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

L11a169 L11a171

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a170 Visit L11a170's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 10q-9/2 + 18q-7/2 - 24q-5/2 + 28q-3/2 - 29q-1/2 + 24q1/2 - 19q3/2 + 11q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 4q-14 - 5q-12 + 2q-10 + q-8 - 5q-6 + 5q-4 - 4q-2 + 6 + q2 - q4 + 6q6 - 3q8 + 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-4z4 - a-4z6 + a-3z-1 - a-3z - 4a-3z3 + 9a-3z5 - 5a-3z7 - a-2 + a-2z2 - 10a-2z4 + 20a-2z6 - 10a-2z8 + a-1z-1 - 4a-1z + a-1z5 + 14a-1z7 - 10a-1z9 + 6z2 - 28z4 + 44z6 - 14z8 - 4z10 - 5az + 10az3 - 21az5 + 39az7 - 21az9 + 12a2z2 - 38a2z4 + 48a2z6 - 17a2z8 - 4a2z10 - a3z - 2a3z3 + 11a3z7 - 11a3z9 + 6a4z2 - 17a4z4 + 21a4z6 - 13a4z8 + a5z - 7a5z3 + 12a5z5 - 9a5z7 - a6z2 + 4a6z4 - 4a6z6 + a7z3 - 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                     4   j = 6                   7 1   j = 4                 12 4     j = 2               13 8       j = 0             16 11         j = -2           13 14           j = -4         11 15             j = -6       7 13               j = -8     3 11                 j = -10   1 7                   j = -12   3                     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, 170]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 170]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[12, 7, 13, 8], X[22, 15, 7, 16], > X[14, 6, 15, 5], X[6, 14, 1, 13], X[16, 21, 17, 22], X[18, 10, 19, 9], > X[20, 11, 21, 12], X[4, 18, 5, 17], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 5, -6}, > {3, -1, 8, -2, 9, -3, 6, -5, 4, -7, 10, -8, 11, -9, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 10 18 24 28 29 -q + ----- - ---- + ---- - ---- + ---- - ------- + 24 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 > 19 q + 11 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 4 5 2 -8 5 5 4 2 4 6 + q - q - q + --- - --- + --- + q - -- + -- - -- + q - q + 14 12 10 6 4 2 q q q q q q 6 8 10 12 14 > 6 q - 3 q + 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 170]][a, z]
Out[8]=	3 3 1 1 2 z 3 5 z 2 z 3 3 3 5 3 -(----) + --- + --- - a z - a z + a z + -- - ---- + a z - 3 a z + a z - 3 a z a 3 a a z a 5 2 z 5 3 5 7 > ---- + 2 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 170]][a, z]
Out[9]=	2 -2 1 1 z 4 z 3 5 2 z 2 2 -a + ---- + --- - -- - --- - 5 a z - a z + a z + 6 z + -- + 12 a z + 3 a z 3 a 2 a z a a 3 4 4 2 6 2 4 z 3 3 3 5 3 7 3 4 z > 6 a z - a z - ---- + 10 a z - 2 a z - 7 a z + a z - 28 z + -- - 3 4 a a 4 5 5 10 z 2 4 4 4 6 4 9 z z 5 5 5 > ----- - 38 a z - 17 a z + 4 a z + ---- + -- - 21 a z + 12 a z - 2 3 a a a 6 6 7 7 7 5 6 z 20 z 2 6 4 6 6 6 5 z 14 z > a z + 44 z - -- + ----- + 48 a z + 21 a z - 4 a z - ---- + ----- + 4 2 3 a a a a 8 7 3 7 5 7 8 10 z 2 8 4 8 > 39 a z + 11 a z - 9 a z - 14 z - ----- - 17 a z - 13 a z - 2 a 9 10 z 9 3 9 10 2 10 > ----- - 21 a z - 11 a z - 4 z - 4 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 3 1 7 3 11 7 13 16 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 11 15 13 2 2 2 4 2 4 3 > ----- + ---- + ---- + 11 t + 13 q t + 8 q t + 12 q t + 4 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a170

L11a169 L11a171