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

L11n88 L11n90

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n89 Visit L11n89's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 2q-13/2 + q-9/2 - 2q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 - q-24 + q-22 - q-20 - q-18 - q-14 + 3q-12 + 3q-8 + q-6 + q-2 + 1 + q2 - q4

HOMFLY-PT Polynomial:

3az3 + az5 - 2a3z-1 - 6a3z - 7a3z3 - 5a3z5 - a3z7 + 4a5z-1 + 8a5z + 6a5z3 + a5z5 - 3a7z-1 - 4a7z + a9z-1

Kauffman Polynomial:

- z2 + 3z4 - z6 + 2az - 10az3 + 12az5 - 3az7 - a2 + 2a2z2 - 3a2z4 + 7a2z6 - 2a2z8 - 2a3z-1 + 15a3z - 32a3z3 + 25a3z5 - 5a3z7 - 2a4 + 15a4z2 - 28a4z4 + 18a4z6 - 3a4z8 - 4a5z-1 + 25a5z - 40a5z3 + 14a5z5 + 3a5z7 - a5z9 - 3a6 + 18a6z2 - 36a6z4 + 21a6z6 - 3a6z8 - 3a7z-1 + 16a7z - 24a7z3 + 6a7z5 + 4a7z7 - a7z9 - a8 + 6a8z2 - 14a8z4 + 11a8z6 - 2a8z8 - a9z-1 + 4a9z - 6a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     2   j = 0                   1 1   j = -2               1 4 2     j = -4             1 2 2       j = -6           1 3 3         j = -8         2 3 2           j = -10       1 2 2             j = -12     1 2 3               j = -14   1 1                   j = -16   1                     j = -18 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, 89]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 89]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[9, 15, 10, 14], X[19, 10, 20, 11], X[21, 9, 22, 8], X[13, 18, 14, 19], > X[15, 21, 16, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 7, -5, 6, 11, -2, -8, 5, -9, 3, -4, 8, > -6, 9, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 2 -(9/2) 2 3 4 3 -q + ----- - ----- + q - ---- + ---- - ---- + ------- - 15/2 13/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 > 3 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 -22 -20 -18 -14 3 3 -6 -2 2 1 + q + --- - q + q - q - q - q + --- + -- + q + q + q - 26 12 8 q q q 4 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 89]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 3 3 3 ----- + ---- - ---- + -- - 6 a z + 8 a z - 4 a z + 3 a z - 7 a z + z z z z 5 3 5 3 5 5 5 3 7 > 6 a z + a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 89]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 2 a 4 a 3 a a 3 5 -a - 2 a - 3 a - a - ---- - ---- - ---- - -- + 2 a z + 15 a z + 25 a z + z z z z 7 9 2 2 2 4 2 6 2 8 2 3 > 16 a z + 4 a z - z + 2 a z + 15 a z + 18 a z + 6 a z - 10 a z - 3 3 5 3 7 3 9 3 4 2 4 4 4 > 32 a z - 40 a z - 24 a z - 6 a z + 3 z - 3 a z - 28 a z - 6 4 8 4 5 3 5 5 5 7 5 9 5 > 36 a z - 14 a z + 12 a z + 25 a z + 14 a z + 6 a z + 5 a z - 6 2 6 4 6 6 6 8 6 7 3 7 > z + 7 a z + 18 a z + 21 a z + 11 a z - 3 a z - 5 a z + 5 7 7 7 9 7 2 8 4 8 6 8 8 8 5 9 > 3 a z + 4 a z - a z - 2 a z - 3 a z - 3 a z - 2 a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 1 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q q t q t q t q t q t q t q t 3 2 2 2 3 1 2 3 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ----- + 12 4 10 4 8 4 10 3 8 3 6 3 8 2 6 2 4 2 q t q t q t q t q t q t q t q t q t 3 2 1 2 t 2 2 2 4 3 > ---- + ---- + ---- + t + --- + t + 2 q t + q t 6 4 2 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n89

L11n88 L11n90