arXiv subjects

Alexander Zupan

Publications and source records attributed to Alexander Zupan.

At least 19 recordsLinked to original sources

arXiv2026

Symmetric ribbon numbers of low-complexity knots

Every knot $K \subset S^3$ that admits a symmetric union presentation bounds an immersed ribbon disk in $S^3$, while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number $r_s(K)$ of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of knots with at most 12 crossings. Along the way, we exhibit novel lower bounds for $r_s(K)$ arising from knot determinants, Alexander polynomials, Jones polynomials, and Kauffman polynomials.

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arXiv2026

A homological generalized Property R conjecture is false

The generalized Property R conjecture (GPRC) predicts that if framed surgery on an $n$-component link $L$ in $S^3$ produces $\#^{n} (S^1\times S^2)$, then $L$ is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an $n$-component link in $S^3$ surgers to the connected sum of $n$ three-manifolds with the homology of $S^1 \times S^2$, then this link should be handleslide equivalent to an $n$-component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in $S^3$ that surger to a connected sum of homology $S^1\times S^2$'s but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.

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arXiv2025

Colorings of symmetric unions and partial knots

Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If both $J$ and $J'$ are partial knots for different symmetric union presentations of the same ribbon knot $K$, the knots $J$ and $J'$ are said to be symmetrically related. Lamm proved that if $J$ and $J'$ are symmetrically related, then $\det J = \det J'$, asking whether the converse is true. In this article, we give a negative answer to Lamm's question, constructing for any natural number $m$ a family of $2^m$ knots with the same determinant but such that no two knots in the family are symmetrically related. This result is a corollary to our main theorem, that if $J$ is the partial knot in a symmetric union presentation for $K$, then $\text{col}_p(J) \leq \text{col}_p(K) \leq \frac{(\text{col}_p(J))^2}{2}$, where $\text{col}_p(\cdot )$ denotes the number of $p$-colorings of a knot.

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arXiv2025

Manifolds with weakly reducible genus-three trisections are standard

Heegaard splittings stratify 3-manifolds by complexity; only $S^3$ admits a genus-zero splitting, and only $S^3$, $S^1 \times S^2$, and lens spaces $L(p,q)$ admit genus-one splittings. In dimension four, the second author and Jeffrey Meier proved that only a handful of simply-connected 4-manifolds have trisection genus two or less, while Meier conjectured that if $X$ admits a genus-three trisection, then $X$ is diffeomorphic to a spun lens space $S_p$ or its sibling $S_p'$, $S^4$, or a connected sum of copies of $\pm \mathbb{CP}^2$, $S^1 \times S^3$, and $S^2 \times S^2$. We prove Meier's conjecture in the case that $X$ admits a weakly reducible genus-three trisection, where weak reducibility is a new idea adapted from Heegaard theory and is defined in terms of disjoint curves bounding compressing disks in various handlebodies. The tools and techniques used to prove the main theorem borrow heavily from 3-manifold topology. Of independent interest, we give a trisection-diagrammatic description of 4-manifolds obtained by surgery on loops and spheres in other 4-manifolds.

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arXiv2024

Ribbon numbers of 12-crossing knots

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera.

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arXiv2024

Bounding the ribbon numbers of knots and links

The ribbon number $r(K)$ of a ribbon knot $K \subset S^3$ is the minimal number of ribbon intersections contained in any ribbon disk bounded by $K$. We find new lower bounds for $r(K)$ using $\det(K)$ and $\Delta_K(t)$, and we prove that the set $\mathfrak{R}_r~=~\{\Delta_K(t)~:~r(K)~\leq~r\}$ is finite and computable. We determine $\mathfrak{R}_2$ and $\mathfrak{R}_3$, applying our results to compute the ribbon numbers for all ribbon knots with 11 or fewer crossings, with three exceptions. Finally, we find lower bounds for ribbon numbers of links derived from their Jones polynomials.

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arXiv2023

Knots bounding non-isotopic ribbon disks

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 answer to a stronger version of an old question of Hitt and Sumners. The examples arise from our main result: a classification of fibered, homotopy-ribbon disks for each generalized square knot $T_{p,q}\# \overline{T}_{p,q}$ up to isotopy. Precisely, we show that each generalized square knot bounds infinitely many pairwise non-isotopic fibered, homotopy-ribbon disks, all of whose exteriors are diffeomorphic. When $q=2$, we prove further that infinitely many of these disks are also ribbon; whether the disks are always ribbon is an open problem.

