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.

There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c 1 n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1-skeleton of the interior of a compact triangulated orientable PL 3-manifold M . There is a positive constant c2 such that for each t 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c 2 t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M . We obtain explicit values for c1 and c2 . Keywords: knot theory, knot diagram, Reidemeister move, normal surfaces, computational complexity This paper grew out of work begun while the authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996/7. Research at MSRI is supported in part by NSF grant DMS-9022140. The first au...

Abstract. We compute the Ozsváth-Szabó Floer homologies HF ± and ̂ HF for threemanifolds obtained by integer surgery on a two-bridge knot. 1.

A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links. Furthermore, it is shown that the polynomial satisfies a Conway-type skein relation -- in contrast to the Alexander polynomial derived from the virtual link group. Keywords: Virtual Knot Theory, Conway Skein Relation, Alexander Invariants AMS classification: 57M25

It is shown that the multiplicative monoids of Temperley-Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. Thereby one obtains a representation of braid groups in matrices, which, though different and presumably new, is related to the standard representation of braid groups in Temperley-Lieb algebras. Mathematics Subject Classification (2000): 57M99, 20F36, 18A40 1

two-bridge link complements

Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S 3 which are vertex-separating, i.e., each vertex can be separated by a non-trivial 2-string tangle from the rest of the diagram. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. Keywords: Knotted Graph, Alternating Diagram, Flyping Conjecture AMS classification: 57M25; 57M15

Abstract. For p�3 and for p�5 we prove that there are exactly p equivalence classes of p-coloured knots modulo 1–framed surgeries along unknots in the kernel of a p–colouring. These equivalence classes are represented by connect-sums of n left-hand p,2-torus knots with a given colouring when n�1, 2,..., p. This gives us a 3–colour and a 5–colour analogue of the surgery presentation of a knot in the complement of an unknot. 1.

A Chebyshev knot is a knot which admits a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + ϕ), where a, b, c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and ϕ ∈ R. Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with ϕ = 0. We also show that every knot is a Chebyshev knot.

We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + deg C = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for