By Inga Johnson, Allison K. Henrich
This well-written and fascinating quantity, meant for undergraduates, introduces knot thought, a space of growing to be curiosity in modern arithmetic. The hands-on process positive factors many workouts to be accomplished via readers. necessities are just a uncomplicated familiarity with linear algebra and a willingness to discover the topic in a hands-on manner.
The beginning bankruptcy bargains actions that discover the area of knots and hyperlinks — together with video games with knots — and invitations the reader to generate their very own questions in knot conception. 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
Instead, we ask the reader to give an argument that a statement is mathematically valid. We specifically use the term ‘argument’ rather than ‘proof to allow for what we recognize is a tension between the desire for a formal proof with an initial lack of the formal definitions that are needed to make such a proof rigorous. An argument, in this setting, may be viewed as being less formal than a proof, but it should be as clear and complete an explanation as possible. In Chapter 2, you will notice a shift from play to formalism.
If so, give an example. If not, explain why not. 6. 3, which is known as the knot 819. 3: The knot 819 is a torus knot. 7. Investigate whether or not the knot 819 is an alternating knot. 8. Investigate whether or not the knot T2,3 is equivalent to T3,2. Can your findings be generalized? 4Closed Braids The family of links called closed braids are a generalization of torus links. We define closed braids by considering a diagram of n concentric strands. 1 with 4 concentric strands. A small disc-like subregion of the diagram is called a replaceable region if it contains arcs from two adjacent concentric strands and nothing else.
We’ve just seen that there are no nontrivial knots that can be drawn with just one or two crossings. In fact, the smallest nontrivial knot is a knot that can be drawn with three crossings. Any knot that can be drawn with three crossings and no fewer is typically referred to as a trefoil knot. 1 can be manipulated in space and then redrawn with fewer crossings. The crossing number of a link L is the minimum number of crossings needed in a diagram of L. For instance, if the crossing number of L is five, then it is impossible to draw a diagram of L that has four or fewer crossings.
An Interactive Introduction to Knot Theory by Inga Johnson, Allison K. Henrich