arXiv subjects

Ryan Stees

Publications and source records attributed to Ryan Stees.

4 recordsLinked to original sources

arXiv2025

Triple linking and rational homology cobordism

If a rational homology 3-sphere $M$ bounds a rational homology 4-ball $W$, then the kernel of the inclusion-induced homomorphism $H_1(M;\mathbb{Z})\to H_1(W;\mathbb{Z})$ is a Lagrangian for the $\mathbb{Q}/\mathbb{Z}$-valued torsion linking form $\lambda_2$ on $H_1(M;\mathbb{Z})$. In this short paper, we prove that the Freedman-Krushkal triple torsion linking form $\lambda_3$ (arXiv:2506.11941v3) vanishes on this Lagrangian under the assumption that $H_2(W;\mathbb{Z})=0$. We then pose several questions about topological rational homology cobordism.

math.GT↗

arXiv2025

Almost-concordance of knots in aspherical 3-manifolds

In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this conjecture for any nontrivial class $x$ in any aspherical $M$ and provide computations that prove the conjecture in a large family of open cases. Our technique employs an extension of Milnor's link invariants to knots and links in non-simply-connected 3-manifolds (arXiv:2310.10918v2). We exhibit a large family of examples where, in a precise sense, we maximize the number of almost-concordance classes distinguished by these invariants.

math.GT↗

arXiv2025

Classical invariants of spiral knots

Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined and first studied by Brothers et al., are a braid-theoretic generalization of torus knots, but comparatively not much is known about this broader family of knots. We give a general recursive formula for the Alexander polynomials of spiral knots, and from this we derive several properties of spiral knots, including a simple genus formula. Additionally, we investigate the consequences these results have on classification questions.

math.GT↗

arXiv2023

Milnor's invariants for knots and links in closed orientable 3-manifolds

In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^3$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants, now known as Milnor's $\overline{\mu}$-invariants, were later shown to be topological link concordance invariants and have since inspired decades of consequential research. Milnor's invariants have many interpretations, and there have been numerous attempts to extend them to other settings. In this paper, we extend Milnor's invariants to topological concordance invariants of knots and links in general closed orientable 3-manifolds. These invariants unify and generalize all previous versions of Milnor's invariants in dimension 3, including Milnor's original invariants for links in $S^3$.

math.GT↗