Narrow Results

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^{\ast }$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^{\ast }$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

We show how an incompressible Seifert surface F for a knot K in S³ can be used to create an essential lamination ℒ_F in the complement of each of an infinite class of knots associated to F. This lamination is persistent for these knots; it remains essential under all non-trivial Dehn fillings of the knot complement. This implies a very strong form of Property P for each of these knots.

We shall study several circles in the 3-sphere called a link which has "high" splitness properties. We offer several kind of those links and study relations among them. Alexander polynomial, Conway polynomial and Milnor μ and invariants did not work for those links as vanishing cause of high splitness. We use higher order elementary ideals to distinguish those links.

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

We describe a procedure for creating infinite families of hyperbolic knots, each having unique minimal genus Seifert surface which cannot be the sole compact leaf of a depth one foliation.

We show that a knot in S³ with an infinite number of incompressible Seifert surfaces contains a closed non-peripheral incompressible surface in its complement.

This paper presents a new algorithm for constructing Seifert surfaces from n-bridge projections of links. The algorithm, 𝔄, produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which canonical genus is strictly greater than genus, (g_c(K) > g(K)), and show that 𝔄 builds surfaces realizing the knot genus g(K). We also present a generalization of Seifert's algorithm which constructs surfaces representing arbitrary relative second homology classes in a link complement.

Any link has a flat braidzel presentation. We introduce a new integral invariant of links, named the flat braidzel length, with respect to the flat braidzel presentation. We give a lower bound for the flat braidzel length, and determine the flat braidzel length of an infinite family of links. Moreover, we give the table of knots with the flat braidzel length five or less.

The flat plumbing basket presentation of a link is introduced by Furihata, Hirasawa and Kobayashi. In this paper, we show that the pass-equivalence and an equivalence introduced by using the flat plumbing basket presentation are the same relation. Furthermore, we obtain an evaluation of the minimal number of bands used for the flat pluming basket presentation for a knot coming from the degree of the Alexander polynomial and the three genus of the knot.

For a knot $K\subset S^3$, its exterior $E(K)=S^3\setminus \eta (K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f:E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E(K)$. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E(K)$ cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy.

A spatial surface is a compact surface embedded in the $3$-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla $\delta$-invariant to obstruct a knot from being a stabilization of another.

We introduce the notion of alteration of a surface embedded in a 3-manifold extending that of compression. We see that any two Seifert surfaces of a link are related to each other by a “single” alteration, even if they are not by compression.

In this paper, we answer a question raised in “Peripheral elements in reduced Alexander modules” [J. Knot Theory Ramifications 31 (2022) 2250058]. We also correct a minor error in that paper.

We show that every oriented link is the boundary of a surface obtained from an embedded disk by plumbing a flat annulus a finite number of times.

We show that the differences between canonical genus and free genus, and differences between free genus and usual genus of a knot can be arbitrarily large.

It is shown that for a positive integer g, there is a free genus one knot which admits g + 1 mutually disjoint and mutually non-equivalent incompressible Seifert surfaces each of genus g. As an application we show that for a positive integer g, the genus two handlebody contains g + 1 mutually disjoint and mutually non-isotopic separating incompressible surfaces each of genus g.