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L11n178 L11n180

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DrawMorseLink

PD Presentation:

X8192 X3,10,4,11 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X11,4,12,5 X6,18,1,17 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 8q-7/2 - 10q-5/2 + 10q-3/2 - 9q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 3q-14 - q-12 + 2q-10 - q-8 - q-6 + q-4 - q-2 + 4 + q4 + 2q6 - q8

HOMFLY-PT Polynomial:

- a-1z-1 + a-1z3 + 2az-1 + az - az3 - az5 - 2a3z-1 - 3a3z - 2a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

a-2z2 - a-2z4 - a-1z-1 + a-1z + 5a-1z3 - 4a-1z5 + 3z2 - 2z4 - z8 - 2az-1 + 7az - 5az3 - az9 - a2 + 6a2z2 - 15a2z4 + 12a2z6 - 5a2z8 - 2a3z-1 + 11a3z - 23a3z3 + 18a3z5 - 5a3z7 - a3z9 + 3a4z2 - 8a4z4 + 9a4z6 - 4a4z8 - a5z-1 + 5a5z - 11a5z3 + 13a5z5 - 5a5z7 - a6z2 + 6a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   1 j = 4                 3   j = 2               3 1   j = 0             6 3     j = -2           5 4       j = -4         5 5         j = -6       4 6           j = -8     2 4             j = -10   1 4               j = -12   2                 j = -14 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, 179]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 179]]
Out[4]=	PD[X[8, 1, 9, 2], X[3, 10, 4, 11], X[16, 7, 17, 8], X[20, 10, 21, 9], > X[22, 15, 7, 16], X[14, 6, 15, 5], X[18, 14, 19, 13], X[12, 20, 13, 19], > X[11, 4, 12, 5], X[6, 18, 1, 17], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -2, 9, 6, -10}, > {3, -1, 4, 2, -9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 8 10 10 9 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 4 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 3 -12 2 -8 -6 -4 -2 4 6 4 + q - q + q + --- - q + --- - q - q + q - q + q + 2 q - 14 10 q q 8 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 179]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 3 5 z 3 3 3 5 3 -(---) + --- - ---- + -- + a z - 3 a z + a z + -- - a z - 2 a z + a z - a z z z z a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 179]][a, z]
Out[9]=	3 5 2 2 1 2 a 2 a a z 3 5 2 z -a - --- - --- - ---- - -- + - + 7 a z + 11 a z + 5 a z + 3 z + -- + a z z z z a 2 a 3 2 2 4 2 6 2 5 z 3 3 3 5 3 7 3 > 6 a z + 3 a z - a z + ---- - 5 a z - 23 a z - 11 a z + 2 a z - a 4 5 4 z 2 4 4 4 6 4 4 z 3 5 5 5 > 2 z - -- - 15 a z - 8 a z + 6 a z - ---- + 18 a z + 13 a z - 2 a a 7 5 2 6 4 6 6 6 3 7 5 7 8 2 8 > a z + 12 a z + 9 a z - 3 a z - 5 a z - 5 a z - z - 5 a z - 4 8 9 3 9 > 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 4 2 4 4 6 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 5 5 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 3 q t + q t + 3 q t + q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n179

L11n178 L11n180