Related papers: Basket, flat plumbing and flat plumbing basket num…
The existence of basket, flat plumbing and flat plumbing basket surfaces of a link was first proven from a braid representative of the link. In the present article, we show the existence of such surfaces from an induced graph of the link.…
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. In present article, we study links of the flat plumbing basket numbers $4$ or less using a special presentation of the flat…
A flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3-sphere. In this paper, we examine close connections between flat plumbing baskets and the…
A flat plumbing basket is a surface consisting a disk and finitely many bands which are contained in distinct pages of the trivial open book decomposition of $\mathbf{S}^{3}$. In this paper, we construct a Legendrian link from a flat…
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by…
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…
Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of…
We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions,…
We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical…
While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a…
We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram $A_{n}$, are isotopic to the torus link $T(2,n+1)$. In addition, we provide examples of links, constructed by…
In an earlier note [arXiv:2301.00295] it was shown that there is an upper bound to the number of disjoint Hopf links (and certain related links) that can be embedded in the unit cube where there is a fixed separation required between the…
Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed…
We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…
We introduce a "deformation" of plumbing. We also define a structure of data used in a calculation by computer aid of the crosscap numbers of alternating knots.
For a positive Hopf plumbed arborescent Seifert surface $S$, we study the set of Hopf bands $H\subset S$, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely…
We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we…
We prove that the Laurent polynomial in $\mathbb{Z}[q^{\pm 1}]$ that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under plumbing of surfaces. We deduce that the Links-Gould…
We define the Witt coindex of a link with non-trivial Alexander polynomial, as a concordance invariant from the Seifert form. We show that it provides an upper bound for the (locally flat) slice Euler characteristic of the link, extending…
In recent work, the author and others have studied compositional algebras of Petri nets. Here we consider mathematical aspects of the pure linking algebras that underly them. We characterise composition of nets without places as the…