Narrow Results

Twisted Alexander invariants of knots are well defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.

In this note, we show that there exists an infinite family of knots with the free genus one, the braidzel genus two and the canonical genus n for any positive odd integer n greater than one.

We show that the differences between canonical genus and free genus, and differences between free genus and usual genus of a knot can be arbitrarily large.

In this paper, we show that free genus one knots do not admit essential tangle decompositions.

We characterize satellite knots of free genus one, and know that the set of satellite knots of free genus one coincides with that of satellite knots of tunnel number one.