几何拓扑

There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's…

We give the first known topological model for the HOMFLY-PT polynomial constructed directly from link diagrams. More precisely, we prove that this invariant is given by graded intersections between explicit Lagrangian submanifolds in a…

We construct geometrically a universal Jones invariant as a limit of invariants given by graded intersections in configuration spaces. For any fixed level $\mathscr N$, we define a new knot invariant, called ``$\mathscr N^{th}$ Unified…

Recently a set of $q$-series invariants, labelled by $\operatorname{Spin}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the $\widehat{Z}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading…

We modify the proof of the disc embedding theorem for $4$-manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the…

I prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^4$ become smoothly isotopic rel. boundary after pushing them into $D^5$.

We extend the skein lasagna theory of Morrison-Walker-Wedrich to 4-manifolds with corners and formulate gluing formulas for 4-manifolds with boundary and, more generally, with corners. As an application, we develop a categorical framework…

Given a knot $K$ and a generic slope $r$, we study the Kauffman bracket skein module (KBSM) $S(E_K (r) , \mathbb{Q} (A))$ of the Dehn filling $E_K (r)$ of slope $r$ along $K$, assuming that the KBSM $S(E_K , \mathbb{Q} [A^{\pm 1}])$ of the…

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to…

We develop the study of the twelve intersection polynomials of long virtual knots, previously introduced in our preceding paper. We define two geometric invariants, the $1$- and $2$-supporting genera, using two distinct surface…

We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed…

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that…

In a recent breakthrough, Ren and Willis gave the first analysis-free proof of the existence of exotic compact, orientable 4-manifolds; their main tool is the Khovanov skein lasagna module defined by Morrison, Walker, and Wedrich. In this…

We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove…

Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of…

A correspondence, by way of Heegaard splittings, between closed oriented 3-manifolds and pairs of surjections from a surface group to a free group has been studied by Stallings, Jaco, and Hempel. This correspondence, by way of trisections,…

We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichm\"uller space $\mathcal{T}(S_g)$ that remain geodesics after adding arbitrary…

In this paper, we use the skein exact sequence and other techniques to compute the second-to-top term of HFK of closed 3-braids. We do it case-by-case according to Xu's classification. We also verify the rank inequality conjectured by…

We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension $2$; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's…

We provide a partial answer to a question of Ekholm, Honda, and K\'alm\'an about the relationship between Khovanov homology and decomposable Lagrangian cobordisms. We also utilize previously defined filtered invariants to give obstructions…