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

L11a270 L11a272

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a271 Visit L11a271's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 6q-17/2 - 9q-15/2 + 11q-13/2 - 14q-11/2 + 13q-9/2 - 11q-7/2 + 8q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - 2q-26 + 2q-24 - q-22 + q-20 + 3q-18 + 3q-14 - q-12 + q-10 + 2q-8 - 2q-6 + 2q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - a3z-1 - 2a3z + a3z3 + a3z5 + a5z-1 + 4a5z + 5a5z3 + 2a5z5 - a7z + a7z3 + a7z5 - a9z - a9z3

Kauffman Polynomial:

- 2az + 3az3 - az5 - a2z2 + 4a2z4 - 2a2z6 - a3z-1 + 4a3z - 5a3z3 + 6a3z5 - 3a3z7 + a4 + 2a4z2 - 5a4z4 + 5a4z6 - 3a4z8 - a5z-1 + 6a5z - 12a5z3 + 6a5z5 - 2a5z9 + 2a6z2 - 8a6z4 + 6a6z6 - 2a6z8 - a6z10 - 3a7z + 6a7z3 - 13a7z5 + 12a7z7 - 5a7z9 + 3a8z2 - 9a8z4 + 11a8z6 - 3a8z8 - a8z10 - 2a9z + 4a9z3 - 3a9z5 + 6a9z7 - 3a9z9 + 2a10z2 - 7a10z4 + 11a10z6 - 4a10z8 + a11z - 6a11z3 + 9a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     1   j = -2                   4 1   j = -4                 5 2     j = -6               6 3       j = -8             7 5         j = -10           7 6           j = -12         5 8             j = -14       4 6               j = -16     2 5                 j = -18   1 4                   j = -20   2                     j = -22 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, 271]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 271]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[14, 5, 15, 6], X[18, 8, 19, 7], X[22, 15, 9, 16], X[20, 17, 21, 18], > X[16, 21, 17, 22], X[6, 20, 7, 19], X[8, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 6, -11}, > {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 9 11 14 13 11 8 5 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 2 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 2 -22 -20 3 3 -12 -10 2 2 -q + q - --- + --- - q + q + --- + --- - q + q + -- - -- + 26 24 18 14 8 6 q q q q q q 2 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 271]][a, z]
Out[8]=	3 5 a a 3 5 7 9 3 3 3 5 3 -(--) + -- - 2 a z - 2 a z + 4 a z - a z - a z - a z + a z + 5 a z + z z 7 3 9 3 3 5 5 5 7 5 > a z - a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 271]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 a - -- - -- - 2 a z + 4 a z + 6 a z - 3 a z - 2 a z + a z - a z + z z 4 2 6 2 8 2 10 2 12 2 3 3 3 > 2 a z + 2 a z + 3 a z + 2 a z - 2 a z + 3 a z - 5 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 12 a z + 6 a z + 4 a z - 6 a z + 4 a z - 5 a z - 8 a z - 8 4 10 4 12 4 5 3 5 5 5 7 5 > 9 a z - 7 a z + 3 a z - a z + 6 a z + 6 a z - 13 a z - 9 5 11 5 2 6 4 6 6 6 8 6 10 6 > 3 a z + 9 a z - 2 a z + 5 a z + 6 a z + 11 a z + 11 a z - 12 6 3 7 7 7 9 7 11 7 4 8 6 8 > a z - 3 a z + 12 a z + 6 a z - 3 a z - 3 a z - 2 a z - 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 3 a z - 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 6 5 8 7 6 7 5 6 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 5 t 2 2 > ---- + t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a271

L11a270 L11a272