English
Eri Hatakenaka
Department of Mathematical Sciences, Institute of Engineering, Tokyo University of Agriculture and Technology.
Research area: low-dimensional topology, including knot theory, 3- and 4-dimensional manifolds, surface-knots, branched covering presentations, quandles, cocycle invariants, and the Dijkgraaf-Witten invariant.
Contents
Research Overview
My research field is low-dimensional topology.
Topology, also called “isou-kikagaku” in Japanese, is a branch of mathematics that studies figures and spaces by focusing not on their lengths or angles themselves, but on properties that remain unchanged under continuous deformation.
In everyday life, we usually think of two figures as having “the same shape” when they fit exactly on top of one another. In topology, however, we imagine figures as if they were made of elastic rubber, and we regard figures that can be transformed into one another by stretching and shrinking as being “the same shape.”
For example, a mug and a doughnut look completely different. In topology, however, both are regarded as the same type of shape, because each has one hole. The appeal of topology lies in looking beyond small differences in shape—such as lengths, angles, and bumps—and finding the essential properties that remain after deformation.
A Topological Way of Thinking
In topology, everyday problems can turn into mathematical problems.
A famous example is the problem of the Seven Bridges of Königsberg. The question is whether there is a walking route that crosses each of the seven bridges exactly once. This problem was solved by ignoring the detailed map and extracting only the essential information about places and the bridges connecting them, turning the situation into a problem about a graph.
In this way, taking an essential structure out of a concrete shape or arrangement and treating it mathematically is one of the important ways of thinking in topology.
Knot Theory
One of the important objects in my research is a knot.
In mathematics, a knot is obtained by tying a string and then joining its two ends to form a closed loop. If two such knots can be transformed into each other by moving and stretching the string, without cutting or reattaching it, they are considered to be the same knot. If this is impossible, they are considered to be different knots.
A knot diagram is a drawing of a three-dimensional knot on a plane. Even when two diagrams look quite different, they may represent the same knot.
Example: Different diagrams of the same knot
These two diagrams were long treated as representing different knots in classical knot tables. Perko showed that they actually represent the same knot.
This example illustrates an important point in knot theory: looking different as diagrams is not the same as being different as knots. If one diagram can be transformed into the other without cutting or reattaching the string, then the two diagrams represent the same knot.
In knot theory, we consider questions such as the following.
- Are two knots the same?
- How can we show that two different-looking diagrams actually represent the same knot?
- How can we prove that two similar-looking knots are genuinely different?
To show that two diagrams represent the same knot, one looks for a sequence of deformations from one diagram to the other. On the other hand, to show that two knots are different, mathematical tools called invariants are useful. An invariant is information that does not change when a knot is deformed.
References: D. Rolfsen, Knots and Links, Appendix C, Publish or Perish, 1976; K. A. Perko Jr., “On the classification of knots,” Proceedings of the American Mathematical Society 45 (1974), 262–266.
Low-Dimensional Topology and Knots
Low-dimensional topology mainly studies spaces of dimensions three and four. We live in a three-dimensional world, but in mathematics we can study the structure of three-dimensional spaces themselves, and also extend our viewpoint to four-dimensional spaces.
Knots are not only fascinating objects in their own right, but also important tools for understanding 3-manifolds and 4-manifolds. For example, knots and links can be used to describe 3-manifolds, and diagrams involving knots can be used to study the structure of 4-manifolds. Surface-knots, which are knotted surfaces in four-dimensional space, are also important objects in low-dimensional topology.
One of the great attractions of this field is that spaces which are difficult to see directly can be understood through knots, diagrams, and invariants.
My Research Themes
I have studied invariants related to knots and 3-manifolds.
In particular, I am interested in invariants obtained from branched covering presentations of 3-manifolds, quandles and their cocycle invariants, and relations with the Dijkgraaf-Witten invariant.
A branched covering presentation is a way to describe one space as lying over another space. Using such presentations, one can read information about 3-manifolds from diagrams of knots and links.
A quandle is an algebraic structure that appears when considering colorings of knot diagrams. From the seemingly simple operation of coloring a knot diagram, one can construct invariants of knots and surface-knots.
The Dijkgraaf-Witten invariant is a topological invariant of 3-manifolds. I study relationships between invariants of 3-manifolds and invariants arising from knots and surface-knots.
Research Keywords
- Low-dimensional topology
- Knot theory
- 3-manifolds
- 4-dimensional topology
- Surface-knots
- Branched covering presentations
- Quandles
- Cocycle invariants
- Dijkgraaf-Witten invariant
The Appeal of Topology
Topology has both an intuitive side that can be enjoyed visually and a rigorous, deeply theoretical side.
The idea that a mug and a doughnut can be regarded as the same shape, the question of whether a route can cross each bridge exactly once, and the problem of whether a knot can be untied all begin from familiar ways of thinking. Eventually, however, they lead to deep mathematics for understanding spaces of dimensions three and four.
For me, the appeal of low-dimensional topology lies in the way it starts from our everyday sense of “shape” and leads us toward the hidden structure of spaces that are difficult to see directly.