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L11n428 L11n430

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DrawMorseLink

PD Presentation:

X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X19,4,20,5 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

q-8 - 2q-7 + 5q-6 - 5q-5 + 7q-4 - 6q-3 + 6q-2 - 4q-1 + 3 - q

A2 (sl(3)) Invariant:

q-28 - q-26 + 2q-24 + 2q-22 + 4q-20 + 5q-18 + 3q-16 + 5q-14 + 3q-10 + q-8 + q-6 + q-4 + 1 - q2

HOMFLY-PT Polynomial:

- 3a2z2 - 4a2z4 - a2z6 + a4z-2 + 4a4 + 11a4z2 + 12a4z4 + 6a4z6 + a4z8 - 2a6z-2 - 5a6 - 7a6z2 - 5a6z4 - a6z6 + a8z-2 + a8 + a8z2

Kauffman Polynomial:

- az + 4az3 - 4az5 + az7 + a2 - 7a2z2 + 18a2z4 - 14a2z6 + 3a2z8 - 3a3z + 10a3z3 - 2a3z5 - 6a3z7 + 2a3z9 - a4z-2 + 8a4 - 26a4z2 + 44a4z4 - 32a4z6 + 7a4z8 + 2a5z-1 - 8a5z + 10a5z3 - 3a5z5 - 5a5z7 + 2a5z9 - 2a6z-2 + 12a6 - 29a6z2 + 31a6z4 - 18a6z6 + 4a6z8 + 2a7z-1 - 6a7z + 6a7z3 - 5a7z5 + 2a7z7 - a8z-2 + 6a8 - 9a8z2 + 5a8z4 + 2a9z3 + a10z2

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                   1 j = 1                 2   j = -1               2 1   j = -3             4 2     j = -5           3 3       j = -7         4 3         j = -9       2 4           j = -11     3 3             j = -13     3               j = -15 1 2                 j = -17 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, 429]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 429]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 20, 10, 21], X[14, 5, 15, 6], X[12, 14, 7, 13], > X[16, 8, 17, 7], X[22, 18, 13, 17], X[3, 10, 4, 11], X[18, 11, 19, 12], > X[6, 15, 1, 16], X[19, 4, 20, 5], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -7, 10, 3, -9}, {5, -1, -2, 7, 8, -4}, > {4, -3, 9, -5, 6, -8, -10, 2, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 2 5 5 7 6 6 4 3 + q - -- + -- - -- + -- - -- + -- - - - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 2 4 5 3 5 3 -8 -6 -4 2 1 + q - q + --- + --- + --- + --- + --- + --- + --- + q + q + q - q 24 22 20 18 16 14 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 429]][a, z]
Out[8]=	4 6 8 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 4 a - 5 a + a + -- - ---- + -- - 3 a z + 11 a z - 7 a z + a z - 2 2 2 z z z 2 4 4 4 6 4 2 6 4 6 6 6 4 8 > 4 a z + 12 a z - 5 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 429]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 3 a + 8 a + 12 a + 6 a - -- - ---- - -- + ---- + ---- - a z - 3 a z - 2 2 2 z z z z z 5 7 2 2 4 2 6 2 8 2 10 2 > 8 a z - 6 a z - 7 a z - 26 a z - 29 a z - 9 a z + a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > 4 a z + 10 a z + 10 a z + 6 a z + 2 a z + 18 a z + 44 a z + 6 4 8 4 5 3 5 5 5 7 5 2 6 > 31 a z + 5 a z - 4 a z - 2 a z - 3 a z - 5 a z - 14 a z - 4 6 6 6 7 3 7 5 7 7 7 2 8 > 32 a z - 18 a z + a z - 6 a z - 5 a z + 2 a z + 3 a z + 4 8 6 8 3 9 5 9 > 7 a z + 4 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 2 3 3 3 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 5 3 17 6 15 6 15 5 13 4 11 4 11 3 9 3 9 2 q q q t q t q t q t q t q t q t q t 2 4 3 3 2 t 2 t t 2 3 3 > ----- + ---- + ---- + --- + --- + -- + 2 q t + q t 7 2 7 5 3 q q q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n429

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