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

L11a245 L11a247

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a246 Visit L11a246's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 15q-9/2 + 21q-7/2 - 25q-5/2 + 25q-3/2 - 23q-1/2 + 17q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - q-18 + q-16 + 5q-14 - 3q-12 + 5q-10 - q-8 - q-6 + 3q-4 - 4q-2 + 6 - 2q2 + 2q6 - 3q8 + q10

HOMFLY-PT Polynomial:

a-1z - a-1z3 - a-1z5 - 3az - az3 + 2az5 + az7 - a3z-1 + 3a3z3 + 3a3z5 + a3z7 + a5z-1 - 2a5z3 - a5z5

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - a-1z - 4a-1z3 + 15a-1z5 - 11a-1z7 - 3z4 + 17z6 - 13z8 - 2az - 3az3 + 12az5 + 3az7 - 9az9 + 4a2z2 - 21a2z4 + 36a2z6 - 15a2z8 - 3a2z10 - a3z-1 + 3a3z + a3z3 - 17a3z5 + 30a3z7 - 16a3z9 + a4 + 7a4z2 - 29a4z4 + 32a4z6 - 9a4z8 - 3a4z10 - a5z-1 + 4a5z - 6a5z3 - 3a5z5 + 12a5z7 - 7a5z9 + 3a6z2 - 13a6z4 + 17a6z6 - 7a6z8 - 6a7z3 + 10a7z5 - 4a7z7 + 2a8z4 - a8z6

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 = 8                       1 j = 6                     4   j = 4                   7 1   j = 2                 10 4     j = 0               13 7       j = -2             13 11         j = -4           12 12           j = -6         9 13             j = -8       6 12               j = -10     3 10                 j = -12   1 5                   j = -14   3                     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, Alternating, 246]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 246]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[20, 15, 21, 16], X[18, 14, 19, 13], X[6, 22, 7, 21], X[22, 18, 9, 17], > X[4, 19, 5, 20], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -7, 4, -11}, > {10, -1, 11, -2, 6, -3, 5, -4, 8, -6, 9, -5, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 15 21 25 25 23 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 -16 5 3 5 -8 -6 3 4 2 6 - q + --- - q + q + --- - --- + --- - q - q + -- - -- - 2 q + 20 14 12 10 4 2 q q q q q q 6 8 10 > 2 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 246]][a, z]
Out[8]=	3 5 3 5 a a z z 3 3 3 5 3 z 5 -(--) + -- + - - 3 a z - -- - a z + 3 a z - 2 a z - -- + 2 a z + z z a a a 3 5 5 5 7 3 7 > 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 246]][a, z]
Out[9]=	3 5 4 a a z 3 5 2 2 4 2 6 2 a - -- - -- - - - 2 a z + 3 a z + 4 a z + 4 a z + 7 a z + 3 a z - z z a 3 4 4 z 3 3 3 5 3 7 3 4 4 z 2 4 > ---- - 3 a z + a z - 6 a z - 6 a z - 3 z + ---- - 21 a z - a 2 a 5 5 4 4 6 4 8 4 z 15 z 5 3 5 5 5 > 29 a z - 13 a z + 2 a z - -- + ----- + 12 a z - 17 a z - 3 a z + 3 a a 6 7 7 5 6 5 z 2 6 4 6 6 6 8 6 11 z > 10 a z + 17 z - ---- + 36 a z + 32 a z + 17 a z - a z - ----- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 3 a z + 30 a z + 12 a z - 4 a z - 13 z - 15 a z - 9 a z - 6 8 9 3 9 5 9 2 10 4 10 > 7 a z - 9 a z - 16 a z - 7 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 5 3 10 6 12 13 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 9 13 12 12 13 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 7 t + 10 q t + 4 q t + 7 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a246

L11a245 L11a247