Download An invitation to knot theory: virtual and classical by Heather A. Dye PDF

By Heather A. Dye

The basically Undergraduate Textbook to educate either Classical and digital Knot Theory

An Invitation to Knot thought: digital and Classical offers complicated undergraduate scholars a gradual creation to the sphere of digital knot conception and mathematical learn. It presents the root for college kids to analyze knot concept and browse magazine articles on their lonesome. every one bankruptcy comprises various examples, difficulties, tasks, and advised readings from examine papers. The proofs are written as easily as attainable utilizing combinatorial methods, equivalence periods, and linear algebra.

The textual content starts with an creation to digital knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and different skein invariants earlier than discussing algebraic invariants, similar to the quandle and biquandle. The e-book concludes with functions of digital knots: textiles and quantum computation.

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Additional info for An invitation to knot theory: virtual and classical

Example text

Initially, virtual knot theory was studied as an extension of classical knot theory. In this book, we study virtual knot theory as a topic that includes classical knot theory. From this viewpoint, the first goal is to enumerate—give a complete list of the mathematical objects that we are studying. The second goal is to determine a method of classification. This means that if someone was to present you with a random virtual link diagram then you could determine its place in the enumerated list. These questions drive the initial research forward and, in the process, introduce other questions.

2. The statement P ⇒ Q and the contrapositive ¬Q ⇒ ¬P have the same values in the truth table, as do the pair Q ⇒ P and ¬P ⇒ ¬Q. These pairs of statements are logically equivalent. If a statement is true then any logically equivalent statement is also true. 2 The converse, contrapositive, and inverse of P ⇒ Q P Q P ⇒ Q Q ⇒ P �P �Q �Q ⇒ ¬P �P ⇒ ¬Q T T T T F F T T T F F T F T F T F T T F T F T F F F T T T T T T We consider examples of these types of statements. We are interested in the set of integers, so we begin with the quantifier “For all integers x”.

Notice that the diagrammatic moves can change the number of crossings and dramatically change the appearance of the diagram. 18. 19, we see several diagrams that are equivalent to the unknot. The diagrammatic moves define an equivalence relation on virtual link diagrams. 2. 2. The diagrammatic moves define an equivalence relation on virtual link diagrams. Proof. A virtual link diagram D is related to itself by a sequence of length zero, so the relation is reflexive. If D1 ~ D2, then the finite sequence of moves transforming D1 into D2 can be reversed, so that D2 ~ D1.

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