Narrow Results

Any ribbon torus-knot in 4-space is naturally associated with a virtual knot. We investigate a relationship between ribbon torus-knots and virtual knots from this viewpoint. We also give a new example of a non-classical virtual knots which can not be detected by the group and the Z-polynomial.

For an arbitrary 1-knot k¹, the spun 2-knot of k¹, denoted by spun(k¹), is a ribbon 2-knot in R⁴. Hence for a ribbon 2-knot K², we can also induce a notion corresponding to the crossing number on a 1-knot, and it is said to be the crossing number of K², denoted by cr(K²). In this note, we will show that the Alexander polynomial plays an important role in determining the crossing number of a ribbon 2-knot. Lastly, we will prove the following: If k¹ is a (p,q)-torus knot, then cr(spun(k¹)) is equal to (p - 1)(q - 1).

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.