Journal of Knot Theory and Its Ramifications

It is well-known that the focal set (i.e. the image of the caustic) of a given convex closed curve γ admits singular points. In this paper, we classify the diffeomorphic type of focal sets of convex curves which admit at most four cusps.

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

By a fixed continuous map from a 3-space to itself, a knot in the 3-space may be mapped to another knot in the 3-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.

In 1990, Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Δ-move. We extend the Okada's result for virtual knots by using a Vassiliev invariant v₂ of virtual knots of degree 2 which is induced from the Kauffman polynomial of a virtual knot. We show that v₂(K₁) - v₂(K₂) = ±48, if K₂ is a virtual knot obtained from a virtual knot K₁ by applying a Δ-move. From this we have a lower bound for the number of Δ-moves if two virtual knots K₁ and K₂ are related by a sequence of Δ-moves.

In this paper we compute the sharp lower bounds for the crossing number of n-string k-loop essential tangles. For essential tangles with only string components, we characterize the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings.