arXiv subjects

William W. Menasco

Publications and source records attributed to William W. Menasco.

At least 19 recordsLinked to original sources

arXiv2024

A Seifert algorithm for integral homology spheres

From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary technique comes from taking a regular projection of any knot and employing Seifert's constructive algorithm. In this note we give a natural generalization of Seifert's algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well known example of such a planar presentation, see the Poincaré homology sphere planar presentation in {\em Knots and Links} by D. Rolfsen \cite{Rolfsen}.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a ``Seifert circle" and associated half-twisted bands.

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arXiv2024

Studying links via plats: split and composite links

Our main results concern changing an arbitrary plat presentation of a split or composite link to one which is obviously recognizable as being split or composite. Pocket moves, first described in \cite{unlinkviaplats}, are utilized -- a pocket move alters a plat presentation without changing its link type, its bridge index or the double coset. A plat presentation of a split link is split if the planar projection of the plat presentation is not connected. We prove that pocket moves are the only obstruction to representing split links by split plat presentations. Since any pocket move corresponds to a sequence of double coset moves, we have the corollary that the double coset of every plat presentation of a split link has a split plat presentation. We obtain an analogous result for composite links by utilizing flip moves, which were also first described in the second author's work, arXiv:2308.00732 [math.GT].

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arXiv2023

Super efficiency of efficient geodesics in the complex of curves

We show that efficient geodesics have the strong property of "super efficiency". For any two vertices, $v , w \in \mathcal{C}(S_g)$, in the complex of curves of a closed oriented surface of genus $g \geq 2 $, and any efficient geodesic, $v = v_1 , \cdots , v_{\text d}=w$, it was previously established by Birman, Margalit and the second author (see arXiv:1408.4133) that there is an explicitly computable list of at most ${\text d}^{(6g-6)}$ candidates for the $v_1$ vertex. In this note we establish a bound for this computable list that is independent of ${\text d}$-distance and only dependent on genus -- the super efficiency property. The proof relies on a new intersection growth inequality between intersection number of curves and their distance in the complex of curves, together with a thorough analysis of the dot graph associated with the intersection sequence.

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arXiv2023

A construction of minimal coherent filling pairs

Let $S_g$ denote the genus $g$ closed orientable surface. A \emph{coherent filling pair} of simple closed curves, $(α,β)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A \emph{minimally intersecting} filling pair, $(α,β)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this paper, we give a simple geometric procedure for constructing minimal intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or {\em origamis}, and we discuss the origami obtained from the construction.

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arXiv2022

Surface embeddings in $\mathbb{R}^2\times\mathbb{R}$

This is an investigation into a classification of embeddings of a surface in Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $π:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ be the natural projection map onto the Euclidean plane. Let $ \varepsilon : S_g \hookrightarrow \mathbb{R}^2 \times \mathbb{R}$ be a smooth embedding of a closed oriented genus $g$ surface such that the set of critical points for the map $π\circ \varepsilon$ is a smooth (possibly multi-component) $1$-manifold, $\mathscr{C} \subset S_g$. We say $\mathscr{C}$ is the crease set of $\varepsilon$ and two embeddings are in the same isotopy class if there exists an isotopy between them that has $\mathscr{C}$ being an invariant set. The case where $π\circ \varepsilon|_\mathscr{C}$ restricts to an immersion is readily accessible, since the turning number function of a smooth curve in $\mathbb{R}^2$ supplies us with a natural map of components of $\mathscr{C}$ into $\mathbb{Z}$. The Gauss-Bonnet Theorem beautifully governs the behavior of $π\circ \varepsilon (\mathscr{C})$, as it implies $χ(S_g) = 2 \sum_{γ\in \mathscr{C}} t(π\circ \varepsilon (γ))$, where $t$ is the turning number function. Focusing on when $S_g \cong S^2$, we give a necessary and sufficient condition for when a disjoint collection of curves $\mathscr{C} \subset S^2$ can be realized as the crease set of an embedding $\varepsilon: S^2 \hookrightarrow \mathbb{R}^2 \times \mathbb{R}$. From there, we give the classification of all isotopy classes of embeddings when $\mathscr{C} \subset S^2$ and $|\mathscr{C}|=3$ -- a simple yet enlightening case. As a teaser of future work, we give an application to knot projections and discuss directions for further investigation.

