几何拓扑

Gompf-Scharlemann-Thompson and Meier-Zupan constructed an infinite family of R-links that are potential counterexamples of the generalized property R conjecture. Their works also show that whether these links are stably handleslide trivial…

Suppose that $\gamma \subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,\pi)$ of measure at least $A/R^2$ such that if $\theta \in I$, then there…

In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we…

There are only three known strongly invertible hyperbolic L-space knots with braid index four and tunnel number two. They are t09284, t10496 and o9_34409 in the SnapPy census. In this paper, we give the first infinite family of strongly…

For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old…

The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when…

For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain…

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes…

We show that there is a topology on certain groups of loops in Euclidean space such that these groups are embedded in a Fr\'echet-Lie group which is the structural group of a principal bundle with connection whose holonomy coincides with…

For classical knots, Murasugi showed that the determinant modulo $8$ is classified by the Arf invariant. Boden and Karimi introduced a determinant for checkerboard colorable virtual knots. We prove that this determinant modulo $8$ is…

Some years ago, it was conjectured by the first author that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series…

Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$)…

We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement $M$. We show that it is rigid, in the sense that it does not depend on the choice of holonomy…

For the genus-$4$ Heegaard surface in the $3$-sphere, we present a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere for the surface. As a consequence, we reduce the connectivity problem in…

We study topological rigidity of real moment-angle manifolds associated to flag simplicial complexes. Using the cubical geometry arising from the Davis construction, we identify the universal cover with the Davis complex and deduce that it…

We analyse the action of the basic Dehn twists on the essential curves, $\gamma$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in…

Real Heegaard Floer homology is an invariant associated to a three-manifold equipped with an involution with nonempty fixed set of codimension two. We show that when the image of the fixed point set is nullhomologous in the quotient, the…

This article gives a self-contained proof of Mostow Rigidity, at least modulo undergrad real analysis. The proof should be accessible to grad students interested in geometry and topology. It has no new research, but I think that this is an…

This paper shows that the complex projective plane $\mathbb{P}^2$ can be realized as the underlying space for a closed hyperbolic $4$-orbifold. This is the first example of a closed hyperbolic $4$-orbifold whose underlying space is…

This paper introduces a quaternionic analogue of toric geometry by developing the theory of local $Q^n := Sp(1)^n$-actions on 4n-dimensional manifolds, modeled on the regular representation. We identify obstructions that measure the failure…