In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid

Abstract. In [23] we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (compare [24]). Applications include new restrictions on the Alexander polynomial of alternating knots. 1.

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

Let M be a connected, compact, orientable 3-manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler characteristic of embedded surfaces. (A similar result holds when b1 (M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings. Contents 1

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Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach algebra situs. When looking at the panorama of skein modules 1, we see, past the rolling hills of homologies and homotopies, distant mountains- the Kauffman bracket skein module, and farther off in the distance skein modules based on other quantum invariants. We concentrate here on the basic properties of the Kauffman bracket skein module; properties fundamental in further development of the theory. In particular we consider the relative Kauffman bracket skein module, and we analyze skein modules of I- bundles over surfaces. History of skein modules from my personal perspective I would like to use this opportunity, of informal presentation, 2 to give my personal history of algebraic topology based on knots (a more formal account was given in [Pr-7]). In July 1986 I left Poland invited by Dale Rolfsen for a visiting position at UBC. In January of 1987, Jim Hoste gave a talk at the first Cascade Mountains Conference (in Vancouver) and described his work on multivariable generalization of the Jones-Conway ([HOMFLY][PT]) polynomial of links in S 3. He was convinced that his construction works for 2 colors when the first color is represented only by a trivial component. He had already succeeded in the case of 2-component 2-bridge links. His method, following Nakanishi, was to analyze link diagrams in an annulus (the trivial component being z axis). We immediately noticed (with Jim) that the analogous construction for the Kauffman bracket polynomial has an easy solution [H-P-1]. In March

We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(2|1)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration. 1

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

This paper describes a method for the automatic evaluation of the Links– Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are presented for the invariant, for all prime knots of up to 10 crossings and various other links. LG distinguishes between these links, and also detects the chirality of those that are chiral. In this sense, it is more sensitive than the well-known two-variable HOMFLY and Kauffman polynomials. When applied to examples which defeat the HOMFLY invariant, interestingly, LG ‘almost’ fails. The automatic method is in fact applicable to the evaluation of any such state sum invariant for which an appropriate R matrix and cap and cup matrices have been determined.

Abstract. We define the Conway skein module C(M) of ordered based links in a 3-manifold M. This module gives rise to C(M)-valued invariants of usual links in M. We determine a basis of the Z[z]-module C(Σ×[0, 1])/Tor(C(Σ×[0, 1])) where Σ is the real projective plane or a surface with boundary. For cylinders over the Möbius strip or the projective plane we derive special properties of the Conway skein module, among them a refinement of a theorem of Hartley and Kawauchi about the Conway polynomial of strongly positive amphicheiral knots in S 3. In addition, we determine the Homfly and Kauffman skein modules of Σ × [0, 1] where Σ is an oriented surface with boundary.