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

L11a98 L11a100

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a99 Visit L11a99's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X22,11,5,12 X4,21,1,22 X18,16,19,15 X16,8,17,7 X8,18,9,17 X20,14,21,13 X14,20,15,19

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 10q-7/2 - 15q-5/2 + 16q-3/2 - 18q-1/2 + 15q1/2 - 11q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 + 2q-14 - 3q-12 + 3q-10 + 4q-8 + q-6 + 5q-4 - q-2 + 2 - q2 - 3q4 + 2q6 - 3q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 + a-1z-1 + 2a-1z - a-1z5 - 3az-1 - 5az - 4az3 - 2az5 + 2a3z-1 - a3z3 - a3z5 + a5z + a5z3

Kauffman Polynomial:

2a-4z4 - a-4z6 - 3a-3z3 + 10a-3z5 - 4a-3z7 - a-2 + 4a-2z2 - 16a-2z4 + 21a-2z6 - 7a-2z8 + a-1z-1 - 2a-1z + 6a-1z3 - 13a-1z5 + 16a-1z7 - 6a-1z9 - 3 + 11z2 - 31z4 + 27z6 - 5z8 - 2z10 + 3az-1 - 7az + 20az3 - 38az5 + 30az7 - 10az9 - 3a2 + 7a2z2 - 14a2z4 + 10a2z6 - 2a2z8 - 2a2z10 + 2a3z-1 - 4a3z + 9a3z3 - 9a3z5 + 6a3z7 - 4a3z9 - 3a4z2 + 5a4z4 + 2a4z6 - 4a4z8 + 5a5z5 - 4a5z7 - 3a6z2 + 6a6z4 - 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                     3   j = 6                   5 1   j = 4                 6 3     j = 2               9 5       j = 0             9 6         j = -2           8 10           j = -4         7 8             j = -6       3 8               j = -8     3 7                 j = -10   1 4                   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, 99]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 99]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], > X[22, 11, 5, 12], X[4, 21, 1, 22], X[18, 16, 19, 15], X[16, 8, 17, 7], > X[8, 18, 9, 17], X[20, 14, 21, 13], X[14, 20, 15, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 10, -11, 7, -8, 9, -7, > 11, -10, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 10 15 16 18 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 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 + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 3 3 4 -6 5 -2 2 4 6 2 + q - q + --- - --- + --- + -- + q + -- - q - q - 3 q + 2 q - 14 12 10 8 4 q q q q q 8 12 14 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 99]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 2 z 5 z 3 3 3 5 3 z --- - --- + ---- + --- - 5 a z + a z + -- - 4 a z - a z + a z - -- - a z z z a 3 a a 5 3 5 > 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 99]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 2 z 3 7 2 -3 - a - 3 a + --- + --- + ---- - --- - 7 a z - 4 a z - a z + 11 z + a z z z a 2 3 3 4 z 2 2 4 2 6 2 3 z 6 z 3 3 3 > ---- + 7 a z - 3 a z - 3 a z - ---- + ---- + 20 a z + 9 a z + 2 3 a a a 4 4 5 7 3 4 2 z 16 z 2 4 4 4 6 4 10 z > 2 a z - 31 z + ---- - ----- - 14 a z + 5 a z + 6 a z + ----- - 4 2 3 a a a 5 6 6 13 z 5 3 5 5 5 7 5 6 z 21 z > ----- - 38 a z - 9 a z + 5 a z - a z + 27 z - -- + ----- + a 4 2 a a 7 7 2 6 4 6 6 6 4 z 16 z 7 3 7 5 7 > 10 a z + 2 a z - 3 a z - ---- + ----- + 30 a z + 6 a z - 4 a z - 3 a a 8 9 8 7 z 2 8 4 8 6 z 9 3 9 10 > 5 z - ---- - 2 a z - 4 a z - ---- - 10 a z - 4 a z - 2 z - 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 4 3 7 3 8 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 8 8 2 2 2 4 2 4 3 > ----- + ---- + ---- + 6 t + 9 q t + 5 q t + 6 q t + 3 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 + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a99

L11a98 L11a100