arXiv research

Polynomially inscribe 6 concyclic points into any smooth curve.

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynom…

New invariants defined for knots and links using quandle representations.

Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.

We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…

Study bounds the volume of moduli space for convex RP² structures.

Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conw…

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobas-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the J…

abess efficiently solves various machine learning problems quickly.

A simpler 3D Möbius strip design without twists.

We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the description of a non-empty open subset of C^n on which the group acts properly discon…

Let $\R$ be a real closed field, ${\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and ${\mathcal P} \subset \R[X_1,...,X_k]$ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$. Let $S \subset \R^{\ell+k}$ be a semi-alg…

New proof of Alexander polynomial constraints for lens space surgeries.

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) a…

Classifies special homogeneous curves with polynomial equations.

Algorithm finds safe zones in policy Markov Decision Processes to limit trajectory escape.

This paper investigates symmetric ribbon numbers of low-complexity knots.

Proposes a group-splicing algorithm for efficient BSGS in high-dimensional settings.

In-degree quiver polynomials for surface-links computed.

Suppose the knot group G(K) of a knot K has a non-abelian representation ρon A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to ρis of the form: Δ_K(t)/(1-t) φ(t^3), where Δ_K (t) is the Alexander polynomial of K and φ(t^3) is an integer polynomial in t^3. We prove the conjectur…

Researchers compute and predict knot volumes using colored Jones polynomials.

Efficiently samples arbitrary compact bodies with polynomial complexity.

We give the first polynomial-time algorithm for performing linear or polynomial regression resilient to adversarial corruptions in both examples and labels. Given a sufficiently large (polynomial-size) training set drawn i.i.d. from distribution D and subsequently corrupted on some fraction of points, our algorithm out…

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

The paper characterizes complex projective spaces using Ehrhart polynomials.

In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positi…

We consider a basic problem at the interface of two fundamental fields: submodular optimization and online learning. In the online unconstrained submodular maximization (online USM) problem, there is a universe $[n]=\{1,2,...,n\}$ and a sequence of $T$ nonnegative (not necessarily monotone) submodular functions arrive …

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus …

We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of $n$ objects is chosen, and a linear cost associated with the chosen subset is incurred. To measure the performance of our algorithms, we use the notion of regret which is the difference between the to…

Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.

Let $\R$ be a real closed field, ${\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and ${\mathcal P} \subset \R[X_1,...,X_k]$ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebr…

Paper determines 2-adjacent knots up to 12 crossings.

Optimal experiments tighten causal effect bounds efficiently.

The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

Designs efficient algorithms to maximize the expectation of Gaussian random variables.

New connection found between complex polynomials and surface homeomorphisms.

A fast algorithm selects best subsets in high-dimensional models.

Paper tackles fair low-rank approximation and column subset selection.

New algorithms explain Naive Bayes classifiers in polynomial time and delay.

We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \geq 2$, and study the map $Φ_d:\mathrm{MP}_d\to \widetildeΛ_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which maps each $f \in \mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show t…

The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the $AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has …

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,..., z_n]$ with no negative exponents and with no factor of the form $z_1^{α_1}... z_n^{α_n}$ where $α_i$ are non zero natural integer. If we assume in addicting that $f$ is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton p…

This work connects knot invariants to Chern-Simons theories via factorization homology.

Paper tackles dense subgraph discovery with noisy feedback.

The paper generalizes polynomial functions on Lie groups and their properties.

Polynomials derived from Heegaard diagrams for 3-manifolds.