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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-8 + 2q-7 - 5q-6 + 9q-5 - 11q-4 + 14q-3 - 11q-2 + 11q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

- q-26 - 2q-24 - q-22 - 3q-20 - 2q-18 + 4q-16 + 2q-14 + 7q-12 + 7q-10 + 5q-8 + 7q-6 + 4q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + 2a2z-2 + 5a2 + 5a2z2 + 3a2z4 + a2z6 - 5a4z-2 - 10a4 - 8a4z2 - 3a4z4 + 4a6z-2 + 6a6 + 3a6z2 - a8z-2 - a8

Kauffman Polynomial:

- a-1z3 + a-1z5 + 3z2 - 7z4 + 4z6 + 2az3 - 10az5 + 6az7 + 2a2z-2 - 7a2 + 11a2z2 - 12a2z4 + 4a2z8 - 5a3z-1 + 11a3z - 16a3z5 + 9a3z7 + a3z9 + 5a4z-2 - 14a4 + 19a4z2 - 12a4z4 - 3a4z6 + 6a4z8 - 9a5z-1 + 21a5z - 12a5z3 - 4a5z5 + 5a5z7 + a5z9 + 4a6z-2 - 10a6 + 14a6z2 - 11a6z4 + 3a6z6 + 2a6z8 - 5a7z-1 + 13a7z - 12a7z3 + 2a7z5 + 2a7z7 + a8z-2 - 2a8 + 3a8z2 - 4a8z4 + 2a8z6 - a9z-1 + 3a9z - 3a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                     1 j = 3                   3   j = 1                 4 1   j = -1               7 3     j = -3             7 7       j = -5           7 4         j = -7         4 7           j = -9       5 7             j = -11     1 5               j = -13   1 4                 j = -15   1                   j = -17 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, 124]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 124]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[20, 17, 9, 18], > X[16, 14, 17, 13], X[8, 16, 5, 15], X[14, 8, 15, 7], X[12, 19, 13, 20], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 7, -6}, > {10, -2, 3, -8, 5, -7, 6, -5, 4, -3, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-8 2 5 9 11 14 11 11 2 -7 - q + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 -22 3 2 4 2 7 7 5 7 4 2 -q - --- - q - --- - --- + --- + --- + --- + --- + -- + -- + -- - q + 24 20 18 16 14 12 10 8 6 2 q q q q q q q q q q 4 6 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 124]][a, z]
Out[8]=	2 4 6 8 2 4 6 8 2 a 5 a 4 a a 2 2 2 4 2 5 a - 10 a + 6 a - a + ---- - ---- + ---- - -- - z + 5 a z - 8 a z + 2 2 2 2 z z z z 6 2 4 2 4 4 4 2 6 > 3 a z - z + 3 a z - 3 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 124]][a, z]
Out[9]=	2 4 6 8 3 5 7 2 4 6 8 2 a 5 a 4 a a 5 a 9 a 5 a -7 a - 14 a - 10 a - 2 a + ---- + ---- + ---- + -- - ---- - ---- - ---- - 2 2 2 2 z z z z z z z 9 a 3 5 7 9 2 2 2 4 2 > -- + 11 a z + 21 a z + 13 a z + 3 a z + 3 z + 11 a z + 19 a z + z 3 6 2 8 2 z 3 5 3 7 3 9 3 4 > 14 a z + 3 a z - -- + 2 a z - 12 a z - 12 a z - 3 a z - 7 z - a 5 2 4 4 4 6 4 8 4 z 5 3 5 > 12 a z - 12 a z - 11 a z - 4 a z + -- - 10 a z - 16 a z - a 5 5 7 5 9 5 6 4 6 6 6 8 6 7 > 4 a z + 2 a z + a z + 4 z - 3 a z + 3 a z + 2 a z + 6 a z + 3 7 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 9 a z + 5 a z + 2 a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 7 1 1 1 4 1 5 5 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 4 7 7 4 7 3 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a124

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