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L11n354 L11n356

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,12,19,11 X7,16,8,17 X15,8,16,9 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

2q-7 - 4q-6 + 9q-5 - 10q-4 + 13q-3 - 12q-2 + 10q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

q-24 + 4q-22 + 3q-20 + 3q-18 + 7q-16 + 2q-14 + 4q-12 + 2q-10 - q-8 + 2q-6 - 3q-4 + 3q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2 + 4a2z2 + 3a2z4 + a2z6 + a4z-2 + a4 - 2a4z2 - 2a4z4 - 2a6z-2 - 2a6 + a6z2 + a8z-2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 7z4 + 4z6 + 4az3 - 11az5 + 6az7 - a2 + 5a2z2 - 9a2z4 - a2z6 + 4a2z8 + 11a3z3 - 20a3z5 + 9a3z7 + a3z9 - a4z-2 + 2a4 - a4z2 + 2a4z4 - 7a4z6 + 6a4z8 + 2a5z-1 - 5a5z + 7a5z3 - 7a5z5 + 4a5z7 + a5z9 - 2a6z-2 + 6a6 - 10a6z2 + 7a6z4 - 2a6z6 + 2a6z8 + 2a7z-1 - 5a7z + a7z3 + a7z5 + a7z7 - a8z-2 + 4a8 - 6a8z2 + 3a8z4

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           7 5       j = -5         6 5         j = -7       5 8           j = -9     4 5             j = -11   1 6               j = -13 1 3                 j = -15 2

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, 355]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 355]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[20, 13, 21, 14], X[22, 20, 15, 19], X[12, 21, 13, 22], > X[14, 18, 5, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, > {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 9 10 13 12 10 2 -7 + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 4 3 3 7 2 4 2 -8 2 3 3 2 q + --- + --- + --- + --- + --- + --- + --- - q + -- - -- + -- - q + 22 20 18 16 14 12 10 6 4 2 q q q q q q q q q q 4 6 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 355]][a, z]
Out[8]=	4 6 8 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 a + a - 2 a + -- - ---- + -- - z + 4 a z - 2 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, 355]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 5 7 -a + 2 a + 6 a + 4 a - -- - ---- - -- + ---- + ---- - 5 a z - 5 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 - 10 a z - 6 a z - -- + 4 a z + 11 a z + a 5 5 3 7 3 4 2 4 4 4 6 4 8 4 z > 7 a z + a z - 7 z - 9 a z + 2 a z + 7 a z + 3 a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 11 a z - 20 a z - 7 a z + a z + 4 z - a z - 7 a z - 2 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 3 9 > 6 a z + 9 a z + 4 a z + a z + 4 a z + 6 a z + 2 a z + a z + 5 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 2 1 3 1 6 4 5 5 -- + - + ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t q t 8 6 5 7 3 t 2 3 2 5 3 > ----- + ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 7 2 5 2 5 3 q q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n355

L11n354 L11n356