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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

q-13/2 - 4q-11/2 + 3q-9/2 - 7q-7/2 + 4q-5/2 - 6q-3/2 + 3q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-24 + q-22 + q-20 + 5q-18 + 8q-16 + 11q-14 + 15q-12 + 12q-10 + 13q-8 + 7q-6 + 5q-4 + 3q-2 + 1 + q2 - q4

HOMFLY-PT Polynomial:

- az-3 - 2az-1 + 3az3 + az5 + 3a3z-3 + 4a3z-1 - 2a3z - 7a3z3 - 5a3z5 - a3z7 - 3a5z-3 - 2a5z-1 + 3a5z + 4a5z3 + a5z5 + a7z-3 - a7z

Kauffman Polynomial:

- z2 + 3z4 - z6 - az-3 + 2az-1 + 3az - 12az3 + 12az5 - 3az7 + 3a2z-2 - 4a2 - 3a2z2 + 2a2z4 + 5a2z6 - 2a2z8 - 3a3z-3 + 3a3z-1 + 11a3z - 32a3z3 + 28a3z5 - 7a3z7 + 6a4z-2 - 7a4 - 4a4z2 + 3a4z4 + 4a4z6 - 2a4z8 - 3a5z-3 + 3a5z-1 + 11a5z - 24a5z3 + 16a5z5 - 4a5z7 + 3a6z-2 - 4a6 - 3a6z2 + 4a6z4 - 2a6z6 - a7z-3 + 2a7z-1 + 3a7z - 4a7z3 - a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                 1 j = 2               2   j = 0             1 1   j = -2           5 2     j = -4       1 2 3       j = -6       6 3         j = -8   1 2 5           j = -10   3 2             j = -12   3               j = -14 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 106]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 106]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 11, 10, 14], X[13, 15, 14, 20], X[19, 5, 20, 10], > X[11, 18, 12, 19], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -6, 8}, > {-4, -5, 10, 9, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 3 7 4 6 3 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 5 8 11 15 12 13 7 5 3 2 1 - q + q + q + --- + --- + --- + --- + --- + -- + -- + -- + -- + q - 18 16 14 12 10 8 6 4 2 q q q q q q q q q 4 > q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 106]][a, z]
Out[8]=	3 5 7 3 5 a 3 a 3 a a 2 a 4 a 2 a 3 5 7 -(--) + ---- - ---- + -- - --- + ---- - ---- - 2 a z + 3 a z - a z + 3 3 3 3 z z z z z z z 3 3 3 5 3 5 3 5 5 5 3 7 > 3 a z - 7 a z + 4 a z + a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 106]][a, z]
Out[9]=	3 5 7 2 4 6 3 2 4 6 a 3 a 3 a a 3 a 6 a 3 a 2 a 3 a -4 a - 7 a - 4 a - -- - ---- - ---- - -- + ---- + ---- + ---- + --- + ---- + 3 3 3 3 2 2 2 z z z z z z z z z 5 7 3 a 2 a 3 5 7 2 2 2 4 2 > ---- + ---- + 3 a z + 11 a z + 11 a z + 3 a z - z - 3 a z - 4 a z - z z 6 2 8 2 3 3 3 5 3 7 3 4 > 3 a z - a z - 12 a z - 32 a z - 24 a z - 4 a z + 3 z + 2 4 4 4 6 4 5 3 5 5 5 6 > 2 a z + 3 a z + 4 a z + 12 a z + 28 a z + 16 a z - z + 2 6 4 6 6 6 7 3 7 5 7 2 8 4 8 > 5 a z + 4 a z - 2 a z - 3 a z - 7 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 3 3 1 2 2 5 6 -- + -- + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 4 2 14 5 12 4 10 4 8 4 10 3 8 3 8 2 6 2 q q q t q t q t q t q t q t q t q t 1 3 2 2 t 2 2 2 4 3 > ----- + ---- + ---- + t + --- + t + 2 q t + q t 4 2 6 4 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n106

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