In the context of [[knot theory]], an invariant refers to a mathematical property or quantity associated with a knot that remains unchanged under certain transformations. These transformations typically involve manipulating the knot by bending, stretching, twisting, or rotating it without cutting or gluing any parts. There are various types of invariants in knot theory. Some of the most well-known ones include: 1. Knot type: This invariant simply identifies whether two knots are equivalent or not. Two knots are considered equivalent if one can be continuously deformed into the other without any cutting or gluing. 2. Alexander polynomial: The Alexander polynomial is a powerful tool in knot theory that assigns a polynomial to each knot. It helps distinguish different knots by providing a unique polynomial for each distinct knot type. 3. Jones polynomial: The Jones polynomial is another important invariant that also assigns a polynomial to each knot. It is more refined than the Alexander polynomial and provides additional information about the underlying structure of the knot. 4. Linking number: The linking number is an integer value that measures how many times one component of a link wraps around another component. It is an example of a topological invariant and remains unchanged under continuous deformations of the link. Moving on to invariants in general, they refer to properties or quantities that remain constant under specific operations or transformations. Invariants have applications across various branches of mathematics and other fields, including physics and computer science. In algebra, for example, an invariant may refer to a property that remains unchanged when certain operations are applied to mathematical objects such as matrices, vectors, or polynomials. Invariant theory studies these properties and develops techniques to identify and manipulate them effectively. In physics, conservation laws provide examples of invariants. Conservation laws state that certain quantities such as energy, momentum, and angular momentum remain constant in isolated systems despite changes occurring within those systems. In computer science and programming languages, there are also invariants that help ensure correct program behavior. These invariants may refer to properties that remain true throughout the execution of a program, such as the values of certain variables or the relationships between different program components. Overall, invariants play a fundamental role in various mathematical and scientific disciplines, providing valuable insights into the underlying structure and behavior of objects and systems. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Invariant") or contains(subject, "invariant") ```