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

L11a128 L11a130

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a129 Visit L11a129's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 4q-13/2 - 7q-11/2 + 8q-9/2 - 11q-7/2 + 10q-5/2 - 9q-3/2 + 7q-1/2 - 5q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-28 - 2q-26 + 4q-18 + 2q-16 + 2q-14 + 3q-12 + 2q-8 - q-6 + q-2 - 2 + q2 - q6 + q8

HOMFLY-PT Polynomial:

- a-1z - a-1z3 + az + 2az3 + az5 + a3z3 + a3z5 - 2a5z-1 - 5a5z - 3a5z3 + 3a7z-1 + 3a7z - a9z-1

Kauffman Polynomial:

a-1z - 4a-1z3 + 4a-1z5 - a-1z7 + 5z2 - 15z4 + 13z6 - 3z8 - az3 - 8az5 + 11az7 - 3az9 + 7a2z2 - 23a2z4 + 19a2z6 - 2a2z8 - a2z10 - a3z + 6a3z3 - 16a3z5 + 17a3z7 - 5a3z9 + 2a4z2 - 7a4z4 + 8a4z6 - a4z8 - a4z10 + 2a5z-1 - 5a5z + 7a5z3 - 4a5z5 + 3a5z7 - 2a5z9 - 3a6 + 3a6z2 + a6z4 - 2a6z8 + 3a7z-1 - 7a7z + 7a7z3 - 2a7z5 - 2a7z7 - 3a8 + 5a8z2 - a8z4 - 2a8z6 + a9z-1 - 2a9z + 3a9z3 - 2a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     2   j = 2                   3 1   j = 0                 4 2     j = -2               5 3       j = -4             6 5         j = -6           5 4           j = -8         3 6             j = -10       4 5               j = -12     1 4                 j = -14   1 3                   j = -16   1                     j = -18 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, Alternating, 129]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 129]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[18, 10, 19, 9], X[16, 12, 17, 11], > X[12, 16, 13, 15], X[10, 18, 11, 17], X[22, 19, 5, 20], X[20, 7, 21, 8], > X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 4, -5, 11, -2, 5, -4, 6, -3, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 4 7 8 11 10 9 7 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 5 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 4 2 2 3 2 -6 -2 2 6 8 -2 - q - --- + --- + --- + --- + --- + -- - q + q + q - q + q 26 18 16 14 12 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 129]][a, z]
Out[8]=	5 7 9 3 -2 a 3 a a z 5 7 z 3 3 3 5 3 ----- + ---- - -- - - + a z - 5 a z + 3 a z - -- + 2 a z + a z - 3 a z + z z z a a 5 3 5 > a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 129]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a z 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- + - - a z - 5 a z - 7 a z - 2 a z + z z z a 3 2 2 2 4 2 6 2 8 2 10 2 4 z 3 > 5 z + 7 a z + 2 a z + 3 a z + 5 a z + 2 a z - ---- - a z + a 3 3 5 3 7 3 9 3 4 2 4 4 4 > 6 a z + 7 a z + 7 a z + 3 a z - 15 z - 23 a z - 7 a z + 5 6 4 8 4 10 4 4 z 5 3 5 5 5 7 5 > a z - a z - a z + ---- - 8 a z - 16 a z - 4 a z - 2 a z - a 7 9 5 6 2 6 4 6 8 6 z 7 3 7 > 2 a z + 13 z + 19 a z + 8 a z - 2 a z - -- + 11 a z + 17 a z + a 5 7 7 7 8 2 8 4 8 6 8 9 3 9 > 3 a z - 2 a z - 3 z - 2 a z - a z - 2 a z - 3 a z - 5 a z - 5 9 2 10 4 10 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 1 1 1 3 1 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 5 3 6 5 4 6 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 4 t + --- + 2 t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a129

L11a128 L11a130