A polynomial lemniscate is a type of curve defined by a polynomial equation, which typically takes the form of a lemniscate—a figure-eight or infinity-shaped curve.

A Prym variety is an important concept in the field of algebraic geometry, particularly in the study of algebraic curves and their Jacobians. Specifically, a Prym variety is associated with a double cover of algebraic curves.

S-equivalence, in the context of formal languages, particularly in the theory of automata, refers to a specific type of equivalence between state machines (such as finite automata, pushdown automata, etc.) concerning the languages they recognize. Two automata are considered S-equivalent if they accept the same set of input strings.

Cayley graphs are a type of graph used in group theory to represent the structure of a group in a visual and geometric way. Named after the mathematician Arthur Cayley, these graphs provide insight into the group's properties, including symmetries and relationships among its elements. ### Definition: A Cayley graph is constructed from a group \( G \) and a generating set \( S \) of that group.

A **conference graph** is a specific type of graph studied in graph theory, related to combinatorial designs.

Frucht's theorem is a result in graph theory that states that for any finite group \( G \), there exists a finite undirected graph (called a "Frucht graph") that is a Cayley graph of \( G \) and is also vertex-transitive (meaning that for any two vertices in the graph, there is some automorphism of the graph that maps one vertex to the other).

The Bloch group is a mathematical construct in the field of algebraic K-theory and number theory. It is named after the mathematician Spencer Bloch. The main idea behind the Bloch group is to provide a way to study the properties of values of certain functions, particularly the behavior of rational numbers and algebraic numbers within the context of abelian varieties and algebraic cycles.

A **half-transitive graph** is a type of graph that is related to the concept of transitive graphs in the field of graph theory. To understand half-transitive graphs, it's helpful to first clarify what a transitive graph is.

Hierarchical closeness typically refers to a concept in social network analysis and organizational theory that measures how closely related individuals or entities are within a hierarchical structure based on their positions. It can be used to assess the proximity of nodes (which could represent people, departments, or other entities) within social or organizational hierarchies.

The matching polynomial is a well-defined polynomial associated with a graph that encapsulates information about its matchings—sets of edges without shared vertices.

Spectral graph theory is a branch of mathematics that studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with them. These matrices include the adjacency matrix, the degree matrix, and the Laplacian matrix, among others. Spectral graph theory connects combinatorial properties of graphs with linear algebra and provides powerful tools for analyzing graphs in various contexts.

A Moufang set is a concept from the field of mathematics, specifically in the context of algebra and geometry. It is related to the study of certain types of algebraic structures that exhibit properties reminiscent of groups but without necessarily adhering to all the group axioms.

The Tutte matrix is a mathematical construct used in the study of graph theory, particularly in the context of understanding the properties of bipartite graphs and the presence of perfect matchings. It is named after the mathematician W. T. Tutte.

Cylindric algebra is a mathematical structure that arises in the study of multi-dimensional logics and is particularly relevant in the fields of model theory and algebraic logic. It is an extension of Boolean algebras to accommodate more complex relationships involving multiple dimensions or "cylindrical" structures. A cylindric algebra can be thought of as an algebraic structure that captures the properties of relations in multiple dimensions, enabling the representation of various logical operations and relations.

Double torus knots and links are concepts from the field of knot theory, which is a branch of topology. In topology, knots are considered as embeddings of circles in three-dimensional space, and links are collections of such embeddings. ### Double Torus A double torus is a surface that is topologically equivalent to two tori (the plural of torus) connected together. It's often visualized as the shape of a "figure eight" or a surface with two "holes.

Topological methods in algebraic geometry refer to the application of topological concepts and techniques to study problems and objects that arise in algebraic geometry. This interdisciplinary area combines elements from both topology (the study of properties of space that are preserved under continuous transformations) and algebraic geometry (the study of geometric objects defined by polynomial equations).

A 3-sphere, often denoted as \( S^3 \), is a higher-dimensional analogue of a sphere. In simple terms, it is the set of points in four-dimensional Euclidean space (\( \mathbb{R}^4 \)) that are at a constant distance (the radius \( r \)) from a central point (the origin).

Alexander duality is a fundamental theorem in algebraic topology, specifically in the study of topological spaces and their homological properties. Named after mathematician James W. Alexander, the duality provides a relationship between the topology of a space and the topology of its complement. In its most basic form, Alexander duality applies to a locally finite CW complex, particularly when considering a subcomplex (or a subset) of a sphere.

Aspherical space is a term used in topology, a branch of mathematics that studies the properties of space that are preserved under continuous transformations. Specifically, an aspherical space is a manifold (or more generally, a topological space) whose universal covering space is contractible. This means that the universal cover does not have any "holes"; it can be continuously shrunk to a point without leaving the space.

A comodule over a Hopf algebroid is a mathematical structure that generalizes the notion of a comodule over a Hopf algebra. Hopf algebras are algebraic structures that combine aspects of both algebra and coalgebra with additional properties (like the existence of an antipode). A Hopf algebroid is a more general structure that facilitates the study of categories and schemes over a base algebra.