arXiv research

Alexander trick applied to homology spheres for manifold homeomorphisms.

Study of skateboard flips as continuous curves in $SO(3)$ group.

Nash's theorem proved with Günther's trick

Explains Conway's tangle trick and its mathematical origins.

Unified framework for gradient estimation in combinatorial spaces.

We prove all knots can be transformed into a trefoil using special diagrams.

Geometric trick simplifies link homotopy and concordance.

Outlier based Robust Principal Component Analysis (RPCA) requires centering of the non-outliers. We show a "bias trick" that automatically centers these non-outliers. Using this bias trick we obtain the first RPCA algorithm that is optimal with respect to centering.

The Gumbel-max trick and its extensions simplify sampling from categorical distributions in machine learning.

A new gradient estimator for categorical distributions reduces bias and variance.

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

Retail Product Image Classification is an important Computer Vision and Machine Learning problem for building real world systems like self-checkout stores and automated retail execution evaluation. In this work, we present various tricks to increase accuracy of Deep Learning models on different types of retail product …

Study of twisted Alexander matrices for certain quandles and their invariants.

Triple-point Whitney trick classifies ornaments of 3-manifolds.

Establishes connection between Alexander polynomials and triangulations.

Embolic volume of compact manifolds is defined in terms of Berger's embolic inequality. In this paper, we show a result of relating embolic volume to the first Betti number. The proof relies on Gromov's covering argument appeared in systolic geometry. Berger called this method covering trick. We exploit and present mor…

Expands Bredon's trick for applications in geometry and topology.

Researchers extend Alexander polynomial to knotoids and linkoids.

Study Alexander matrices for link quandles and their relation to knot invariants.

Bredon's trick helps extend local properties to global topological spaces.

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

Paper discusses groups where twisted Alexander polynomials vanish.

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterizati…

We introduce a family of pairwise stochastic gradient estimators for gradients of expectations, which are related to the log-derivative trick, but involve pairwise interactions between samples. The simplest example of our new estimator, dubbed the fundamental trick estimator, is shown to arise from either a) introducin…

The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each time solving for the most likely configuration. We derive an entire family of r…

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…

An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of …

Study calculates twisted Alexander polynomials for Montesinos knots.

The paper studies twisted Alexander polynomials for knot groups in various extensions.

Formula for Alexander polynomial of links with twists.

Study on a knot invariant's vanishing order.

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…

A simplified proof of the Alexander-Conway polynomial exists.

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexand…

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize an…

New Alexander polynomial for singular knots improves upon existing methods.

New Alexander invariants for knot groups computed using $K_1$-groups.

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $Δ_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $Δ_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexa…

New methods compute Alexander polynomials for complex knots.

Paper computes Alexander polynomials for arborescent links.

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal is a polynomial $H_K(s,t,q)$ in three variables which we call the virtual Alexa…

Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regre…

Alexander quandles can be embedded into groups.

Explicit formulas for pretzel knots' Alexander polynomials.

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots w…

Homology handles with trivial Alexander polynomial bound a 3D sphere.

Study of embedding calculus using infinite operads.

Study Alexander polynomials of links in 3-torus.