前橋組みひもセミナー
Maebashi Braid Seminar
支援：2024年度科学研究費補助金(基盤研究(C))「ブレイドシステムのHurwitz同値不変量の列の構成と曲面ブレイドへの応用」研究代表者：矢口義朗（前橋工科大学）研究課題番号:19K03508.

日時: 令和6年11月9日(土)14時00分～16時30分
会場: 前橋工科大学 １号館４階 142教室
（JR前橋駅北口４番乗り場から バス で15分ほどです。）
Date & Time: Saturday, November 9, 2024, 14:00-16:30
Venue: Maebashi Institute of Technology, Room 142, 4th floor of No.1 Bldg.

14:00-15:00 Talk 1
Speaker: Komal Negi (Indian Institute of Technology, Ropar)
Title: Numerical invariants of twisted knots [PDF]
Abstract: Knots in oriented thickened surfaces are known as Twisted knots, which was introduced by Mario O Bourgoin [1] in 2008.
　In this talk, we discuss the invariants of twisted knots derived from the invariants of virtual knots, such as the odd writhe and arc shift number [2]. We present a class of twisted knots where the arc shift number of every member is 1. This is a joint work with my supervisor M. Prabhakar.
　Additionally, we discuss one more numerical invariant for twisted knots obtained from warping degree [3], and we use the warping degree to develop a labeling scheme for twisted knots, known as ‘warping labeling’. We have further generalized this scheme to ‘up-down labeling’ for twisted knots. This is a collaborative work with A. Shimizu and M. Prabhakar.

[1] M. O. Bourgoin, Twisted link theory, Algebr. Geom. Topol. 8 (2008), no. 3, 1249–1279.
[2] K. Negi and M. Prabhakar, Generalization of Arc Shift for twisted knots, J. Knot theory and its Ramifications, Vol. 33, No. 02, 2450004 (2024).
[3] K. Negi, A. Shimizu, M. Prabhakar, Warping Labeling for twisted knots and twisted virtual braids, arXiv:2406.08505, 2024.
15:00-15:30 Tea Time

15:30-16:30 Talk 2
Speaker: Komal Negi (Indian Institute of Technology, Ropar)
Title: All about twisted virtual braids [PDF]
Abstract: Twisted virtual braids are a combinatorial generalization of virtual braids. In this talk, I will discuss the relationship between twisted virtual braids and twisted links, by stating the theorems for twisted virtual braids corresponding to the Alexander theorem and the Markov theorem in knot theory. Interestingly, the set of twisted virtual braids on n strands forms a group. I will present a group presentation and also a reduced group presentation of the twisted virtual braid group. This is a collaborative work with S. Kamada and M. Prabhakar [1].
　In the second part of this talk, I will discuss the invariants for twisted virtual braids which is defined by using the concept of warping degree. I will also show that the up-down labeling [2] can be extended to twisted virtual braids. Furthermore, we prove that by restricting the labeling set to Z2, we can construct invariants for twisted virtual braids. This is a joint work with A. Shimizu and M. Prabhakar.

[1] K. Negi, M. Prabhakar, S. Kamada, Twisted virtual braids and twisted links, Osaka Journal of Mathematics, Vol. 61(Issue 4), (2023).
[2] K. Negi, A. Shimizu, M. Prabhakar, Warping Labeling for twisted knots and twisted virtual braids, arXiv:2406.08505, 2024.
　　　
Organizers:
矢口義朗, Yoshiro YAGUCHI (前橋工科大学) y.yaguchi (add @maebashi-it.ac.jp)
吉田はん, Han YOSHIDA (OCAMI)
清水理佳, Ayaka SHIMIZU (OCAMI)