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

L11a524 L11a526

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a525 Visit L11a525's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X12,15,7,16 X22,19,13,20 X16,9,17,10 X10,22,11,21 X20,12,21,11 X18,5,19,6 X2738 X4,13,5,14 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-10 + 4q-9 - 8q-8 + 13q-7 - 16q-6 + 20q-5 - 18q-4 + 17q-3 - 12q-2 + 7q-1 - 3 + q

A2 (sl(3)) Invariant:

- q-32 - q-30 + 3q-28 - q-26 + q-24 + 7q-22 + 5q-18 + 4q-16 + 2q-14 + 5q-12 - q-10 + 4q-8 - 3q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2 + 2a2z2 - a2z4 + a4z-2 + 2a4 - a4z2 - 3a4z4 - 2a6z-2 - 5a6 - 4a6z2 - 3a6z4 + a8z-2 + 3a8 + 4a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 - a2 + 4a2z2 - 6a2z4 + 6a2z6 + 9a3z3 - 13a3z5 + 9a3z7 - a4z-2 + 4a4 - 8a4z2 + 15a4z4 - 18a4z6 + 10a4z8 + 2a5z-1 - 6a5z + 12a5z3 - 9a5z5 - 7a5z7 + 7a5z9 - 2a6z-2 + 12a6 - 41a6z2 + 66a6z4 - 56a6z6 + 14a6z8 + 2a6z10 + 2a7z-1 - 8a7z + 8a7z3 + 12a7z5 - 30a7z7 + 12a7z9 - a8z-2 + 10a8 - 34a8z2 + 59a8z4 - 46a8z6 + 8a8z8 + 2a8z10 - 3a9z + 10a9z3 + 2a9z5 - 13a9z7 + 5a9z9 + 2a10 - 6a10z2 + 15a10z4 - 14a10z6 + 4a10z8 - a11z + 3a11z3 - 3a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     2   j = -1                   5 1   j = -3                 8 3     j = -5               9 4       j = -7             10 9         j = -9           10 8           j = -11         7 11             j = -13       6 9               j = -15     3 8                 j = -17   1 5                   j = -19   3                     j = -21 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, 525]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 525]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[12, 15, 7, 16], X[22, 19, 13, 20], > X[16, 9, 17, 10], X[10, 22, 11, 21], X[20, 12, 21, 11], X[18, 5, 19, 6], > X[2, 7, 3, 8], X[4, 13, 5, 14], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 7, -3}, > {10, -2, 3, -5, 11, -8, 4, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-10 4 8 13 16 20 18 17 12 7 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 3 -26 -24 7 5 4 2 5 -10 -1 - q - q + --- - q + q + --- + --- + --- + --- + --- - q + 28 22 18 16 14 12 q q q q q q 4 3 4 2 4 > -- - -- + -- - q + q 8 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 525]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 2 4 2 a + 2 a - 5 a + 3 a - a + -- - ---- + -- + z + 2 a z - a z - 2 2 2 z z z 6 2 8 2 2 4 4 4 6 4 > 4 a z + 4 a z - a z - 3 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 525]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 5 -a + 4 a + 12 a + 10 a + 2 a - -- - ---- - -- + ---- + ---- - 6 a z - 2 2 2 z z z z z 7 9 11 2 2 2 4 2 6 2 8 2 > 8 a z - 3 a z - a z - z + 4 a z - 8 a z - 41 a z - 34 a z - 10 2 3 3 3 5 3 7 3 9 3 11 3 > 6 a z - 2 a z + 9 a z + 12 a z + 8 a z + 10 a z + 3 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > z - 6 a z + 15 a z + 66 a z + 59 a z + 15 a z + 3 a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 13 a z - 9 a z + 12 a z + 2 a z - 3 a z + 6 a z - 18 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 56 a z - 46 a z - 14 a z + 9 a z - 7 a z - 30 a z - 13 a z + 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > a z + 10 a z + 14 a z + 8 a z + 4 a z + 7 a z + 12 a z + 9 9 6 10 8 10 > 5 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 3 1 5 3 8 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 9 7 11 10 8 10 9 9 4 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 8 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a525

L11a524 L11a526