Tangles. Some Category-theoretical Results. The Categories OTa and ODi. Representations of the Category OTa. The Operator T. On Reidemeister Moves. 35.
AARHUS UNIVERSITET
U(14E BACKGROUND TO V.
0.
TUBAEVs
IFOLDS QUANTUM INVARIANTS OF 3-MAN
Dieter van de Griend
September,
1993
Various Publications Series No. 42
A
MATEMATISK INSTITUT
Some Background to V.G Turaev’ s Quantum Invariants of 3-Manifolds
Pieter van de Griend
Abstract In this paper we shed some light on the background of the central Theorem 2.5 in V.G Turaev’s Paper Quantum Invari ants of 3-Manfo1ds. We do so via a study of hierarchical gen eralizations of a procedure for obtaining linkinvariants from representations of the braidgroup B 71 and some earlier work by the same author and collaborations with N. Reshetikhin.
iv
Contents Preface Chapter I Section 1 Section 2 Section 3 Chapter II Section 1 Section 2 Section 3 Chapter III Section 1 Section 2 Section 3 Section 4 Section 5 Appendix A Bibliography Index
11
First to Braids and then to Link Invariants A Little about Knots and Links About Geometrical and Algebraical Braids From Braids to Links and Vice Versa Turaev’s Machine Preliminaries Yang-Baxter Operators An Example A Whiff of Tangle Categories Tangles Some Category-theoretical Results The Categories OTa and ODi Representations of the Category OTa The Operator T On Reidemeister Moves .
.
1 2 3 7
7 8 9 11 12 14 18 23 31 33 35 36
Preface In [my] opinion the discovery of the categories of tan gles and ribbon graphs was one of the major steps to wards the construction of topological field theories in dimension 3. Vladimir Turaev 1992.
This paper arose due to a few rather dense packed seminars, which were part of the 1993 Fall course on quantum invariants at Aarhus Universitet by Henning Haahr Andersen. Their purpose was to illustrate the background of the pivotal Theorem 2.5 in Vladimir Turaev’s Quantum Invariants of 3-Manifolds [25].
Theorem 2.5: Let V be a strict ribbon category with braiding c, twist &, and compatible duality (*, b, d). There exists a unique covariant functor F : Rihv —* V preserving tensor product and satisfying the following conditions: (i) F transforms any object (V, 1) into V and any object (V, —1) into V. (ii) For any objects V W E V
F (x ) 7
=
c,
F 7)
=
F ( U)
O,
7 b
F ( fly)
=
(iii) For any elementary v-colored ribbon graph F we have F(f) f is the color of the only coupon of F.
dv
=
where
The functor F has the following properties: F (x ) 17
=
(cTz)
) 47 F(Z F (TTt7)
Cy* 147*,
1,
=
F (Yf17)
(c7*)
) 17 F(Z,
, 1 (cr.,y)
F (Tw)
=
=
(cv. w)
1,
F (y)
=
cy.,w,
=
—1,
F (y)
=
(Ov
The layout of this paper is that we proceed in three phases. In Chapter I we encounter link invariants, derivable from characters of representations of the braid group B,. In Chapter II we explain how this perspective can be generalized into the equivalent framework of Yang-Baxter operators. In Chapter III we see how the successive generalization of the trace map leads to the emergence of link invariants as functors between tangle categories. This is the essence of the proofs leading to Theorem 5.1, which in turn has evolved to become Theorem 2.5 of [25]. The latter represents II
the central tool in the (currently) final steps in terms of the hierarchical abstractions of link- and 3-manifold invariants derivable from the trace map. In the Appendix A some remarks on the so-called Reidemeister Moves can be found. We shall allow ourselves the liberty to both take an intuitive and rather leisurely approach to links and draw on results from braid- and category theory without giving all of their proofs.
Pieter van de Griend October 6, 1993.
Matematisk Institut Ny Munkegade Aarhus Denmark
111
Some Background to V.G Turaev’s Quantum Invariants of 3-Manifolds
Chapter I First to Braids and then to Link Invariants We establish relations between braids and links, and vice versa, via the Alexanderand Markov Theorems. Their combined action is to reduce the link isotopy problem to an algebraic problem of classifying certain equivalence classes. We conclude this chapter by showing how the foregoing enables us to construct an invariant. This will form the basis for the successive generalizations of trace invariants of the next chapters.
1.1 A Little about Knots and Links The notion of a knot is easy to convey, however the mathematical cord into which a knot materializes is assumed to be infinitely elastic. This forces us to bring in the definitions for cord and elasticity. Mathematically we can formulate precisely what we understand by these rather vague notions along with that of “a knotted structure”. Definition 1.1: A (proper) knot K is the trajectory y* of an embedding ‘y of the unit circle S 1 into S . i.e: 3 3 : 7 c 1 S _S
and
(3)
Definition 1.2: An n-link L is the trajectory disjoint union of n unit circles into 3• i.e: :
JJs
3,
fl ( 7 S)
=
of an embedding ‘y of the ordered
0
and
7*
=
L
(4)
In the following we shall refer to the set of all such n-links as L The property that any n-link can be deformed into quite differently appearing structures via sequences of smooth local transformations is called (ambient) isotopy. This induces an equivalence relation - on L. We formulate our
Problem 1: Classify the isotopy classes of IZ/.
Knots can be brought into a certain normal form, which relates them to braids. Let’s have a look at such objects first.
Pieter van de Griend
1.2 About Geometrical and Algebraical Braids In Euclidean 3-space, with coordinates (cr, y, z), consider the planes z = 0 and 2 z = 1. Call x {1} the upper plane and 2 x {0} the lower plane. Mark the points p = ((i, 0), 1), 1 < i < n in the upper plane and project them orthogonally onto the lower plane points p. , p ,
..
.
An arc is a map A, : [0. 1] —* II2 x [0, 1] such that t (1) = p. (0) = p and ((.f (t). ,fj (t)) , t) with smooth
Definition 2.1:
A geometrical v-braid /3 is a system of v embedded arcs A = x [0. 1], where the i-th arc A, connects the point P1 on the upper {A,, , A} plane to the point 74(i) on the lower plan for some permutation r e S, such that Definition 2.2: ..
.
i. ii.
Each arc A 1 intersects each intermediate z-plane, 0 < z < 1 exactly once. The arcs A 1 A, intersect each intermediate z-plane in exactly u distinct points.
The permutation r is called the permutation of the braid. Definition 2.3: The arc A, is called the i-th braid strand. The number u is called the strand index. Definition 2.4:
Two geometrical n-braids A° = {A?,. , A,} and A’ = are equivalent if there is a homotopy through geometrical braids, A} Al respecting their permutation r, from A° to A’, i.e if there exist v. level-preserving .
.
{
continuous maps :[0,1]x[0,1] 1
