1.    Geometric Invariants of Surface-knots (with A. Al Kharusi)

A surface-knot is a closed surface embedded in 4-space. The aim of this project is to investigate a geometric surface-knot invariant for surface-knots. The projected image of a surface-knot under the orthogonal projection is called a surface-knot diagram. The diagram may contain some triple points. There exists a surface-knot diagram which has the least number of triple points for all possible diagrams. The number of triple points is a geometric surface-knot invariant called a triple point number. Currently, we investigate the lower bounds of triple point numbers.

Main Results

1. T. Yashiro. Pseudo-cycles of surface-knots. Journal of Knot Theory and its Ramifications. 25 1650068 (2016) (18 pages).
2. A. Al Kharusi and T. Yashiro. On crossing changes for surface-knots. Kyungpook Mathematical Journal. 6 (2016) 1247-1257.
3. T. Yashiro. Covering diagrams over surface-knot diagrams. Journal of Knot Theory and its Ramifications. 27 1850042 (2018) (11 pages). DOI: 10.1142/S0218216518500426.
4. A. Al Kharusi and T. Yashiro, No Surface-knot of genus one has triple point number two. Journal of Knot Theory and its Ramifications. 27 1850063 (2018)(24 pages).

2. Construction of Surface-knot Diagrams（wiht A. A. Mohamad)

The aim of this project is to investigate how a surface-knot diagram is constructed. We have constructed a diagram by pasting simple diagrams.

Main Result

1. A. Mohamad and T. Yashiro. On contruction of surface-knots. Journal of Knot Theory and its Ramifications. 25 1650053 (2016) (18 pages). DOI: 10.1142/S021821651650053X

3.   Marked graph diagrams and triple points (with A. Al Kharusi, A. Kawauchi, Z. Al Maamari)

A surface embedded in four-space can be described as a sequence of link diagrams with some marked crossings. Yoshikawa introduced a marked graph diagram in which all marked graphs are concentrated in the middle of the sequence. We can construct a surface-knot diagram from a marked graph diagram. The aim of this project to investigate a relation between the marked graph diagram and the number triple points in the diagram induced from the marked graph diagram.

4. Topological model of DNA replication (with A. A. Mohamad)

A double strand DNA has a double helical structure and it is modeled by a thin long twisted ribbon fixed at the both ends. A DNA-link is a topological model of such a DNA segment in the nuclear of a eukaryotic cell. In the cell cycle, the DNA is replicated and distributed into new cells. The complicated replication process follows the semi-conservative scheme in which each backbone string is preserved in the replicated DNA. This is interpreted in terms of splitting process of the DNA-link. In order to split the DNA-link, unknotting operations are required. The aim of this project is to obtain a topological model for finding a mechanism of the splitting process.

Main Results

1. A. A. Mohamad and T. Yashiro. A Topological Model of DNA Replication with DNA-links. Far East Journal of Mathematical Sciences. 107 (2018). 241-255.
2. A. A. Mohamad and T. Yashiro. A rewinding model for replicons with DNA-links. BIOMATH 9 (2020). 2001047. URL: http://dx.doi.org/10.11145/j.biomath.2020.01.047

5.  Generalized metric and the set of knots (with A. A. Mohamad)

A generalized metric function can have a partial ordered set as its codomain. The set of knots can be viewed as a partial ordered set regarding to the number of crossings.  This research investigates properties of a generalized metric function with the set of knots as its co-domain.

Main Result

1. A. A. Mohamad and T. Yashiro. Topological study of generalized metric spaces. JP Journal of Geometry and Topology. 22 (2019) 165-188.