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arXiv2021

Origami edge-paths in the curve graph

An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are "origami pairs of curves" -- filling pairs of simple closed curves, $ (α,β)$, in $S_g$ such that their minimal intersection is equal to their algebraic intersection -- they are "coherent". An origami pair of curves is naturally associated with an origami on $S_g$. Our main result establishes that for any origami pair of curves there exists an "origami edge-path", a sequence of curves, $α=α_0, α_1, α_2, \cdots, α_n = β$, such that: $α_i$ intersects $α_{i+1}$ at exactly once; any pair $(α_i, α_j)$ is coherent; and thus, any filling pair, $(α_i, α_j)$, is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.

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arXiv2019

Alternating Knots

This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.

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arXiv2015

MICC: A tool for computing short distances in the curve complex

The complex of curves $\mathcal{C}(S_g)$ of a closed orientable surface of genus $g \geq 2$ is the simplicial complex having its vertices, $\mathcal{C}^0(S_g)$, are isotopy classes of essential curves in $S_g$. Two vertices co-bound an edge of the $1$-skeleton, $\mathcal{C}^1(S_g)$, if there are disjoint representatives in $S_g$. A metric is obtained on $\mathcal{C}^0(S_g)$ by assigning unit length to each edge of $\mathcal{C}^1(S_g)$. Thus, the distance between two vertices, $d(v,w)$, corresponds to the length of a geodesic---a shortest edge-path between $v$ and $w$ in $\mathcal{C}^1 (S_g)$. Recently, Birman, Margalit and the second author introduced the concept of {\em initially efficient geodesics} in $\mathcal{C}^1(S_g)$ and used them to give a new algorithm for computing the distance between vertices. In this note we introduce the software package MICC ({\em Metric in the Curve Complex}), a partial implementation of the initially efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, we give examples of distance four vertex pairs, for $g=2$ and 3. Previously, there was only one known example, in genus $2$, due to John Hempel.

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arXiv2014

The curve complex has dead ends

It is proved that the curve graph $C^1(Σ)$ of a surface $Σ_{g,n}$ has a local pathology that had not been identified as such: there are vertices $α,β$ in $C^1(Σ)$ such that $β$ is a dead end of every geodesic joining $α$ to $β$. It also has double dead-ends. Every dead end has depth 1.

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arXiv2013

Embedded annuli and Jones' conjecture

We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro.

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arXiv2012

Recognizing destabilization, exchange moves and flypes

The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed---destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"---determining when a given closed $n$-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for these three key isotopies of the MTWS calculus. The algorithm is "directed" by a complexity measure that can be {\em monotonically simplified} by the application of "elementary moves".

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arXiv2012

Monotonic Simplification and Recognizing Exchange Reducibility

The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed--destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"--determining when a given closed n-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for three isotopies of the MTWS calculus--destabilization, exchange moves and braid preserving flypes. The algorithm is directed by a complexity measure that can be "monotonic simplified" by that application of "elementary moves".

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arXiv2009

Climbing a Legendrian mountain range without Stabilization

We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal tb value without having to use Legendrian stabilization. In doing so we obtain new ways to visualize convex tori and Legendrian divides and rulings, using tilings and braided rectangular diagrams.

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arXiv2008

A note on closed 3-braids

Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. We also use it as an opportunity to review what is known and suggest some open questions.

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arXiv2008

A note on transversal knots that are closed 3-braids

The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids.

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arXiv2007

On rectangular diagrams, Legendrian knots and transverse knots

A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show Alexander and Markov Theorems for Legendrian links in 3-space.

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arXiv2006

Erratum: On iterated torus knots and transversal knots

In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pin points the error. In "An addendum on iterated torus knots" a more subtle result is proven resulting in giving a geometric realization of the Honda-Etnyre transverse (2,3)-cable of the (2,3)-torus knot example--Appendix joint with H. Matsuda. (See math.SG/0306330 and math.GT/0610566.)

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arXiv2006

An addendum on iterated torus knots

In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is false, and the Erratum math.GT/0610565 identifies the gap. The purpose of this paper is to explore the situation more deeply, in order to pinpoint exactly which cable knots are {\it not} transversally simple. The class is subtle and interesting. We will recover the strength of the main theorem in math.GT/0002110, in the sense that we will be able to prove a strong theorem about cable knots, but the theorem itself is more subtle than Theorem 1.2 of math.GT/0002110. In particular, we give a geometric realization of the Honda-Etnyre transverse (2,3)-cable of the (2,3)-torus knot example (Appendix joint with H. Matsuda). (See math.SG/0306330)

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