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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 7q-5/2 - 13q-3/2 + 13q-1/2 - 18q1/2 + 14q3/2 - 13q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-10 + q-8 + 9q-6 + 8q-4 + 13q-2 + 16 + 10q2 + 13q4 + 3q6 + 5q8 + 2q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

- a-3z-3 - a-3z-1 + a-3z + 2a-3z3 + a-3z5 + 3a-1z-3 + 3a-1z-1 - 2a-1z - 6a-1z3 - 4a-1z5 - a-1z7 - 3az-3 - 3az-1 + 2az + 5az3 + 2az5 + a3z-3 + a3z-1 - a3z - a3z3

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - a-4z2 + 7a-4z4 - 8a-4z6 - a-3z-3 + 2a-3z-1 + 3a-3z - 12a-3z3 + 17a-3z5 - 11a-3z7 + 3a-2z-2 - 4a-2 - 3a-2z2 + 6a-2z4 + 6a-2z6 - 8a-2z8 - 3a-1z-3 + 3a-1z-1 + 11a-1z - 44a-1z3 + 53a-1z5 - 18a-1z7 - 2a-1z9 + 6z-2 - 7 - 4z2 - 3z4 + 24z6 - 13z8 - 3az-3 + 3az-1 + 11az - 40az3 + 43az5 - 11az7 - 2az9 + 3a2z-2 - 4a2 - 3a2z2 + a2z4 + 9a2z6 - 5a2z8 - a3z-3 + 2a3z-1 + 3a3z - 10a3z3 + 11a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   3   j = 8                 5 1   j = 6               8 3     j = 4             8 7       j = 2           10 6         j = 0         7 12           j = -2       6 6             j = -4     3 9               j = -6   1 4                 j = -8   3                   j = -10 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, 170]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 170]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[14, 19, 11, 20], X[20, 13, 15, 14], X[18, 10, 19, 9], > X[10, 12, 5, 11], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 8, -9}, > {4, -5, 10, -8, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 13 13 3/2 5/2 q - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 14 q - 13 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 -8 9 8 13 2 4 6 8 16 - q + q + --- + q + -- + -- + -- + 10 q + 13 q + 3 q + 5 q + 10 6 4 2 q q q q 10 12 14 16 > 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 170]][a, z]
Out[8]=	3 3 1 3 3 a a 1 3 3 a a z 2 z 3 -(-----) + ---- - --- + -- - ---- + --- - --- + -- + -- - --- + 2 a z - a z + 3 3 3 3 3 3 a z z z 3 a a z a z z z a z a 3 3 5 5 7 2 z 6 z 3 3 3 z 4 z 5 z > ---- - ---- + 5 a z - a z + -- - ---- + 2 a z - -- 3 a 3 a a a a
In[9]:=	Kauffman[Link[10, Alternating, 170]][a, z]
Out[9]=	3 2 4 2 1 3 3 a a 6 3 3 a 2 3 -7 - -- - 4 a - ----- - ---- - --- - -- + -- + ----- + ---- + ---- + --- + 2 3 3 3 3 3 2 2 2 2 3 a z a a z a z z z z a z z a z 3 2 2 3 a 2 a 3 z 11 z 3 2 z 3 z 2 2 > --- + ---- + --- + ---- + 11 a z + 3 a z - 4 z - -- - ---- - 3 a z - z z 3 a 4 2 a a a 3 3 3 4 4 4 2 2 z 12 z 44 z 3 3 3 4 z 7 z > a z + ---- - ----- - ----- - 40 a z - 10 a z - 3 z - -- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 z 2 4 4 4 4 z 17 z 53 z 5 3 5 > ---- + a z + 2 a z - ---- + ----- + ----- + 43 a z + 11 a z + 2 5 3 a a a a 6 6 7 7 6 8 z 6 z 2 6 4 6 11 z 18 z 7 3 7 > 24 z - ---- + ---- + 9 a z - a z - ----- - ----- - 11 a z - 4 a z - 4 2 3 a a a a 8 9 8 8 z 2 8 2 z 9 > 13 z - ---- - 5 a z - ---- - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 4 3 9 6 7 12 + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 6 2 4 4 2 6 2 6 3 8 3 8 4 > ---- + 6 q t + 8 q t + 7 q t + 8 q t + 3 q t + 5 q t + q t + 2 q t 10 4 12 5 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a170

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