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L11n60 L11n62

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X19,5,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-1z-1 - 4a-1z - 2a-1z3 + 3az-1 + 9az + 8az3 + 2az5 - 4a3z-1 - 8a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-3z + a-2 - a-2z2 - a-2z4 - a-1z-1 + 3a-1z - 8a-1z3 + 7a-1z5 - 2a-1z7 + 3 - z2 - 9z4 + 11z6 - 3z8 - 3az-1 + 13az - 28az3 + 21az5 - 2az7 - az9 + 3a2 - a2z2 - 15a2z4 + 19a2z6 - 5a2z8 - 4a3z-1 + 16a3z - 29a3z3 + 19a3z5 - a3z7 - a3z9 + 2a4 - a4z2 - 7a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 6                 2 j = 4               2   j = 2             3 2   j = 0           4 3     j = -2         2 3 1     j = -4       3 4         j = -6     2 2           j = -8   1 4             j = -10   1               j = -12 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, 61]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 61]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 11, 19, 12], > X[19, 5, 20, 22], X[15, 21, 16, 20], X[21, 17, 22, 16], X[12, 17, 13, 18], > X[8, 14, 9, 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, -9, 11, -2, 4, -8, 9, -3, -6, 7, 8, -4, > -5, 6, -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 5 5 6 6 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 4 q - 2 q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 3 -10 -8 -6 2 4 8 -2 + q + --- + q + --- + q + q + q - -- + q + 2 q 16 12 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 61]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 4 z 3 5 2 z 3 -(---) + --- - ---- + ---- - --- + 9 a z - 8 a z + a z - ---- + 8 a z - a z z z z a a 3 3 5 > 3 a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 61]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a z 3 z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- - -- + --- + 13 a z + 16 a z + a z z z z 3 a a 2 3 5 2 z 2 2 4 2 8 z 3 3 3 5 3 > 7 a z - z - -- - a z - a z - ---- - 28 a z - 29 a z - 9 a z - 2 a a 4 5 4 z 2 4 4 4 7 z 5 3 5 5 5 > 9 z - -- - 15 a z - 7 a z + ---- + 21 a z + 19 a z + 5 a z + 2 a a 7 6 2 6 4 6 2 z 7 3 7 5 7 8 > 11 z + 19 a z + 8 a z - ---- - 2 a z - a z - a z - 3 z - a 2 8 4 8 9 3 9 > 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 4 2 2 3 3 + q + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 4 2 4 3 2 4 6 2 > ----- + ----- + - + ---- + 2 q t + 2 q t + 2 q t 4 2 2 2 t 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n61

L11n60 L11n62