We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface.
Quandles are algebraic structures whose defining operation encodes the Reidemeister moves of link diagrams. They arise naturally in diverse areas such as knot theory, group theory, surface theory, quantum algebra, symmetric spaces, and Hopf algebras. A (co)homology theory for quandles was first introduced by Fenn, Rourke, and Sanderson [16] via a homotopytheoretic approach using classifying spaces. Carter et al. [8] subsequently applied quandle cohomology to knot theory through the construction of state-sum invariants. These theories were later generalized in [1,24]. More recently, Szymik [30] interpreted quandle cohomology within the framework of Quillen cohomology.
Motivated by bounded cohomology of groups, as introduced by Johnson [20] and later developed by Gromov [19], Kędra [23] recently introduced a bounded cohomology theory for quandles and related it to a natural metric based on their inner symmetries. He showed that a quandle is bounded with respect to this metric if and only if the comparison map from second bounded cohomology to ordinary second quandle cohomology is injective. Furthermore, he proved that fundamental quandles of non-trivial knots and free quandles are unbounded, implying the non-triviality of their second bounded cohomology. More recently, it was shown in [29] that the second bounded cohomology of the fundamental quandle of any non-split link with a non-solvable link group, as well as that of any split link, is infinite-dimensional. As a consequence, the second bounded cohomology of the fundamental quandle of a knot detects the unknot. This paper is motivated by a family of quandles arising from surfaces and their bounded cohomology. Let S g be a closed orientable surface of genus g ≥ 1 and D g the set of isotopy classes of simple closed curves on S g . The binary operation α * β = T β (α), where α, β ∈ D g and T β is the Dehn twist along β, equips D g with the structure of a quandle, known as the Dehn quandle of the surface S g . These quandles originally appeared in the work of Zablow [35,36]. Notably, Niebrzydowski and Przytycki [26] proved that the Dehn quandle of the torus is isomorphic to the fundamental quandle of the trefoil knot. However, this phenomenon is exceptional and does not persist in general [34]. In [10], two approaches were developed to obtain explicit presentations for such quandles.
In this paper, we introduce new families of quandles that enrich the classical Dehn quandles, arising from collections of closed curves on closed orientable surfaces and from integral weighted multicurves. Since the basis of the Goldman Lie algebra [18] is the set of all closed curves on these surfaces, our motivation for defining a quandle structure on these collections of curves comes from examining a connection between integral quandle rings and the Goldman Lie algebra. It is known that the quandle structure on the classical Dehn quandle extends to a structure on the set of measured foliations on the torus [9]. For surfaces of higher genus, since multicurves with positive real weights are dense in the space of measured foliations [31], we define a quandle structure on integral weighted multicurves with the hope that a quandle structure might also arise on the entire space of measured foliations. Remarkably, these quandles classify closed orientable surfaces of genus at least three. We prove their core algebraic properties, compute their second bounded quandle cohomology, and, along the way, we establish an analogue of the Gromov Mapping Theorem for abelian quandle extensions. Finally, we study idempotents in the integral quandle rings associated with Dehn quandles.
This paper is organized as follows. In Section 2, we review the basic concepts of quandles, bounded cohomology of groups, and bounded cohomology of quandles. In Section 3, we introduce two new families of quandles containing D g for each g ≥ 1. The first, denoted by C g , is defined on the set of isotopy classes of all unoriented closed curves on S g , while the second, denoted by W g , is defined on the set of all integral weighted multicurves on S g (see Propositions 3.1 and 3.3). For the subquandle W + g of W g consisting of positively weighted multicurves, we prove the existence of a quandle covering C g → W + g (Theorem 3.7). As a consequence of our description of the inner automorphism groups of these quandles, we show that they classify surfaces of genus greater than two (Proposition 3.10). In Section 4, we study metrics on these quandles and prove that they are unbounded with respect to the quandle metric (Proposition 4.2). For the quandle D g , we further demonstrate that the quandle metric and the curve complex metric are not comparable when g ≥ 2 (Proposition 4.5). In Section 5, we investigate the bounded cohomology of these families of quandles, and prove that the second bounded cohomology of each of these quandles is infinite-dimensional (Theorem 5.5). We also establish an analogue of the Gromov Mapp
This content is AI-processed based on open access ArXiv data.