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

L11a330 L11a332

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a331 Visit L11a331's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 3q-19/2 - 6q-17/2 + 8q-15/2 - 11q-13/2 + 12q-11/2 - 12q-9/2 + 10q-7/2 - 8q-5/2 + 5q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 - q-28 + 2q-26 + 2q-22 + 2q-20 + 4q-16 - 2q-14 + 2q-12 - q-10 + q-6 - q-4 + q-2 - 1

HOMFLY-PT Polynomial:

a3z-1 + 3a3z + 7a3z3 + 5a3z5 + a3z7 - 3a5z-1 - 13a5z - 22a5z3 - 18a5z5 - 7a5z7 - a5z9 + 2a7z-1 + 6a7z + 8a7z3 + 5a7z5 + a7z7

Kauffman Polynomial:

- a2 + 5a2z2 - 8a2z4 + 5a2z6 - a2z8 + a3z-1 - 3a3z + 16a3z3 - 27a3z5 + 16a3z7 - 3a3z9 - 3a4 + 20a4z2 - 32a4z4 + 10a4z6 + 5a4z8 - 2a4z10 + 3a5z-1 - 16a5z + 39a5z3 - 62a5z5 + 42a5z7 - 9a5z9 - 3a6 + 18a6z2 - 43a6z4 + 31a6z6 - 2a6z8 - 2a6z10 + 2a7z-1 - 8a7z + 10a7z3 - 12a7z5 + 17a7z7 - 6a7z9 + a8z2 - 6a8z4 + 18a8z6 - 8a8z8 + 3a9z - 7a9z3 + 17a9z5 - 9a9z7 - 2a10z2 + 10a10z4 - 8a10z6 - 2a11z + 5a11z3 - 6a11z5 - 3a12z4 - a13z3

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 = 2                       1 j = 0                     2   j = -2                   3 1   j = -4                 5 2     j = -6               6 4       j = -8             6 4         j = -10           6 6           j = -12         5 6             j = -14       3 6               j = -16     3 5                 j = -18     3                   j = -20 1 3                     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, 331]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 331]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 5, 21, 6], > X[22, 18, 9, 17], X[18, 14, 19, 13], X[14, 22, 15, 21], X[16, 7, 17, 8], > X[8, 9, 1, 10], X[6, 15, 7, 16], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 8, -9}, > {9, -1, 2, -3, 6, -7, 10, -8, 5, -6, 11, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 8 11 12 12 10 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 3 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 2 2 2 4 2 2 -10 -6 -4 -2 -1 + q - q + --- + --- + --- + --- - --- + --- - q + q - q + q 26 22 20 16 14 12 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 331]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 5 3 7 3 -- - ---- + ---- + 3 a z - 13 a z + 6 a z + 7 a z - 22 a z + 8 a z + z z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 5 a z - 18 a z + 5 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 331]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - 3 a z - 16 a z - 8 a z + 3 a z - z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 > 2 a z + 5 a z + 20 a z + 18 a z + a z - 2 a z + 16 a z + 5 3 7 3 9 3 11 3 13 3 2 4 4 4 > 39 a z + 10 a z - 7 a z + 5 a z - a z - 8 a z - 32 a z - 6 4 8 4 10 4 12 4 3 5 5 5 > 43 a z - 6 a z + 10 a z - 3 a z - 27 a z - 62 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 12 a z + 17 a z - 6 a z + 5 a z + 10 a z + 31 a z + 18 a z - 10 6 3 7 5 7 7 7 9 7 2 8 4 8 > 8 a z + 16 a z + 42 a z + 17 a z - 9 a z - a z + 5 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 2 a z - 8 a z - 3 a z - 9 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 3 3 3 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 8 20 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 6 5 6 6 6 6 4 6 2 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + --- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 3 t 2 t 2 3 > --- + 2 t + -- + q t 2 2 q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a331

L11a330 L11a332