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Dehn quandles of surfaces and their bounded cohomology

Pankaj Kapari, Deepanshi Saraf, Mahender Singh

Abstract

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.

Dehn quandles of surfaces and their bounded cohomology

Abstract

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.
Paper Structure (11 sections, 30 theorems, 81 equations, 9 figures, 1 table)

This paper contains 11 sections, 30 theorems, 81 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quandles
  4. Bounded cohomology of groups
  5. Bounded cohomology of quandles
  6. New quandle structures on surfaces
  7. Metrics on Dehn quandles of surfaces
  8. Bounded cohomology of Dehn quandles and Gromov Mapping Theorem
  9. Bounded cohomology of Dehn quandles of surfaces
  10. An analogue of the Gromov Mapping Theorem
  11. Idempotents in integral quandle rings of Dehn quandles

Key Result

Proposition 2.3

Let $X$ be a quandle. Then $X$ is isomorphic to $\sqcup_{i\in I}(\operatorname{Inn}(X)/H_i,S_{x_i})$, where $I$ has the cardinality of the number of connected components of $X$, $S_{x_i}\in Inn(X)$ is the inner automorphism of $X$ corresponding to $x_i\in X$ and $H_i= \mathrm{Stab}_{\operatorname{In

Figures (9)

  • Figure 1: Resolution of a self-intersection point of a non-simple closed curve and independence of the resultant multicurve on the chosen orientation.
  • Figure 2: Resolution of self-intersection points of a non-simple closed curve on $S_2$ into simple closed curves.
  • Figure 3: Resolution of a closed curve leading to two isotopic simple closed curves.
  • Figure 4: Connecting two simple closed curves $\alpha_i$ and $\alpha_j$ to construct a non-simple closed curve whose resolution gives back $\alpha_i$ and $\alpha_j$.
  • Figure 5: Two non-isotopic curves with identical associated multicurves.
  • ...and 4 more figures

Theorems & Definitions (62)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 52 more