55 citations · 242 across the 28 of their papers we have counts for
31 papers
Derivative links in contact topology
Joseph Breen, Alexander Zupan
We import the theory of -links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of…
A homological generalized Property R conjecture is false
Tye Lidman, Trevor Oliveira-Smith, Alexander Zupan
The generalized Property R conjecture (GPRC) predicts that if framed surgery on an -component link in produces , then is handleslide equiva…
Colorings of symmetric unions and partial knots
Ben Clingenpeel, Zongzheng Dai, Gabriel Diraviam +12
Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot by introducing additional crossings to…
Manifolds with weakly reducible genus-three trisections are standard
Román Aranda, Alexander Zupan
Heegaard splittings stratify 3-manifolds by complexity; only admits a genus-zero splitting, and only , , and lens spaces admit genus-one splitti…
Knots bounding non-isotopic ribbon disks
Jeffrey Meier, Alexander Zupan
We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise non-isotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive…
A family of Andrews-Curtis trivializations via 4-manifold trisections
Ethan Romary, Alexander Zupan
An R-link is an -component link in such that Dehn surgery on yields . Every R-link gives rise to a geometrically simply-connected homotop…
Tri-plane diagrams for simple surfaces in
Wolfgang Allred, Manuel Aragón, Zack Dooley +8
Meier and Zupan proved that an orientable surface in admits a tri-plane diagram with zero crossings if and only if is unknotted, so that the cross…
Bridge trisections and Seifert solids
Jason Joseph, Jeffrey Meier, Maggie Miller +1
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construct…
Hexagonal lattice diagrams for complex curves in
Alexander Zupan
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in are closely related: We prove that a positive genus surface $\…
Bridge trisections and classical knotted surface theory
Jason Joseph, Jeffrey Meier, Maggie Miller +1
We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be c…
Cubic graphs induced by bridge trisections
Jeffrey Meier, Abigail Thompson, Alexander Zupan
Every embedded surface in the 4-sphere admits a bridge trisection, a decomposition of into three simple pieces. In this case, the surface $\mathca…
Equivalent characterizations of handle-ribbon knots
Maggie Miller, Alexander Zupan
The stable Kauffman conjecture posits that a knot in is slice if and only if it admits a slice derivative. We prove a related statement: A knot is handle-ribbon (also called…
The Powell Conjecture and reducing sphere complexes
Alexander Zupan
The Powell Conjecture offers a finite generating set for the genus Goeritz group, the group of automorphisms of that preserve a genus Heegaard surface , generali…
Generalized square knots and homotopy 4-spheres
Jeffrey Meier, Alexander Zupan
The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing -component links with a Dehn surgery realizing . We cal…
Bridge trisections of knotted surfaces in 4--manifolds
Jeffrey Meier, Alexander Zupan
We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, afte…
Characterizing Dehn surgeries on links via trisections
Jeffrey Meier, Alexander Zupan
We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the po…
Natural properties of the trunk of a knot
Derek Davies, Alexander Zupan
The trunk of a knot in , defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots a…
Independence of volume and genus bridge numbers
Jessica S. Purcell, Alexander Zupan
A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge sur…
New examples of Brunnian theta graphs
Byoungwook Jang, Anna Kronaeur, Pratap Luitel +3
The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a n…
Bridge trisections of knotted surfaces in
Jeffrey Meier, Alexander Zupan
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the c…
Classification of trisections and the Generalized Property R Conjecture
Jeffrey Meier, Trent Schirmer, Alexander Zupan
We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisec…
Genus two trisections are standard
Jeffrey Meier, Alexander Zupan
We show that the only closed 4-manifolds admitting genus two trisections are and connected sums of , , and $\overline{\mathbb{CP}}^2…
Bridge spectra of twisted torus knots
R. Sean Bowman, Scott Taylor, Alex Zupan
We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result…
Knots with compressible thin levels
Ryan Blair, Alexander Zupan
We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.