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L10n17 L10n19

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - q-7/2 - 2q-1/2 + q1/2 - 2q3/2 + 2q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-14 - q-12 - q-10 + q-8 + q-6 + 2q-4 + 3q-2 + 2 + 3q2 + q4 + q6 - q10 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 4a-1z-1 - 8a-1z - 5a-1z3 - a-1z5 + 4az-1 + 9az + 6az3 + az5 - a3z-1 - 3a3z - a3z3

Kauffman Polynomial:

a-4 - 3a-4z2 + 4a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 9a-3z3 + 9a-3z5 - 2a-3z7 + 4a-2 - 10a-2z2 + 4a-2z4 + 3a-2z6 - a-2z8 - 4a-1z-1 + 15a-1z - 26a-1z3 + 17a-1z5 - 3a-1z7 + 7 - 14z2 + 3z4 + 4z6 - z8 - 4az-1 + 17az - 26az3 + 14az5 - 2az7 + 4a2 - 12a2z2 + 8a2z4 - a2z6 - a3z-1 + 6a3z - 9a3z3 + 6a3z5 - a3z7 + a4 - 5a4z2 + 5a4z4 - a4z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n18

L10n17 L10n19