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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-38 + 2q-36 + q-34 + 4q-32 + 4q-30 + q-28 + 5q-26 + 2q-24 + 2q-22 + 2q-20 - q-18 + 3q-16 - q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial:

a4z2 + a4z4 + 2a6 + 6a6z2 + 3a6z4 + a8z-2 + 2a8 + 3a8z2 + 2a8z4 - 2a10z-2 - 5a10 - 3a10z2 + a12z-2 + a12

Kauffman Polynomial:

- a4z2 + a4z4 - 2a5z3 + 3a5z5 - 2a6 + 7a6z2 - 9a6z4 + 6a6z6 + a7z + 2a7z3 - 7a7z5 + 6a7z7 - a8z-2 + 3a8 - 4a8z2 + 2a8z4 - 3a8z6 + 4a8z8 + 2a9z-1 - 8a9z + 12a9z3 - 16a9z5 + 7a9z7 + a9z9 - 2a10z-2 + 9a10 - 16a10z2 + 14a10z4 - 13a10z6 + 6a10z8 + 2a11z-1 - 8a11z + 12a11z3 - 12a11z5 + 3a11z7 + a11z9 - a12z-2 + 3a12 + a12z2 - 2a12z4 - 3a12z6 + 2a12z8 + a13z + 4a13z3 - 6a13z5 + 2a13z7 - 2a14 + 5a14z2 - 4a14z4 + a14z6

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 = -3                     1 j = -5                   3 1 j = -7                 4     j = -9               5 3     j = -11             6 4       j = -13           6 6         j = -15         5 5           j = -17       2 6             j = -19     3 5               j = -21   1 4                 j = -23   1                   j = -25 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, 143]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 143]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[12, 15, 5, 16], X[18, 9, 19, 10], > X[16, 7, 17, 8], X[20, 11, 13, 12], X[10, 17, 11, 18], X[8, 19, 9, 20], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -8, 4, -7, 6, -3}, > {10, -2, 3, -5, 7, -4, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-12 2 5 7 11 11 11 9 7 3 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 -34 4 4 -28 5 2 2 2 -18 3 q + --- + q + --- + --- + q + --- + --- + --- + --- - q + --- - 36 32 30 26 24 22 20 16 q q q q q q q q -14 -12 2 2 -6 > q + q + --- - -- + q 10 8 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 143]][a, z]
Out[8]=	8 10 12 6 8 10 12 a 2 a a 4 2 6 2 8 2 2 a + 2 a - 5 a + a + -- - ----- + --- + a z + 6 a z + 3 a z - 2 2 2 z z z 10 2 4 4 6 4 8 4 > 3 a z + a z + 3 a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 143]][a, z]
Out[9]=	8 10 12 9 11 6 8 10 12 14 a 2 a a 2 a 2 a 7 -2 a + 3 a + 9 a + 3 a - 2 a - -- - ----- - --- + ---- + ----- + a z - 2 2 2 z z z z z 9 11 13 4 2 6 2 8 2 10 2 12 2 > 8 a z - 8 a z + a z - a z + 7 a z - 4 a z - 16 a z + a z + 14 2 5 3 7 3 9 3 11 3 13 3 4 4 > 5 a z - 2 a z + 2 a z + 12 a z + 12 a z + 4 a z + a z - 6 4 8 4 10 4 12 4 14 4 5 5 7 5 > 9 a z + 2 a z + 14 a z - 2 a z - 4 a z + 3 a z - 7 a z - 9 5 11 5 13 5 6 6 8 6 10 6 > 16 a z - 12 a z - 6 a z + 6 a z - 3 a z - 13 a z - 12 6 14 6 7 7 9 7 11 7 13 7 8 8 > 3 a z + a z + 6 a z + 7 a z + 3 a z + 2 a z + 4 a z + 10 8 12 8 9 9 11 9 > 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 1 1 4 3 5 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 6 5 5 6 6 6 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 3 4 3 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a143

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