By Inga Johnson, Allison K. Henrich
This well-written and interesting quantity, meant for undergraduates, introduces knot thought, a space of becoming curiosity in modern arithmetic. The hands-on procedure positive aspects many routines to be accomplished by means of readers. necessities are just a easy familiarity with linear algebra and a willingness to discover the topic in a hands-on manner.
The establishing bankruptcy bargains actions that discover the realm of knots and hyperlinks — together with video games with knots — and invitations the reader to generate their very own questions in knot thought. next chapters advisor the reader to find the formal definition of a knot, households of knots and hyperlinks, and numerous knot notations. Additional themes comprise combinatorial knot invariants, knot polynomials, unknotting operations, and digital knots.
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Additional info for An Interactive Introduction to Knot Theory
1: Examples of twist knots. 2. 1 are twist knots. 1? 3. Prove that every twist knot is invertible. 4. 4) on the projection of twist knots Tn where n is even. Prove the following propositions. If Kenya (the knotter) plays second on a twist knot Tn, where n is even, then Kenya has a winning strategy. If Ulysses (the unknotter) plays second on a twist knot Tn, where n is even, then Ulysses has a winning strategy. 5. Investigate the Knotting-Unknotting game on the projection of twist knots Tn where n is odd.
Chapter 3 Families of Links and Braids Whenever you learn a new mathematical concept, it is useful to have a collection of simple examples to apply the new concept to in order to gain intuition. For instance, is a nice example of a vector space, is a useful example of a group, and the torus is an interesting, yet simple, example of a topological space. In this chapter, we introduce a few families of knots and links with particularly nice properties. These examples will provide some context for us to think about the knot invariants we will introduce in Chapter 5.
6: A generating set of oriented Reidemeister moves. 7. 6. Show how this R2 move can be derived from a sequence of moves in the generating set. 7: An oriented R2 move. 4Nonequivalence and Invariants Now that we have a well-defined notion of equivalence for links and link diagrams, we can begin to explore the idea of nonequivalence. To show that two diagrams represent equivalent knots or links, we need only find a sequence of Reidemeister moves that transforms one into the other. But how is it possible to show that two diagrams fail to be equivalent?