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

L11n2 L11n4

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n3 Visit L11n3's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 4q-11/2 - 6q-9/2 + 6q-7/2 - 7q-5/2 + 6q-3/2 - 5q-1/2 + 2q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - 2q-18 + q-16 + 2q-14 + q-12 + 3q-10 + q-6 + 3 + q4 + q6

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z + 2az-1 + 3az + 2az3 - 3a3z-1 - 5a3z - 3a3z3 - a3z5 + 3a5z-1 + 4a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-1z-1 + 2a-1z - a-1z3 + 2z2 - 2z4 - 2az-1 + 7az - 9az3 + 3az5 - az7 - 2a2 + 8a2z2 - 15a2z4 + 8a2z6 - 2a2z8 - 3a3z-1 + 12a3z - 15a3z3 + 5a3z5 + a3z7 - a3z9 + 6a4z2 - 16a4z4 + 14a4z6 - 4a4z8 - 3a5z-1 + 10a5z - 13a5z3 + 9a5z5 - a5z9 + 2a6 - 4a6z2 + a6z4 + 5a6z6 - 2a6z8 - a7z-1 + 3a7z - 6a7z3 + 7a7z5 - 2a7z7 + a8 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n3

L11n2 L11n4