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

L11a328 L11a330

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a329 Visit L11a329's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 11q-7/2 - 15q-5/2 + 17q-3/2 - 18q-1/2 + 14q1/2 - 11q3/2 + 7q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-14 - q-12 + 2q-10 + 2q-8 - 2q-6 + 5q-4 - 2q-2 + 5 - q4 + q6 - 3q8 + q10 - q12

HOMFLY-PT Polynomial:

a-1z-1 + 7a-1z + 9a-1z3 + 5a-1z5 + a-1z7 - 3az-1 - 16az - 25az3 - 19az5 - 7az7 - az9 + 2a3z-1 + 7a3z + 9a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 - 4a-3z3 + 8a-3z5 - 3a-3z7 - a-2 + 7a-2z2 - 12a-2z4 + 14a-2z6 - 5a-2z8 + a-1z-1 - 8a-1z + 18a-1z3 - 18a-1z5 + 14a-1z7 - 5a-1z9 - 3 + 18z2 - 29z4 + 20z6 - 3z8 - 2z10 + 3az-1 - 17az + 34az3 - 45az5 + 31az7 - 10az9 - 3a2 + 14a2z2 - 28a2z4 + 19a2z6 - 4a2z8 - 2a2z10 + 2a3z-1 - 6a3z + 7a3z3 - 10a3z5 + 9a3z7 - 5a3z9 + 4a4z2 - 9a4z4 + 11a4z6 - 6a4z8 + 2a5z - 3a5z3 + 8a5z5 - 5a5z7 - a6z2 + 5a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     2   j = 6                   5 1   j = 4                 6 2     j = 2               8 5       j = 0             10 6         j = -2           8 9           j = -4         7 9             j = -6       4 8               j = -8     3 7                 j = -10   1 5                   j = -12   2                     j = -14 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, 329]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 329]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 5, 21, 6], > X[16, 8, 17, 7], X[18, 14, 19, 13], X[22, 16, 9, 15], X[8, 18, 1, 17], > X[14, 22, 15, 21], X[6, 9, 7, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 5, -8}, > {10, -1, 2, -3, 6, -9, 7, -5, 8, -6, 11, -4, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 11 15 17 18 -q + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 -12 2 2 2 5 2 4 6 8 10 5 + q - q + --- - q + --- + -- - -- + -- - -- - q + q - 3 q + q - 14 10 8 6 4 2 q q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 329]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 7 z 3 9 z 3 3 3 5 z --- - --- + ---- + --- - 16 a z + 7 a z + ---- - 25 a z + 9 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 19 a z + 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 329]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 8 z 3 5 7 -3 - a - 3 a + --- + --- + ---- - --- - 17 a z - 6 a z + 2 a z - a z + a z z z a 2 2 3 3 2 2 z 7 z 2 2 4 2 6 2 4 z 18 z 3 > 18 z - ---- + ---- + 14 a z + 4 a z - a z - ---- + ----- + 34 a z + 4 2 3 a a a a 4 4 3 3 5 3 7 3 4 3 z 12 z 2 4 4 4 > 7 a z - 3 a z + 2 a z - 29 z + ---- - ----- - 28 a z - 9 a z + 4 2 a a 5 5 6 4 8 z 18 z 5 3 5 5 5 7 5 6 > 5 a z + ---- - ----- - 45 a z - 10 a z + 8 a z - a z + 20 z - 3 a a 6 6 7 7 z 14 z 2 6 4 6 6 6 3 z 14 z 7 > -- + ----- + 19 a z + 11 a z - 3 a z - ---- + ----- + 31 a z + 4 2 3 a a a a 8 9 3 7 5 7 8 5 z 2 8 4 8 5 z 9 > 9 a z - 5 a z - 3 z - ---- - 4 a z - 6 a z - ---- - 10 a z - 2 a a 3 9 10 2 10 > 5 a z - 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 3 7 4 8 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 7 9 8 2 2 2 4 2 4 3 > ----- + ---- + ---- + 6 t + 8 q t + 5 q t + 6 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a329

L11a328 L11a330