Knots with proprety R+

Bradd Evans Clark

doi:10.1155/s0161171283000460

Abstract

If we consider the set of manifolds that can be obtained by surgery on a fixed knot K, then we have an associated set of numbers corresponding to the Heegaard genus of these manifolds. It is known that there is an upper bound to this set of numbers. A knot K is said to have Property R+ if longitudinal surgery yields a manifold of highest possible Heegaard genus among those obtainable by surgery on K. In this paper we show that torus knots, 2‐bridge knots, and knots which are the connected sum of arbitrarily many (2, m)‐torus knots have Property R+ It is shown that if K is constructed from the tangles (B1, t1), (B2, t2), …, (Bn, tn) then where T(K) is the tunnel of K and T(Bi, ti) is the tunnel number of the tangle (Bi, ti). We show that there exist prime knots of arbitrarily high tunnel number that have Property R+ and that manifolds of arbitrarily high Heegaard genus can be obtained by surgery on prime knots.

Highlights

INTRODUCTIONB.E. CLARK and Property R, have been studied
A traditional method of constructing 3-manifolds is to perform Dehn surgery3. on a knot or link in S As a result of this relationship between knots and3-manifolds, several negatively defined properties of knots, namely Property PB.E
We show that there exist prime knots of arbitrarily high tunnel number that have Property R+ and that manifolds of arbitrarily high Heegaard genus can be obtained by surgery on prime knots

Summary

INTRODUCTION

B.E. CLARK and Property R, have been studied. 3-manifolds obtainable by surgery on K in terms of Heegaard genus. We shall demonstrate that infinitely many knots have Property R+. If X is a point set, we shall use cl(X) for the closure of X, int(X) for the interior of X and 8X for the boundary of X. The genus of a 3-manifold is defined to be the minimal genus of a Heegaard splitting of the manifold. If X is a polyhedron contained in the P.L. 3-manifold M, N(X) c M is called a regular neighborhood of X in M if X N(X) and N(X) is a 3-manifold which can be simplicially collapsed to X. All manifolds in this paper are assumed to be simplicial and all maps to be piecewise linear

Let K

As a rule

The infinite cyclic covering of M has the same n

Hence the