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

L11n59 L11n61

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n60 Visit L11n60's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,13,18,12 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-15/2 - 3q-13/2 + 6q-11/2 - 9q-9/2 + 9q-7/2 - 11q-5/2 + 9q-3/2 - 7q-1/2 + 4q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + 2q-16 + 3q-14 + q-12 + 5q-10 + q-8 + 3q-6 - 2q-2 + 1 - 3q2 + q6

HOMFLY-PT Polynomial:

- a-1z + az-1 + 4az + 3az3 - 4a3z-1 - 10a3z - 7a3z3 - 2a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-1z - a-1z3 + 1 + 2z2 - 4z4 - az-1 + 4az - 8az3 + 3az5 - 2az7 + 4a2 - 4a2z2 - 6a2z4 + 6a2z6 - 3a2z8 - 4a3z-1 + 14a3z - 28a3z3 + 23a3z5 - 6a3z7 - a3z9 + 7a4 - 14a4z2 + 4a4z4 + 11a4z6 - 6a4z8 - 4a5z-1 + 15a5z - 29a5z3 + 29a5z5 - 7a5z7 - a5z9 + 4a6 - 11a6z2 + 9a6z4 + 4a6z6 - 3a6z8 - a7z-1 + 4a7z - 8a7z3 + 9a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   1 j = 2                 3   j = 0               4 1   j = -2             6 4     j = -4           5 3       j = -6         4 6         j = -8       5 5           j = -10     2 5             j = -12   1 4               j = -14   2                 j = -16 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, 60]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 60]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[11, 19, 12, 18], > X[22, 19, 5, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], X[17, 13, 18, 12], > 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]=	-(15/2) 3 6 9 9 11 9 7 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 2 3 -12 5 -8 3 2 2 1 - q - q + q - --- + --- + --- + q + --- + q + -- - -- - 3 q + 18 16 14 10 6 2 q q q q q q 6 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 60]][a, z]
Out[8]=	3 5 7 a 4 a 4 a a z 3 5 7 3 3 3 - - ---- + ---- - -- - - + 4 a z - 10 a z + 6 a z - a z + 3 a z - 7 a z + z z z z a 5 3 3 5 > 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 60]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a z 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + - + 4 a z + 14 a z + z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 z > 15 a z + 4 a z + 2 z - 4 a z - 14 a z - 11 a z - 3 a z - -- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 8 a z - 28 a z - 29 a z - 8 a z - 4 z - 6 a z + 4 a z + 6 4 8 4 5 3 5 5 5 7 5 2 6 > 9 a z + 3 a z + 3 a z + 23 a z + 29 a z + 9 a z + 6 a z + 4 6 6 6 8 6 7 3 7 5 7 7 7 > 11 a z + 4 a z - a z - 2 a z - 6 a z - 7 a z - 3 a z - 2 8 4 8 6 8 3 9 5 9 > 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 4 2 5 5 5 4 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 4 6 5 3 6 2 4 2 > ----- + ----- + ----- + ---- + ---- + t + 3 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n60

L11n59 L11n61