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L10n101 L10n103

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X15,20,16,17 X19,16,20,9 X17,12,18,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-23/2 + q-21/2 - 4q-19/2 + 2q-17/2 - 4q-15/2 + q-13/2 - 3q-11/2 + q-9/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 + 4q-36 + 6q-34 + 8q-32 + 12q-30 + 11q-28 + 11q-26 + 9q-24 + 6q-22 + 5q-20 + 2q-18 + 3q-16 + q-14 + q-10 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 4a5z - 5a5z3 - a5z5 - a7z-3 - 2a7z-1 - a7z - a7z3 + 3a9z-3 + 7a9z-1 + 5a9z - 3a11z-3 - 4a11z-1 + a13z-3

Kauffman Polynomial:

a5z-1 - 4a5z + 5a5z3 - a5z5 - a6 - a6z2 + a6z4 + a7z-3 - 3a7z-1 + 4a7z - 3a7z3 + a7z5 - 3a8z-2 + 11a8 - 21a8z2 + 16a8z4 - 3a8z6 + 3a9z-3 - 12a9z-1 + 26a9z - 36a9z3 + 22a9z5 - 4a9z7 - 6a10z-2 + 24a10 - 36a10z2 + 19a10z4 - a10z8 + 3a11z-3 - 14a11z-1 + 31a11z - 41a11z3 + 26a11z5 - 5a11z7 - 3a12z-2 + 13a12 - 16a12z2 + 4a12z4 + 3a12z6 - a12z8 + a13z-3 - 6a13z-1 + 13a13z - 13a13z3 + 6a13z5 - a13z7

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 = -4                     1 j = -6                     1 j = -8               2 1     j = -10             2         j = -12           3 6 1       j = -14         1 1 3         j = -16       1 3             j = -18     3 1               j = -20   1 4                 j = -22                       j = -24 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, 102]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 102]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 18, 12, 19], X[15, 20, 16, 17], X[19, 16, 20, 9], X[17, 12, 18, 13], > 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}, {-8, 5, -7, 6}, > {10, -2, -5, 8, 4, -3, -6, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(21/2) 4 2 4 -(13/2) 3 -(9/2) -q + q - ----- + ----- - ----- + q - ----- + q - 19/2 17/2 15/2 11/2 q q q q -(5/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 4 6 8 12 11 11 9 6 5 2 3 -14 q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + q + 36 34 32 30 28 26 24 22 20 18 16 q q q q q q q q q q q -10 -8 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 102]][a, z]
Out[8]=	7 9 11 13 5 7 9 11 a 3 a 3 a a a 2 a 7 a 4 a 5 7 -(--) + ---- - ----- + --- - -- - ---- + ---- - ----- - 4 a z - a z + 3 3 3 3 z z z z z z z z 9 5 3 7 3 5 5 > 5 a z - 5 a z - a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 102]][a, z]
Out[9]=	7 9 11 13 8 10 6 8 10 12 a 3 a 3 a a 3 a 6 a -a + 11 a + 24 a + 13 a + -- + ---- + ----- + --- - ---- - ----- - 3 3 3 3 2 2 z z z z z z 12 5 7 9 11 13 3 a a 3 a 12 a 14 a 6 a 5 7 9 > ----- + -- - ---- - ----- - ------ - ----- - 4 a z + 4 a z + 26 a z + 2 z z z z z z 11 13 6 2 8 2 10 2 12 2 5 3 > 31 a z + 13 a z - a z - 21 a z - 36 a z - 16 a z + 5 a z - 7 3 9 3 11 3 13 3 6 4 8 4 10 4 > 3 a z - 36 a z - 41 a z - 13 a z + a z + 16 a z + 19 a z + 12 4 5 5 7 5 9 5 11 5 13 5 8 6 > 4 a z - a z + a z + 22 a z + 26 a z + 6 a z - 3 a z + 12 6 9 7 11 7 13 7 10 8 12 8 > 3 a z - 4 a z - 5 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 4 3 1 1 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 24 10 20 9 20 8 18 8 18 7 16 7 16 6 q t q t q t q t q t q t q t 1 1 3 3 6 2 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- 14 6 14 5 12 5 14 4 12 4 10 4 12 3 8 3 8 2 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n102

L10n101 L10n103