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We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror reflections. This allow us to characterise completely extended welded knots by the parity of their number of wens, that we can always reduce to 0 or 1., Comment: All comments are welcome!

We study virtualized Delta, sharp, and pass moves for oriented virtual links, and give necessary and sufficient conditions for two oriented virtual links to be related by the local moves. In particular, they are unknotting operations for oriented virtual knots. We provide lower bounds for the unknotting numbers and prove that they are best possible., Comment: 20 pages

We introduce a local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized $\Delta$-moves., Comment: 10 pages

The 3rd Anti-UAV Workshop & Challenge aims to encourage research in developing novel and accurate methods for multi-scale object tracking. The Anti-UAV dataset used for the Anti-UAV Challenge has been publicly released. There are two main differences between this year's competition and the previous two. First, we have expanded the existing dataset, and for the first time, released a training set so that participants can focus on improving their models. Second, we set up two tracks for the first time, i.e., Anti-UAV Tracking and Anti-UAV Detection & Tracking. Around 76 participating teams from the globe competed in the 3rd Anti-UAV Challenge. In this paper, we provide a brief summary of the 3rd Anti-UAV Workshop & Challenge including brief introductions to the top three methods in each track. The submission leaderboard will be reopened for researchers that are interested in the Anti-UAV challenge. The benchmark dataset and other information can be found at: https://anti-uav.github.io/., Comment: Technical report for 3rd Anti-UAV Workshop and Challenge. arXiv admin note: text overlap with arXiv:2108.09909

We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. The calculations of intersection polynomials are given up to crossing number four. We also study several properties of intersection polynomials., Comment: 27 pages, 22 figures. We divide the original version into four papers

The $\Xi$-move is a local move generated by forbidden moves in virtual knot theory. This move was introduced by Taniguchi and the second author, who showed that it characterizes the odd writhe of virtual knots, which is a fundamental invariant defined by Kauffman. In this paper, we extend this result by classifying $2$-component virtual links up to $\Xi$-moves, using refinements of the odd writhe and linking numbers., Comment: 26 pages, 39 figures; v2: substantial revision. To appear in Fundamenta Mathematicae

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links., Comment: 26 pages, 25 figures

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$., Comment: 9 pages, 6 figures

We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.

We prove that any $11$-colorable knot is presented by an $11$-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially $11$-colored diagrams of the knot. We also prove a similar result for any $11$-colorable ribbon $2$-knot., Comment: 20 pages, 27 figures. Submitted

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams., Comment: Dedicated to Professor Louis H. Kauffman for his 60th birthday

We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an unknotting operation, such as ribbon 2-knots and twist-spun knots, have trivial Khovanov-Jacobsson number., Comment: Short paper: 5 pages, 4 Figures

We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images., Comment: 14 pages, eight figures, interesting applications

The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers., Comment: 17 pages, 17 Figures. epsf.sty embedded in Tex File

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we geometrically represent every element of the bordism group uniquely by a certain standard form of a surface-link, a generalization of a Hopf link. The standard forms give rise to an inverse of Sanderson's geometrically defined invariant., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-14.abs.html

Triple linking numbers were defined for 3-component oriented surface-links in 4-space using signed triple points on projections in 3-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple linking numbers and that triple linking numbers are link homology invariants., Comment: 13 Pages; 3 Figures; epsf sty file included in TeX file