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L10n83 L10n85

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X13,20,14,15 X7,16,8,17 X15,8,16,9 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-12 - q-11 + 2q-10 - q-9 + 2q-8 - q-7 + q-6 + q-3

A2 (sl(3)) Invariant:

q-38 + 2q-36 + 2q-34 + 4q-32 + 3q-30 + 3q-28 + 3q-26 + 2q-24 + 2q-22 + q-18 + q-16 + q-14 + q-12 + q-10

HOMFLY-PT Polynomial:

3a6 + 9a6z2 + 6a6z4 + a6z6 + a8z-2 - a8z2 - 2a10z-2 - 4a10 - 2a10z2 + a12z-2 + a12

Kauffman Polynomial:

- 3a6 + 9a6z2 - 6a6z4 + a6z6 + a7z - a7z3 - a8z-2 + 2a8 + a8z2 - a8z4 + 2a9z-1 - 10a9z + 14a9z3 - 7a9z5 + a9z7 - 2a10z-2 + 10a10 - 20a10z2 + 17a10z4 - 7a10z6 + a10z8 + 2a11z-1 - 10a11z + 17a11z3 - 11a11z5 + 2a11z7 - a12z-2 + 4a12 - 6a12z2 + 7a12z4 - 5a12z6 + a12z8 + a13z + 2a13z3 - 4a13z5 + a13z7 - 2a14 + 6a14z2 - 5a14z4 + a14z6

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 = -5                     1 j = -7                     1 j = -9               1 1     j = -11             1         j = -13           2 3 1       j = -15         1             j = -17       1 3 1           j = -19     1 1 1             j = -21     1                 j = -23 1 1                   j = -25 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, NonAlternating, 84]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 84]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[13, 20, 14, 15], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[11, 18, 12, 19], X[19, 12, 20, 13], X[17, 14, 18, 5], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {-5, 4, -8, 6, -7, 3}, > {9, -1, -4, 5, 10, -2, -6, 7, -3, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-12 -11 2 -9 2 -7 -6 -3 q - q + --- - q + -- - q + q + q 10 8 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 2 4 3 3 3 2 2 -18 -16 -14 q + --- + --- + --- + --- + --- + --- + --- + --- + q + q + q + 36 34 32 30 28 26 24 22 q q q q q q q q -12 -10 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 84]][a, z]
Out[8]=	8 10 12 6 10 12 a 2 a a 6 2 8 2 10 2 6 4 3 a - 4 a + a + -- - ----- + --- + 9 a z - a z - 2 a z + 6 a z + 2 2 2 z z z 6 6 > a z
In[9]:=	Kauffman[Link[10, NonAlternating, 84]][a, z]
Out[9]=	8 10 12 9 11 6 8 10 12 14 a 2 a a 2 a 2 a -3 a + 2 a + 10 a + 4 a - 2 a - -- - ----- - --- + ---- + ----- + 2 2 2 z z z z z 7 9 11 13 6 2 8 2 10 2 > a z - 10 a z - 10 a z + a z + 9 a z + a z - 20 a z - 12 2 14 2 7 3 9 3 11 3 13 3 6 4 > 6 a z + 6 a z - a z + 14 a z + 17 a z + 2 a z - 6 a z - 8 4 10 4 12 4 14 4 9 5 11 5 13 5 > a z + 17 a z + 7 a z - 5 a z - 7 a z - 11 a z - 4 a z + 6 6 10 6 12 6 14 6 9 7 11 7 13 7 10 8 > a z - 7 a z - 5 a z + a z + a z + 2 a z + a z + a z + 12 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 1 1 1 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 25 10 23 10 23 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 1 3 1 1 2 3 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 15 6 17 5 13 5 13 4 11 4 13 3 q t q t q t q t q t q t q t q t 1 1 > ----- + ----- 9 3 9 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n84

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