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Alexander's conjecture states that for every two finite triangulations of the same topological space, if they have a common subdivision, then they have a common stellar subdivision. We generalize the recent result of Adiprasito and Pak, who resolved Alexander's conjecture for finite simplicial complexes, to infinite simplicial complexes.

The crucial feature of resurgence theory is the ambiguity of non-perturbative behavior, reflected either in the different choices of integration contours or in the existence of several solutions to Ward identities. This is well illustrated by considering exactly solvable models, of which the prominent example is Chern-Simons theory. Its important chapter, which should have a direct generalization to arbitrary Yang-Mills, is the consideration of Wilson averages in the double-scaling limit of large representation and small coupling. For historical reasons, we call it a Kashaev limit. It possesses a natural interpretation in terms of quasiclassical/WKB approximation, which is, however, somewhat peculiar and thus sheds new light on the old story. The crucial point is the appearance of Alexander polynomials $Δ$ in two seemingly opposite roles: the classical $A$-polynomials have common roots with $Δ$, while Jones polynomials tend to $Δ^{-1}$ in the perturbative expansion. The consistency is provided by the peculiar form of the quantum $A$-polynomial, and the resolution of the puzzle is the co-existence of two different branches (phases) in the quasiclassical limit -- with non-trivial and

In this paper, we prove that the property of being a grape (in any of its variants) is invariant under Alexander duality. The explicitly determined (simple-)homotopy type of a grape can be transferred to its Alexander dual via Combinatorial Alexander Duality in (co)homology. We also provide several applications.

In our previous work, we introduced the notion of a twisted Alexander vanishing (TAV) group, defined as a finite group for which the corresponding twisted Alexander polynomial of a knot vanishes. In this paper, we discuss the orders of TAV groups and construct knots whose twisted Alexander polynomials vanish. Moreover, we show that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.

Alexander polynomial was born one century ago, but explicit formulas have been found for only a few families of links. In this paper, we present an efficient method of computing Alexander polynomial for arborescent links. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain polynomials associated to rational tangles which can be computed recursively. Specifically, we deduce explicit closed formulas for all pretzel links.

In 1928, Alexander defined a sequence of knot polynomials, D_i(K). The first, D_1(K), is the classical Alexander polynomial. These are easily defined in terms of the homology of the infinite cyclic cover of the knot. In theory they can be computed by putting an associated Alexander matrix in Smith normal form. However, standard algorithms for computing the Smith form become impractically slow, even for some 16 crossing knots. Here, methods are developed that can effectively compute the Alexander polynomials of knots with up to 100 crossings.

The aim of this paper is to provide a new characterization of isomorphism classes of generalized Alexander quandles in terms of the underlying groups and their automorphisms. This extends the previous result [4, Theorem 1.4]. Additionally, we compute the number of generalized Alexander quandles up to quandle isomorphism arising from groups up to order 127 and their group automorphisms.

The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any alternating link are trapezoidal -- was settled for special alternating links by the present authors (2023); Kálmán, the second author, and Postnikov gave an alternative proof (2025). The present paper is a study of the special combinatorial and discrete geometric properties that Alexander polynomials of special alternating links possess along with a generalization to all Eulerian graphs, introduced by Murasugi and Stoimenow (2003). We prove that the Murasugi and Stoimenow generalized Alexander polynomials can be expressed in terms of volumes of root polytopes of unimodular matrices. The latter generalizes a result regarding the Alexander polynomials of special alternating links that follows by putting together the work of Li and Postnikov (2013) and Kálmán, the second author, and Postnikov (2025). Furthermore, we conjecture a generalization of Fox's conjecture to the generalized Alexander polynomials of Murasugi and Stoimenow and bijectively rel

The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for generalizations of knotoids. We prove a conjecture on the mock Alexander polynomial for knotoids, which generalizes to uni-linkoids. Afterwards we give constructions for canonical invariants of linkoids derived from the mock Alexander polynomial, using the formalism of generalized knotoids due to Adams et al.

Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix, which is a quandle version of a twisted Alexander matrix and defined an invariant of finitely presented quandles. In this paper, we study $f$-twisted Alexander matrices of certain quandles with the Alexander pair obtained from a quandle 2-cocycle. We show that the 0-th elementary ideal of $f$-twisted Alexander matrix of the knot quandle of a surface knot with the Alexander pair obtained from a quandle 2-cocycle can be described with the Carter-Saito-Satoh's invariant. We also discuss a relationship between $f$-twisted Alexander matrices of connected quandles with the Alexander pair obtained from a quandle 2-cocycle and quandle homology groups.

Based on a vanishing theorem for non-fibered knots due to Friedl and Vidussi, we define the twisted Alexander vanishing order of a knot to be the order of the smallest finite group such that the corresponding twisted Alexander polynomial is zero. In this paper, we show its basic properties, and provide several explicit values for knots with $10$ or fewer crossings. Moreover, we characterize a finite group admitting the zero-twisted Alexander polynomial.

We give an explicit formula of the Alexander polynomial of the link obtained by adding an arbitrary number of full twists to positively oriented parallel n-strands in terms of the Alexander polynomials of the links obtained by adding 0,1,...,n-1 full twists. From this, we see that the Alexander polynomials stabilize after adding sufficiently many full twists. The main tool used in the computation is expressing the Alexander polynomial using the vector space representation of $U_{q}(gl(1|1))$.

In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $ρ_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write one of the standard matrices $A$ whose determinant is its Alexander polynomial, yet instead of computing the determinant we consider a certain quadratic expression in the entries of $A^{-1}$. The proximity of our formulas to the Alexander polynomial suggest that they should have a topological explanation. This we don't have yet.

In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes this, while the Alexander-Conway polynomial, independently, is insufficient. As the Alexander-Conway polynomial on its own is not likely to detect causality an additional piece of information must be supplemented to the link polynomial. This paper aims to examine the ability of Alexander quandles, to distinguish the connected sum of two Hopf links and Allen-Swenberg Links. As the number of homomorphisms given by the Alexander quandles for the connected sum of Hopf links and the number of homomorphisms for the Allen-Swenberg Link are the same, we can conclude that the links are not distinguishable from one another using Alexander quandles and hence these quandles cannot capture causality when added to the Alexander-Conway polynomial.

We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one which proves the same result. However, we show that Lickorish's obstruction does not subsume the obstruction coming from the condition on determinants.

The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we obtain several characterizations in some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to $15$.

Scientists at Nanyang Technological University in Singapore have discovered a surprisingly simple way to create exotic light structures called optical skyrmions using a 200-year-old optical effect known as the Poisson spot。 Instead of relying on expensive, highly engineered materials, they simply shine a laser at a tiny circular disc, producing sta

The asteroid that wiped out the dinosaurs was likely an exceptionally rare CO chondrite from a distant region of the solar system。 Its unusual chemistry suggests that planet-cooling dust and debris, rather than sulfur inside the asteroid, may have delivered the deadliest blow

A catastrophic asteroid breakup may have triggered a huge wave of impacts across the inner solar system about 800 million years ago。 The debris was launched from near a gravitational gateway controlled by Jupiter, sending fragments toward Earth, the Moon, and Mars。 The bombardment may explain ancient lunar craters and could have contributed to majo