We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a $3$-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.

We characterize maps between $n$-dimensional Nöbeling manifolds that can be approximated by homeomorphisms.

For a non-trivial knot in the $3$-sphere, only integral Dehn surgery can create a closed $3$-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451.

In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of a closed $(d+1)$-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R.~Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an $S^1$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D.~Silver and S.~Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated to $G$ in its unmodified form.

Let $A$ be a finite abelian group and $M$ be a branched cover of an homology $3$-sphere, branched over a link $L$, with covering group $A$. We show that $H_1(M;Z[1/|A|])$ is determined as a $Z[1/|A|][A]$-module by the Alexander ideals of $L$ and certain ideal class invariants.