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arXiv2023

A family of Andrews-Curtis trivializations via 4-manifold trisections

An R-link is an $n$-component link $L$ in $S^3$ such that Dehn surgery on $L$ yields $\#^n(S^1 \times S^2)$. Every R-link $L$ gives rise to a geometrically simply-connected homotopy 4-sphere $X_L$, which in turn can be used to produce a balanced presentation of the trivial group. Adapting work of Gompf, Scharlemann, and Thompson, Meier and Zupan produced a family of R-links $L(p,q;c/d)$, where the pairs $(p,q)$ and $(c,d)$ are relatively prime and $c$ is even. For this family, $L(3,2;2n/(2n+1))$ induces the infamous trivial group presentation $\langle x,y \, | \, xyx=yxy, x^{n+1}=y^n \rangle$, a popular collection of potential counterexamples to the Andrews-Curtis conjecture for $n \geq 3$. In this paper, we use 4-manifold trisections to show that the group presentations corresponding to a different family, $L(3,2;4/d)$, are Andrews-Curtis trivial for all $d$.

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arXiv2023

Tri-plane diagrams for simple surfaces in $S^4$

Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa's table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.

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arXiv2022

Bridge trisections and Seifert solids

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 construction of a Seifert solid that is described by a Heegaard diagram. The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles. We conclude with two classification results. The first shows that surfaces admitting doubly-standard shadow diagrams are unknotted. The second says that a $b$-bridge trisection in which some sector contains at least $b-1$ patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors.

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arXiv2022

Bridge trisections and classical knotted surface theory

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 computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity.

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arXiv2022

Hexagonal lattice diagrams for complex curves in $\mathbb{CP}^2$

We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in $\mathbb{CP}^2$ are closely related: We prove that a positive genus surface $\mathcal{K}$ in $\mathbb{CP}^2$ that minimizes genus in its homology class is isotopic to a complex curve $\mathcal{C}_d$ if and only if $\mathcal{K}$ admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meet only at bridge points and tile the central surface of the standard trisection of $\mathbb{CP}^2$ by hexagons. There are eight families of these diagrams, two of which represent surfaces in efficient bridge position. Combined with a result of Lambert-Cole relating symplectic surfaces and bridge trisections, this allows us to provide a purely combinatorial reformulation of the symplectic isotopy problem in $\mathbb{CP}^2$. Finally, we show that that the varieties $\mathcal{V}_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1z_2^{d-1} + z_2z_3^{d-1} + z_3z_1^{d-1} = 0\}$ and $\mathcal{V}'_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1^{d-1}z_2 + z_2^{d-1}z_3 + z_3^{d-1}z_1 = 0\}$ are in efficient bridge position with respect to the standard Stein trisection of $\mathbb{CP}^2$, and their shadow diagrams agree with the two families of efficient hexagonal lattice diagrams. As a corollary, we prove that two infinite families of complex hypersurfaces in $\mathbb{CP}^3$ admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.

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arXiv2020

Cubic graphs induced by bridge trisections

Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $\Gamma$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $\Gamma$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves.

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arXiv2020

Equivalent characterizations of handle-ribbon knots

The stable Kauffman conjecture posits that a knot in $S^3$ is slice if and only if it admits a slice derivative. We prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball $B$ if and only if it admits an R-link derivative; i.e. an $n$-component derivative $L$ with the property that zero-framed surgery on $L$ yields $\#^n(S^1\times S^2)$. We also show that $K$ bounds a handle-ribbon disk $D \subset B$ if and only if the 3-manifold obtained by zero-surgery on $K$ admits a singular fibration that extends over handlebodies in $B \setminus D$, generalizing a classical theorem of Casson and Gordon to the non-fibered case for handle-ribbon knots.

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arXiv2019

The Powell Conjecture and reducing sphere complexes

The Powell Conjecture offers a finite generating set for the genus $g$ Goeritz group, the group of automorphisms of $S^3$ that preserve a genus $g$ Heegaard surface $Σ_g$, generalizing a classical result of Goeritz in the case $g=2$. We study the relationship between the Powell Conjecture and the reducing sphere complex $\mathcal{R}(Σ_g)$, the subcomplex of the curve complex $\mathcal{C}(Σ_g)$ spanned by the reducing curves for the Heegaard splitting. We prove that the Powell Conjecture is true if and only if $\mathcal{R}(Σ_g)$ is connected. Additionally, we show that reducing curves that meet in at most six points are connected by a path in $\mathcal{R}(Σ_g)$; however, we also demonstrate that even among reducing curves meeting in four points, the distance in $\mathcal{R}(Σ_g)$ between such curves can be arbitrarily large. We conclude with a discussion of the geometry of $\mathcal{R}(Σ_g)$.

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arXiv2019

Generalized square knots and homotopy 4-spheres

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diffeomorphic to the standard 4-sphere in the special case that one of the 2-handles is attached along a knot of the form $Q_{p,q} = T_{p,q}\#T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf. Along the way, we use thin position techniques from Heegaard theory to give a characterization of 2R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counterexamples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.

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arXiv2017

Bridge trisections of knotted surfaces in 4--manifolds

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, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside $\mathbb{CP}^2$. Using these examples, we prove that there exist exotic 4--manifolds with $(g,0)$--trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.

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arXiv2017

Characterizing Dehn surgeries on links via trisections

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 potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard. Finally, we give an analog of the Casson- Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection.

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