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DrawMorseLink

PD Presentation:

X8192 X20,10,13,9 X6,20,1,19 X18,7,19,8 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,17,7,18 X2,13,3,14 X14,3,15,4

Gauss Code:

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

Jones Polynomial:

q-7 - 4q-6 + 8q-5 - 12q-4 + 16q-3 - 15q-2 + 16q-1 - 11 + 8q - 4q2 + q3

A2 (sl(3)) Invariant:

q-20 - 2q-18 + 2q-16 - 2q-14 + q-12 + 5q-10 + 2q-8 + 10q-6 + 2q-4 + 6q-2 + 2 - q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial:

z-2 + 1 + 2z2 + 3z4 + z6 - 2a2z-2 - a2 - 4a2z2 - 8a2z4 - 5a2z6 - a2z8 + a4z-2 + 2a4z2 + 3a4z4 + a4z6

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 6a-1z3 - 10a-1z5 + 4a-1z7 - z-2 + 1 - 2z2 + 10z4 - 15z6 + 6z8 + 2az-1 - az + 12az3 - 20az5 + 3az7 + 3az9 - 2a2z-2 + a2 - 4a2z2 + 18a2z4 - 31a2z6 + 14a2z8 + 2a3z-1 - a3z + 12a3z3 - 26a3z5 + 9a3z7 + 3a3z9 - a4z-2 + a4 + 2a4z2 - 3a4z4 - 7a4z6 + 8a4z8 + 4a5z3 - 12a5z5 + 10a5z7 + 3a6z2 - 8a6z4 + 8a6z6 - 2a7z3 + 4a7z5 + 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                   3   j = 3                 5 1   j = 1               6 3     j = -1             10 5       j = -3           7 8         j = -5         9 8           j = -7       5 9             j = -9     3 7               j = -11   1 5                 j = -13   3                   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, 163]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 163]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 10, 13, 9], X[6, 20, 1, 19], X[18, 7, 19, 8], > X[4, 11, 5, 12], X[16, 6, 17, 5], X[10, 16, 11, 15], X[12, 17, 7, 18], > X[2, 13, 3, 14], X[14, 3, 15, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 10, -5, 6, -3}, {4, -1, 2, -7, 5, -8}, > {9, -10, 7, -6, 8, -4, 3, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-7 4 8 12 16 15 16 2 3 -11 + q - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 2 -12 5 2 10 2 6 2 4 2 + q - --- + --- - --- + q + --- + -- + -- + -- + -- - q + 2 q - 18 16 14 10 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 163]][a, z]
Out[8]=	2 4 2 -2 2 a a 2 2 2 4 2 4 2 4 1 - a + z - ---- + -- + 2 z - 4 a z + 2 a z + 3 z - 8 a z + 2 2 z z 4 4 6 2 6 4 6 2 8 > 3 a z + z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 163]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 z 2 2 1 + a + a - z - ---- - -- + --- + ---- - a z - a z - 2 z + -- - 4 a z + 2 2 z z 2 z z a 3 4 2 6 2 6 z 3 3 3 5 3 7 3 4 > 2 a z + 3 a z + ---- + 12 a z + 12 a z + 4 a z - 2 a z + 10 z - a 4 5 2 z 2 4 4 4 6 4 8 4 10 z 5 3 5 > ---- + 18 a z - 3 a z - 8 a z + a z - ----- - 20 a z - 26 a z - 2 a a 6 7 5 5 7 5 6 z 2 6 4 6 6 6 4 z > 12 a z + 4 a z - 15 z + -- - 31 a z - 7 a z + 8 a z + ---- + 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 3 a z + 9 a z + 10 a z + 6 z + 14 a z + 8 a z + 3 a z + 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 10 1 3 1 5 3 7 5 9 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 9 8 7 5 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 5 q t + q t + 3 q t + 5 2 5 3 q q t q t q t 7 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a163

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