arXiv research

Special shadow-complexity equals k+1 for k copies of S1×S3.

New invariant measures complexity of 2-knots in 4D space.

Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.

Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly em…

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-…

In this paper we find infinitely many Mazur type manifolds and corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also find such manifolds among 4-manifolds constructed from Bing's house. Our manifolds …

In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is…

We give a self-contained introduction to the theory of Turaev's shadows as a tool to study 3 and 4-manifolds. The goal of the present paper twofold: on one side it is intended to be a shortcut to a basic use of the theory of shadows, on the other side it gives a sketchy overview of some of the recent results on shadows…

Geometric proof confirms link volume conjecture.

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies th…

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arX…

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the re…

New divide with gleams method simplifies symmetric link representation.

The paper introduces a new complexity measure for 4-manifolds and connects it to the trisection genus.

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

A first order Vassiliev invariant of an oriented knot in an $S^1$-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds relat…

Researchers create links in 3-sphere satisfying volume conjecture.

This paper reveals hidden hyperbolic structures in divide links.

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitab…

A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev's shadow world invariant of links in a circle bundle over a su…

This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for these topics. The thesis uses skein theory and focues on the connected sum of cop…

We extend the definition of the colored Jones polynomials to framed links and trivalent graphs in S^3 # k S^2 X S^1 using a state-sum formulation based on Turaev's shadows. Then, we prove that the natural extension of the Volume Conjecture is true for an infinite family of hyperbolic links.

Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…

Eisermann has shown that the Jones polynomial of a $n$-component ribbon link $L\subset S^3$ is divided by the Jones polynomial of the trivial $n$-component link. We improve this theorem by extending its range of application from links in $S^3$ to colored knotted trivalent graphs in $\#_g(S^2\times S^1)$, the connected …

We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev's shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise here all the closed oriented 4-manifolds that have complexity at most one. T…

This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We sh…

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…

New invariants for singular knots and links defined using shadow structures.

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitel…

A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister …

Paper studies knotoid chirality using shadow quandle colorings and invariants.

The average shadowing property is considered for set-valued dynamical systems, generated by parameterized IFS, which are uniformly contracting, or conjugacy, or products of such ones. We also prove that if a continuous surjective IFS F on a compact metric space X has the aver- age shadowing property, then every point x…

Paper optimizes trading strategies by creating shadow prices for markets with transaction costs.

We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…

Computes Kauffman bracket polynomial for specific 2-tangle shadows.

The paper presents fundamental groups of complements of shadows in 4-balls.

An asset network systemic risk (ANWSER) model is presented to investigate the impact of how shadow banks are intermingled in a financial system on the severity of financial contagion. Particularly, the focus of this study is the impact of the following three representative topologies of an interbank loan network betwee…

Study on inflection points of plane curve shadows with fixed embedded shapes.

We introduce an associative algebra Z[X,S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of Z[X,S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by th…

New forms generalize Whitney forms with rational coefficients for numerical analysis.

Study of quandle coloring quivers with dihedral quandles.

Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.

We introduce a topological combinatorial game called the Link Smoothing Game. The game is played on the shadow of a link diagram and legal moves consist of smoothing precrossings. One player's goal is to keep the diagram connected while the other player's goal is to disconnect the shadow. We make significant progress t…

The paper studies dynamical properties in semigroups modulo ideals.

For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices may fail to exist even i…

We show that for each even integer $m\ge 2$, every reduced shadow with sufficiently many crossings is a shadow of a torus knot T(2,m+1), or of a twist knot $T_m$, or of a connected sum of $m$ trefoil knots.

A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.