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L11n368 L11n370

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,11,19,12 X12,17,13,18 X22,20,17,19 X16,22,5,21 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 9q-4 + 12q-3 - 10q-2 + 10q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

q-22 + 5q-16 + 2q-14 + 5q-12 + 5q-10 + 2q-8 + 5q-6 - q-4 + 3q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2z-2 + 3a2 + 4a2z2 + 3a2z4 + a2z6 - 2a4z-2 - 4a4 - 4a4z2 - 2a4z4 + a6z-2 + a6 + a6z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 7z4 + 4z6 + 2az3 - 11az5 + 6az7 + a2z-2 - 3a2 + 5a2z2 - 6a2z4 - 3a2z6 + 4a2z8 - 2a3z-1 + 4a3z + 3a3z3 - 12a3z5 + 6a3z7 + a3z9 + 2a4z-2 - 4a4 + a4z2 + 6a4z4 - 8a4z6 + 5a4z8 - 2a5z-1 + 4a5z - 3a5z3 + 3a5z5 + a5z9 + a6z-2 - a6 - 4a6z2 + 6a6z4 - a6z6 + a6z8 - 3a7z3 + 3a7z5 + a8 - 2a8z2 + a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                   1 j = 3                 3   j = 1               4 1   j = -1             6 3     j = -3           6 6       j = -5         6 4         j = -7       3 6           j = -9     4 6             j = -11   1 5               j = -13   2                 j = -15 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[11, NonAlternating, 369]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 369]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[20, 16, 21, 15], > X[18, 11, 19, 12], X[12, 17, 13, 18], X[22, 20, 17, 19], X[16, 22, 5, 21], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, > {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 9 12 10 10 2 -7 + q - -- + -- - -- + -- - -- + -- + 4 q - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 5 2 5 5 2 5 -4 3 2 4 6 q + --- + --- + --- + --- + -- + -- - q + -- - q + 2 q - q 16 14 12 10 8 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 369]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 3 a - 4 a + a + -- - ---- + -- - z + 4 a z - 4 a z + a z - z + 2 2 2 z z z 2 4 4 4 2 6 > 3 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 369]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -3 a - 4 a - a + a + -- + ---- + -- - ---- - ---- + 4 a z + 4 a z + 2 2 2 z z z z z 3 2 2 2 4 2 6 2 8 2 z 3 3 3 > 2 z + 5 a z + a z - 4 a z - 2 a z - -- + 2 a z + 3 a z - a 5 5 3 7 3 4 2 4 4 4 6 4 8 4 z > 3 a z - 3 a z - 7 z - 6 a z + 6 a z + 6 a z + a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 11 a z - 12 a z + 3 a z + 3 a z + 4 z - 3 a z - 8 a z - a z + 7 3 7 2 8 4 8 6 8 3 9 5 9 > 6 a z + 6 a z + 4 a z + 5 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 6 1 2 1 5 4 6 3 6 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 6 4 6 3 t 2 3 2 5 3 > ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 5 2 5 3 q q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n369

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