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L11n262 L11n264

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-7 - 2q-6 + 7q-5 - 8q-4 + 11q-3 - 10q-2 + 9q-1 - 7 + 4q - q2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-1z3 + a-1z5 + z2 - 7z4 + 4z6 + 3az3 - 12az5 + 6az7 - a2z4 - 5a2z6 + 4a2z8 + 7a3z3 - 12a3z5 + 5a3z7 + a3z9 - a4z-2 + 3a4 - 10a4z2 + 17a4z4 - 12a4z6 + 5a4z8 + 2a5z-1 - 3a5z + 2a5z3 + 3a5z5 - a5z7 + a5z9 - 2a6z-2 + 5a6 - 12a6z2 + 12a6z4 - 3a6z6 + a6z8 + 2a7z-1 - 3a7z - a7z3 + 2a7z5 - a8z-2 + 3a8 - 3a8z2 + 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             5 3     j = -3           6 5       j = -5         5 4         j = -7       3 6           j = -9     4 5             j = -11   1 6               j = -13   1                 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, 263]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 263]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 18, 12, 19], X[22, 20, 9, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, -5, 9, 4, -3, 7, -8, -9, 5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 7 8 11 10 9 2 -7 + q - -- + -- - -- + -- - -- + - + 4 q - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 3 7 3 4 3 -8 2 3 2 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, 263]][a, z]
Out[8]=	4 6 8 4 6 a 2 a a 2 2 2 4 2 4 2 4 4 4 3 a - 3 a + -- - ---- + -- - z + 3 a z + a z - z + 3 a z - a z + 2 2 2 z z z 2 6 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 263]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 2 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z + z - 2 2 2 z z z z z 3 4 2 6 2 8 2 z 3 3 3 5 3 7 3 > 10 a z - 12 a z - 3 a z - -- + 3 a z + 7 a z + 2 a z - a z - a 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 7 z - a z + 17 a z + 12 a z + a z + -- - 12 a z - 12 a z + a 5 5 7 5 6 2 6 4 6 6 6 7 > 3 a z + 2 a z + 4 z - 5 a z - 12 a z - 3 a z + 6 a z + 3 7 5 7 2 8 4 8 6 8 3 9 5 9 > 5 a z - a z + 4 a z + 5 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 1 1 1 6 4 5 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 5 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 L11n263

L11n262 L11n264