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

L11a282 L11a284

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a283 Visit L11a283's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X20,11,21,12 X8,9,1,10 X22,17,9,18 X12,4,13,3 X18,8,19,7 X6,14,7,13 X14,6,15,5 X4,16,5,15 X16,21,17,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 14q-5/2 + 18q-3/2 - 20q-1/2 + 19q1/2 - 17q3/2 + 11q5/2 - 7q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 2q-12 + 2q-8 - 4q-6 + 3q-4 - 3q-2 + 1 + 2q2 - q4 + 6q6 + 2q10 + 2q12 - q14 + q16

HOMFLY-PT Polynomial:

- 2a-3z-1 - 3a-3z - 3a-3z3 - a-3z5 + 3a-1z-1 + 7a-1z + 7a-1z3 + 4a-1z5 + a-1z7 - az-1 - az + 2az3 + 3az5 + az7 - a3z - 2a3z3 - a3z5

Kauffman Polynomial:

2a-5z - 5a-5z3 + 4a-5z5 - a-5z7 + 3a-4z2 - 12a-4z4 + 11a-4z6 - 3a-4z8 + 2a-3z-1 - 5a-3z + 4a-3z3 - 10a-3z5 + 12a-3z7 - 4a-3z9 - 3a-2 + 11a-2z2 - 26a-2z4 + 23a-2z6 - 3a-2z8 - 2a-2z10 + 3a-1z-1 - 10a-1z + 15a-1z3 - 22a-1z5 + 28a-1z7 - 11a-1z9 - 3 + 12z2 - 27z4 + 37z6 - 12z8 - 2z10 + az-1 - 2az - 4az3 + 14az5 + 2az7 - 7az9 - a2 + a2z2 - 4a2z4 + 16a2z6 - 12a2z8 + a3z - 9a3z3 + 18a3z5 - 13a3z7 - 3a4z2 + 8a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     2   j = 8                   5 1   j = 6                 7 3     j = 4               10 4       j = 2             9 7         j = 0           11 10           j = -2         8 10             j = -4       6 10               j = -6     3 8                 j = -8   1 6                   j = -10   3                     j = -12 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, 283]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 283]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 11, 21, 12], X[8, 9, 1, 10], X[22, 17, 9, 18], > X[12, 4, 13, 3], X[18, 8, 19, 7], X[6, 14, 7, 13], X[14, 6, 15, 5], > X[4, 16, 5, 15], X[16, 21, 17, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -9, 8, -7, 6, -3}, > {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 14 18 20 3/2 q - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 17 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q - 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 2 4 3 3 2 4 6 10 12 1 - q + --- - --- + -- - -- + -- - -- + 2 q - q + 6 q + 2 q + 2 q - 14 12 8 6 4 2 q q q q q q 14 16 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 283]][a, z]
Out[8]=	3 3 5 -2 3 a 3 z 7 z 3 3 z 7 z 3 3 3 z ---- + --- - - - --- + --- - a z - a z - ---- + ---- + 2 a z - 2 a z - -- + 3 a z z 3 a 3 a 3 a z a a a 5 7 4 z 5 3 5 z 7 > ---- + 3 a z - a z + -- + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 283]][a, z]
Out[9]=	3 2 2 3 a 2 z 5 z 10 z 3 2 -3 - -- - a + ---- + --- + - + --- - --- - ---- - 2 a z + a z + 12 z + 2 3 a z z 5 3 a a a z a a 2 2 3 3 3 3 z 11 z 2 2 4 2 5 z 4 z 15 z 3 3 3 > ---- + ----- + a z - 3 a z - ---- + ---- + ----- - 4 a z - 9 a z + 4 2 5 3 a a a a a 4 4 5 5 5 3 4 12 z 26 z 2 4 4 4 6 4 4 z 10 z > a z - 27 z - ----- - ----- - 4 a z + 8 a z - a z + ---- - ----- - 4 2 5 3 a a a a 5 6 6 22 z 5 3 5 5 5 6 11 z 23 z 2 6 > ----- + 14 a z + 18 a z - 4 a z + 37 z + ----- + ----- + 16 a z - a 4 2 a a 7 7 7 8 8 4 6 z 12 z 28 z 7 3 7 8 3 z 3 z > 9 a z - -- + ----- + ----- + 2 a z - 13 a z - 12 z - ---- - ---- - 5 3 a 4 2 a a a a 9 9 10 2 8 4 z 11 z 9 10 2 z > 12 a z - ---- - ----- - 7 a z - 2 z - ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 6 3 8 6 10 11 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 8 2 2 2 4 2 4 3 6 3 6 4 > ---- + 10 t + 9 q t + 7 q t + 10 q t + 4 q t + 7 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a283

L11a282 L11a284