Kengo Kishimoto

THE DEALTERNATING NUMBER AND THE ALTERNATION NUMBER OF A CLOSED 3-BRAID

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

A TABLE OF GENUS TWO HANDLEBODY-KNOTS UP TO SIX CROSSINGS

A handlebody-knot is a handlebody embedded in the 3-sphere. We enumerate all genus two handlebody-knots up to six crossings.

Simple-ribbon fusions on non-split links

In [Simple ribbon fusions and genera for links, J. Math. Soc. Japan 68 (2016) 1033–1045], we introduced special types of fusions, so-called simple-ribbon fusions on links, and showed that they never decrease the genera of links. In this paper, we also gave a geometric condition for a simple-ribbon fusion to preserve a link type, and it turns out that a “complicated” simple-ribbon fusion may preserve a link type if the link is split. In this paper, we show that only “clearly trivial” simple-ribbon fusion preserves a link type if the link is non-split. This enables us to give a lower bound for the difference between genera of knots related by a simple-ribbon fusion.