Weakly minimal area of cubical 2-knots on R~4

摘要

A cubical 2-knot K-2 is an embedding of the 2-sphere in the 2-skeleton of the canonial cubulation of R-4, so K-2 is the union of m(K-2) unit squares, hence its area is m(K-2). In [6] was proved that given two cubical knots of dimension two in R-4, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. We say that a cubical 2-knot K-2 is weakly minimal if the area of K-2 can not be reduced by any single cubical move. One interesting question is the following: Given a knot type, what is the area needed for a cubical 2-knot on the canonical cubulation of R-4 to be a weakly minimal knot with the given knot type? In this paper, we answer this question for the spun trefoil knot. (C) 2019 Elsevier B.V. All rights reserved.