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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 6q-7 - 9q-6 + 12q-5 - 12q-4 + 13q-3 - 10q-2 + 7q-1 - 3 + q

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 2q-26 + 2q-24 + 5q-22 - q-20 + 2q-16 - q-14 + 4q-12 + q-10 + 4q-8 + 3q-6 - q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2z-2 + 3a2 + 2a2z2 - a2z4 - 3a4z-2 - 7a4 - 6a4z2 - 3a4z4 + 4a6z-2 + 8a6 + 6a6z2 - 3a8z-2 - 4a8 + a10z-2

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 + a2z-2 - 4a2 + 8a2z2 - 8a2z4 + 6a2z6 - a3z-1 + a3z + 5a3z3 - 9a3z5 + 7a3z7 + 3a4z-2 - 14a4 + 28a4z2 - 25a4z4 + 5a4z6 + 4a4z8 - a5z-1 + a5z + 11a5z3 - 24a5z5 + 11a5z7 + a5z9 + 4a6z-2 - 21a6 + 38a6z2 - 29a6z4 - a6z6 + 6a6z8 - a7z-1 + a7z + 7a7z3 - 17a7z5 + 6a7z7 + a7z9 + 3a8z-2 - 14a8 + 25a8z2 - 17a8z4 + a8z6 + 2a8z8 - a9z-1 + a9z + 3a9z3 - 5a9z5 + 2a9z7 + a10z-2 - 4a10 + 6a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                     1 j = 1                   2   j = -1                 5 1   j = -3               6 3     j = -5             7 4       j = -7           5 6         j = -9         7 7           j = -11       5 8             j = -13     1 4               j = -15   1 5                 j = -17   1                   j = -19 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, 123]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 123]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[18, 12, 19, 11], X[20, 15, 9, 16], X[16, 19, 17, 20], X[12, 18, 13, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -4}, > {10, -2, 5, -8, 4, -3, 6, -7, 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 6 9 12 12 13 10 7 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 2 5 -20 2 -14 4 -10 4 3 -1 + q + --- + --- + --- + --- - q + --- - q + --- + q + -- + -- - 28 26 24 22 16 12 8 6 q q q q q q q q -4 4 2 4 > q + -- - q + q 2 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 123]][a, z]
Out[8]=	2 4 6 8 10 2 4 6 8 a 3 a 4 a 3 a a 2 2 2 3 a - 7 a + 8 a - 4 a + -- - ---- + ---- - ---- + --- + z + 2 a z - 2 2 2 2 2 z z z z z 4 2 6 2 2 4 4 4 > 6 a z + 6 a z - a z - 3 a z
In[9]:=	Kauffman[Link[10, Alternating, 123]][a, z]
Out[9]=	2 4 6 8 10 3 2 4 6 8 10 a 3 a 4 a 3 a a a -4 a - 14 a - 21 a - 14 a - 4 a + -- + ---- + ---- + ---- + --- - -- - 2 2 2 2 2 z z z z z z 5 7 9 a a a 3 5 7 9 2 2 2 4 2 > -- - -- - -- + a z + a z + a z + a z - z + 8 a z + 28 a z + z z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 38 a z + 25 a z + 6 a z - 2 a z + 5 a z + 11 a z + 7 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 > 3 a z + z - 8 a z - 25 a z - 29 a z - 17 a z - 4 a z + 5 3 5 5 5 7 5 9 5 2 6 4 6 > 3 a z - 9 a z - 24 a z - 17 a z - 5 a z + 6 a z + 5 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > a z + a z + a z + 7 a z + 11 a z + 6 a z + 2 a z + 4 a z + 6 8 8 8 5 9 7 9 > 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 1 1 5 1 4 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 8 7 7 5 6 7 4 6 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a123

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