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L11n232 L11n234

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X6,13,7,14 X14,7,15,8 X8,15,1,16 X17,22,18,9 X21,16,22,17 X19,4,20,5 X5,20,6,21

Gauss Code:

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

Jones Polynomial:

q-29/2 - 3q-27/2 + 4q-25/2 - 5q-23/2 + 5q-21/2 - 4q-19/2 + 3q-17/2 - q-15/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

- 2q-44 + q-42 + q-38 + 2q-36 - q-34 + q-32 - 2q-30 + q-26 + q-24 + 3q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- 2a9z-1 - 19a9z - 37a9z3 - 28a9z5 - 9a9z7 - a9z9 + 3a11z-1 + 19a11z + 23a11z3 + 9a11z5 + a11z7 - a13z-1 - 4a13z - 2a13z3

Kauffman Polynomial:

2a9z-1 - 19a9z + 37a9z3 - 28a9z5 + 9a9z7 - a9z9 - 3a10 + 19a10z2 - 23a10z4 + 9a10z6 - a10z8 + 3a11z-1 - 22a11z + 41a11z3 - 32a11z5 + 10a11z7 - a11z9 - 3a12 + 21a12z2 - 26a12z4 + 9a12z6 - a12z8 + a13z-1 - 5a13z + 6a13z3 - 4a13z5 - a14 + a14z2 + 2a14z4 - 3a14z6 - 3a15z + 7a15z3 - 3a15z5 - a15z7 + 4a16z4 - 3a16z6 - a17z + 5a17z3 - 3a17z5 + a18z2 - a18z4

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                       1 j = -10                       1 j = -12                   1     j = -14               2         j = -16             2 1 1       j = -18           3 2           j = -20         2 2 1           j = -22       3 3               j = -24     2 3                 j = -26   1 2                   j = -28   2                     j = -30 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, 233]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 233]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[9, 18, 10, 19], > X[6, 13, 7, 14], X[14, 7, 15, 8], X[8, 15, 1, 16], X[17, 22, 18, 9], > X[21, 16, 22, 17], X[19, 4, 20, 5], X[5, 20, 6, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 10, -11, -5, 6, -7}, > {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 3 4 5 5 4 3 -(15/2) q - ----- + ----- - ----- + ----- - ----- + ----- - q - 27/2 25/2 23/2 21/2 19/2 17/2 q q q q q q -(13/2) -(9/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 -42 -38 2 -34 -32 2 -26 -24 3 2 -18 --- + q + q + --- - q + q - --- + q + q + --- + --- + q + 44 36 30 22 20 q q q q q -16 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 233]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 9 11 13 9 3 11 3 ----- + ----- - --- - 19 a z + 19 a z - 4 a z - 37 a z + 23 a z - z z z 13 3 9 5 11 5 9 7 11 7 9 9 > 2 a z - 28 a z + 9 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 233]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 9 11 13 -3 a - 3 a - a + ---- + ----- + --- - 19 a z - 22 a z - 5 a z - z z z 15 17 10 2 12 2 14 2 18 2 9 3 > 3 a z - a z + 19 a z + 21 a z + a z + a z + 37 a z + 11 3 13 3 15 3 17 3 10 4 12 4 > 41 a z + 6 a z + 7 a z + 5 a z - 23 a z - 26 a z + 14 4 16 4 18 4 9 5 11 5 13 5 15 5 > 2 a z + 4 a z - a z - 28 a z - 32 a z - 4 a z - 3 a z - 17 5 10 6 12 6 14 6 16 6 9 7 > 3 a z + 9 a z + 9 a z - 3 a z - 3 a z + 9 a z + 11 7 15 7 10 8 12 8 9 9 11 9 > 10 a z - a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 2 1 2 2 3 3 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 30 11 28 10 26 10 26 9 24 9 24 8 22 8 q t q t q t q t q t q t q t 3 2 2 3 1 2 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 20 6 18 6 20 5 18 5 16 5 16 4 q t q t q t q t q t q t q t q t 2 1 1 > ------ + ------ + ------ 14 4 16 3 12 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n233

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