Out[9]=	-6 11 24 1 3 3 a 3 3 6 1 13 - a + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- + ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 7 a a a z a z a z z z a z a z a z 3 12 14 6 a 3 z 3 z 21 z 28 z 2 > ---- - ---- - --- - --- - --- + --- + ---- + ---- + 13 a z - 17 z - 5 3 a z z 7 5 3 a a z a z a a a 2 2 3 3 3 3 3 4 16 z 33 z z 6 z 6 z 30 z 30 z 3 4 3 z > ----- - ----- - -- + ---- - ---- - ----- - ----- - 13 a z + 6 z - ---- + 4 2 9 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 5 6 8 z 11 z 6 z 6 z 13 z 27 z 14 z 5 6 7 z > ---- + ----- + ---- - ---- + ----- + ----- + ----- + 6 a z + 2 z - ---- + 6 4 2 7 5 3 a 6 a a a a a a a 6 6 7 7 8 8 9 9 4 z 13 z 7 z 6 z 7 8 4 z 5 z z z > ---- + ----- - ---- - ---- - a z - z - ---- - ---- - -- - -- 4 2 5 3 4 2 3 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 1 4 q 4 8 q + 4 q + ----- + ----- + ----- + t + ----- + - + ---- + 3 q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 5 q t + 7 q t + 6 q t + 3 q t + 4 q t + 3 q t + 3 q t + 14 5 14 6 16 6 > 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a167

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