English abstract
Abstract
ncer
This thesis is primarily concerned with two main questions that refer to the knottedness problem on surface-knot theory. The first question is: Suppose A
and A' are surface-knot diagrams of equivalent non-trivial surface-knots F and F',
respectively. Assume that a trivial surface-knot is obtained from A by crossing
changes. Can we obtain a trivial surface-knot from Aby making suitable crossing
changes as well? The second question is: What is the minimal triple point number
for orientable surface-knots? We introduce the following contributions related to
11
Yves
the first question: A crossing change operation along double point curves of surface-knot diagrams is defined. We show that any surface-knot diagram has a union of exchangeable double curves. We define a family of surface-knots called du-exchangeable surface-knots and a special sequence of surface-knot diagrams called a t-descendent sequence. We prove that for a du-exchangeable surface knot, there exists a finite t-descendent sequence of surface-knot diagrams, each of
e
which can be unknotted by the crossing changes. As an application, an invariant
for du-exchangeable surface-knots called the du-exchange index is defined. The
answer to the second question is well known for 2-knot case. We estimate the
lower bound for triple point number for genus-one surface-knots. In fact, we show
that there is no orientable surface-knot of genus one with triple point number two.
To prove this result, we use Roseman moves and the oriented intersection number
eman Moves
1 Numi
of simple closed curves in the double decker set. Indeed, we use diagrammatic
methods to discuss the two questions under consideration
