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

L11a368 L11a370

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a369 Visit L11a369's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 4q-13/2 - 7q-11/2 + 8q-9/2 - 10q-7/2 + 9q-5/2 - 8q-3/2 + 6q-1/2 - 4q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 - q-20 + 2q-18 + q-14 + 2q-12 + 3q-8 + q-4 + q-2 - 1 + q2 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 - az-1 - az + 2az3 + az5 + a3z-1 + 4a3z + 6a3z3 + 2a3z5 + 2a5z3 + a5z5 - 2a7z - a7z3

Kauffman Polynomial:

2a-1z - 7a-1z3 + 5a-1z5 - a-1z7 + 3z2 - 11z4 + 9z6 - 2z8 + az-1 - 5az + 8az3 - 10az5 + 8az7 - 2az9 - a2 + 2a2z4 + 2a2z8 - a2z10 + a3z-1 - 9a3z + 27a3z3 - 27a3z5 + 16a3z7 - 4a3z9 + 2a4z4 + a4z8 - a4z10 + 4a5z - 7a5z3 + a5z5 + 3a5z7 - 2a5z9 - 2a6z2 - 5a6z4 + 6a6z6 - 3a6z8 + 6a7z - 16a7z3 + 11a7z5 - 4a7z7 - 3a8z2 + 5a8z4 - 3a8z6 + 3a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     1   j = 2                   3 1   j = 0                 3 1     j = -2               5 3       j = -4             5 4         j = -6           5 4           j = -8         4 6             j = -10       3 4               j = -12     1 4                 j = -14   1 3                   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, Alternating, 369]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 369]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 5, 17, 6], X[14, 3, 15, 4], X[20, 8, 21, 7], > X[22, 17, 11, 18], X[18, 10, 19, 9], X[4, 15, 5, 16], X[8, 20, 9, 19], > X[6, 22, 7, 21], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -7, 2, -9, 4, -8, 6, -11}, > {10, -1, 11, -3, 7, -2, 5, -6, 8, -4, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 4 7 8 10 9 8 6 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 4 Sqrt[q] + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -20 2 -14 2 3 -4 -2 2 8 -1 - q - q + --- + q + --- + -- + q + q + q + q 18 12 8 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 369]][a, z]
Out[8]=	3 3 a a 2 z 3 7 z 3 3 3 5 3 -(-) + -- - --- - a z + 4 a z - 2 a z - -- + 2 a z + 6 a z + 2 a z - z z a a 7 3 5 3 5 5 5 > a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 369]][a, z]
Out[9]=	3 2 a a 2 z 3 5 7 2 6 2 -a + - + -- + --- - 5 a z - 9 a z + 4 a z + 6 a z + 3 z - 2 a z - z z a 3 8 2 10 2 7 z 3 3 3 5 3 7 3 > 3 a z + 2 a z - ---- + 8 a z + 27 a z - 7 a z - 16 a z + a 5 9 3 4 2 4 4 4 6 4 8 4 10 4 5 z > 3 a z - 11 z + 2 a z + 2 a z - 5 a z + 5 a z - a z + ---- - a 5 3 5 5 5 7 5 9 5 6 6 6 > 10 a z - 27 a z + a z + 11 a z - 2 a z + 9 z + 6 a z - 7 8 6 z 7 3 7 5 7 7 7 8 2 8 > 3 a z - -- + 8 a z + 16 a z + 3 a z - 4 a z - 2 z + 2 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > a z - 3 a z - 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 1 3 1 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 4 4 6 5 4 5 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a369

L11a368 L11a370