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DrawMorseLink

PD Presentation:

X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,13,12 X2738 X6,13,1,14

Gauss Code:

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

Jones Polynomial:

q-13 - 3q-12 + 6q-11 - 8q-10 + 11q-9 - 10q-8 + 10q-7 - 7q-6 + 5q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

q-40 + q-38 + 3q-34 + q-32 + 3q-30 + 4q-28 + 2q-26 + 6q-24 + q-22 + 3q-20 + q-18 - q-16 + 2q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

4a6z2 + 4a6z4 + a6z6 + a8z-2 + 7a8 + 13a8z2 + 9a8z4 + 2a8z6 - 2a10z-2 - 9a10 - 10a10z2 - 3a10z4 + a12z-2 + 2a12 + a12z2

Kauffman Polynomial:

4a6z2 - 4a6z4 + a6z6 + 4a7z3 - 6a7z5 + 2a7z7 - a8z-2 + 7a8 - 18a8z2 + 18a8z4 - 11a8z6 + 3a8z8 + 2a9z-1 - 9a9z + 15a9z3 - 14a9z5 + 3a9z7 + a9z9 - 2a10z-2 + 11a10 - 23a10z2 + 22a10z4 - 16a10z6 + 6a10z8 + 2a11z-1 - 9a11z + 17a11z3 - 17a11z5 + 6a11z7 + a11z9 - a12z-2 + 3a12 + 5a12z2 - 8a12z4 + a12z6 + 3a12z8 + 3a13z3 - 6a13z5 + 5a13z7 - 2a14 + 5a14z2 - 7a14z4 + 5a14z6 - 3a15z3 + 3a15z5 - a16z2 + a16z4

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                     1 j = -7                   2 1 j = -9                 3     j = -11               4 2     j = -13             6 3       j = -15           4 4         j = -17         7 6           j = -19       3 6             j = -21     3 5               j = -23   1 4                 j = -25   2                   j = -27 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, 159]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 159]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[12, 17, 7, 18], X[10, 19, 11, 20], X[18, 9, 19, 10], X[20, 11, 13, 12], > X[2, 7, 3, 8], X[6, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -4, 2, -10}, {9, -1, 7, -6, 8, -5}, > {10, -3, 4, -2, 5, -7, 6, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-13 3 6 8 11 10 10 7 5 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 3 -32 3 4 2 6 -22 3 -18 -16 q + q + --- + q + --- + --- + --- + --- + q + --- + q - q + 34 30 28 26 24 20 q q q q q q 2 -12 -10 > --- - q + q 14 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 159]][a, z]
Out[8]=	8 10 12 8 10 12 a 2 a a 6 2 8 2 10 2 7 a - 9 a + 2 a + -- - ----- + --- + 4 a z + 13 a z - 10 a z + 2 2 2 z z z 12 2 6 4 8 4 10 4 6 6 8 6 > a z + 4 a z + 9 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 159]][a, z]
Out[9]=	8 10 12 9 11 8 10 12 14 a 2 a a 2 a 2 a 9 7 a + 11 a + 3 a - 2 a - -- - ----- - --- + ---- + ----- - 9 a z - 2 2 2 z z z z z 11 6 2 8 2 10 2 12 2 14 2 16 2 > 9 a z + 4 a z - 18 a z - 23 a z + 5 a z + 5 a z - a z + 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 4 a z + 15 a z + 17 a z + 3 a z - 3 a z - 4 a z + 18 a z + 10 4 12 4 14 4 16 4 7 5 9 5 11 5 > 22 a z - 8 a z - 7 a z + a z - 6 a z - 14 a z - 17 a z - 13 5 15 5 6 6 8 6 10 6 12 6 14 6 > 6 a z + 3 a z + a z - 11 a z - 16 a z + a z + 5 a z + 7 7 9 7 11 7 13 7 8 8 10 8 12 8 > 2 a z + 3 a z + 6 a z + 5 a z + 3 a z + 6 a z + 3 a z + 9 9 11 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 2 1 4 3 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 6 7 6 4 4 6 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 3 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a159

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