© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11a448 L11a450

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a449 Visit L11a449's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X22,18,13,17 X16,8,17,7 X12,14,5,13 X8,21,9,22 X20,11,21,12 X18,9,19,10 X10,19,11,20 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 10q-3 + 14q-2 - 15q-1 + 17 - 13q + 10q2 - 6q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-18 + q-14 + 3q-12 + 5q-8 + 2q-6 + 3q-4 + 6q-2 + 1 + 6q2 - q4 + q6 + q8 - 2q10 + q12 - q14

HOMFLY-PT Polynomial:

- 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + z-2 + 8 + 15z2 + 14z4 + 6z6 + z8 - 2a2z-2 - 9a2 - 14a2z2 - 9a2z4 - 2a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + a-4z2 - 6a-4z4 + 3a-4z6 - 2a-3z + 7a-3z3 - 11a-3z5 + 5a-3z7 + 3a-2 - 13a-2z2 + 21a-2z4 - 16a-2z6 + 6a-2z8 - 7a-1z + 16a-1z3 - 4a-1z5 - 5a-1z7 + 4a-1z9 - z-2 + 11 - 37z2 + 62z4 - 40z6 + 10z8 + z10 + 2az-1 - 12az + 16az3 + 3az5 - 14az7 + 7az9 - 2a2z-2 + 11a2 - 28a2z2 + 39a2z4 - 30a2z6 + 8a2z8 + a2z10 + 2a3z-1 - 8a3z + 14a3z3 - 13a3z5 - a3z7 + 3a3z9 - a4z-2 + 3a4 - 2a4z2 + a4z4 - 8a4z6 + 4a4z8 - a5z + 5a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   4 1   j = 5                 6 2     j = 3               7 4       j = 1             10 6         j = -1           8 10           j = -3         6 7             j = -5       4 8               j = -7     3 6                 j = -9   1 5                   j = -11   2                     j = -13 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, Alternating, 449]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 449]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[22, 18, 13, 17], X[16, 8, 17, 7], > X[12, 14, 5, 13], X[8, 21, 9, 22], X[20, 11, 21, 12], X[18, 9, 19, 10], > X[10, 19, 11, 20], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 8, -9, 7, -5}, > {5, -2, 11, -4, 3, -8, 9, -7, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 7 10 14 15 2 3 4 5 17 + q - -- + -- - -- + -- - -- - 13 q + 10 q - 6 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -14 3 5 2 3 6 2 4 6 8 10 12 1 + q + q + --- + -- + -- + -- + -- + 6 q - q + q + q - 2 q + q - 12 8 6 4 2 q q q q q 14 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 449]][a, z]
Out[8]=	2 4 2 2 2 4 -2 2 a a 2 5 z 2 2 4 2 8 - -- - 9 a + 3 a + z - ---- + -- + 15 z - ---- - 14 a z + 3 a z + 2 2 2 2 a z z a 4 6 4 4 z 2 4 4 4 6 z 2 6 8 > 14 z - ---- - 9 a z + a z + 6 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 449]][a, z]
Out[9]=	2 4 3 3 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z 11 + -- + 11 a + 3 a - a - z - ---- - -- + --- + ---- - --- - --- - 2 2 2 z z 3 a a z z a 2 2 3 5 2 z 13 z 2 2 4 2 > 12 a z - 8 a z - a z - 37 z + -- - ----- - 28 a z - 2 a z + 4 2 a a 3 3 3 6 2 2 z 7 z 16 z 3 3 3 5 3 4 > 3 a z - ---- + ---- + ----- + 16 a z + 14 a z + 5 a z + 62 z - 5 3 a a a 4 4 5 5 5 6 z 21 z 2 4 4 4 6 4 z 11 z 4 z 5 > ---- + ----- + 39 a z + a z - 3 a z + -- - ----- - ---- + 3 a z - 4 2 5 3 a a a a a 6 6 3 5 5 5 6 3 z 16 z 2 6 4 6 6 6 > 13 a z - 8 a z - 40 z + ---- - ----- - 30 a z - 8 a z + a z + 4 2 a a 7 7 8 5 z 5 z 7 3 7 5 7 8 6 z 2 8 > ---- - ---- - 14 a z - a z + 3 a z + 10 z + ---- + 8 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 4 a z + ---- + 7 a z + 3 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 3 6 4 8 -- + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 7 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a449

L11a448 L11a450