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L11n129 L11n131

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DrawMorseLink

PD Presentation:

X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X19,4,20,5 X21,14,22,15

Gauss Code:

{{1, 4, -3, 10, -5, -6}, {6, -1, -7, 3, -2, 5, -8, 11, -9, 7, -4, 2, -10, 8, -11, 9}}

Jones Polynomial:

- q-19/2 + 3q-17/2 - 5q-15/2 + 8q-13/2 - 9q-11/2 + 9q-9/2 - 9q-7/2 + 5q-5/2 - 4q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 - q-26 + q-24 - 2q-22 - 3q-20 - 2q-16 + 4q-14 + 2q-12 + 4q-10 + 4q-8 + 2q-4 - q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - a3z + 2a3z3 + a3z5 + 3a5z-1 - 2a5z - 8a5z3 - 5a5z5 - a5z7 - a7z-1 + 2a7z + 3a7z3 + a7z5

Kauffman Polynomial:

a2z2 - a2z4 - 2a3z-1 + 6a3z3 - 4a3z5 + 3a4 + 2a4z2 - 4a4z4 + a4z6 - a4z8 - 3a5z-1 - a5z + 10a5z3 - 9a5z5 + 2a5z7 - a5z9 + 3a6 + 3a6z2 - 12a6z4 + 9a6z6 - 4a6z8 - a7z-1 - 3a7z3 + 5a7z5 - 2a7z7 - a7z9 + a8 - a8z2 - 2a8z4 + 5a8z6 - 3a8z8 + a9z - 5a9z3 + 9a9z5 - 4a9z7 - 3a10z2 + 7a10z4 - 3a10z6 + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 0                   1 j = -2                 3   j = -4               3 2   j = -6             6 2     j = -8           4 4       j = -10         5 5         j = -12       3 4           j = -14     2 5             j = -16   1 3               j = -18   2                 j = -20 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[11, NonAlternating, 130]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 130]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 19, 12, 18], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[5, 13, 6, 12], X[6, 7, 1, 8], X[9, 16, 10, 17], X[13, 20, 14, 21], > X[15, 22, 16, 7], X[19, 4, 20, 5], X[21, 14, 22, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, 10, -5, -6}, > {6, -1, -7, 3, -2, 5, -8, 11, -9, 7, -4, 2, -10, 8, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 5 8 9 9 9 5 4 1 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 2 3 2 4 2 4 4 2 -2 q - q + q - --- - --- - --- + --- + --- + --- + -- + -- - q 22 20 16 14 12 10 8 4 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 130]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 5 3 7 3 ----- + ---- - -- - a z - 2 a z + 2 a z + 2 a z - 8 a z + 3 a z + z z z 3 5 5 5 7 5 5 7 > a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 130]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 5 9 2 2 4 2 6 2 3 a + 3 a + a - ---- - ---- - -- - a z + a z + a z + 2 a z + 3 a z - z z z 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > a z - 3 a z + 6 a z + 10 a z - 3 a z - 5 a z + 2 a z - 2 4 4 4 6 4 8 4 10 4 3 5 5 5 > a z - 4 a z - 12 a z - 2 a z + 7 a z - 4 a z - 9 a z + 7 5 9 5 11 5 4 6 6 6 8 6 10 6 > 5 a z + 9 a z - a z + a z + 9 a z + 5 a z - 3 a z + 5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z - 2 a z - 4 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 4 5 5 4 4 6 2 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 6 4 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 L11n130

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