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Untangling the mysteries of Knot Theory
Transcript of Untangling the mysteries of Knot Theory
Can we find simpler projections that represent these knots?
Are these knots equivalent, or is there some property that distinguishes between them? Three famous knots... The trivial knot, also known as the unknot, is a knot that can deformed into a perfectly round circle. Knot equivalence We say that two knots are topologically equivalent if one knot can be deformed into the other by means of stretching, twisting, and bending but without tearing. What is a knot? A knot is a simple closed curve in three-dimensional space. A table of knots The unknot, trefoil, and figure-eight knots are the simplest examples in an infinite progression of increasingly complicated prime knots. The BIG questions in knot theory... Polygonal knot theory Polygonal knots are piecewise-linear curves that can be used to model the rigidity inherent in physical knots. Department of Mathematics & Physics
Ave Maria University Untangling the mysteries of Knot Theory Dr. Jorge Alberto Calvo ? Are these knots equivalent? The trefoil knot is a knot that can drawn with only three crossing points. In this projection, the trefoil looks like
a flower with three petals. The figure-eight knot, also known as Listing's knot, can drawn with four (but no fewer) crossings. A table of knots There are 71 prime knots with nine or fewer crossings...
and 1 701 936 prime knots with sixteen or fewer crossings! Pop Quiz! What is this? It's a knot listing not listing Listing's knot! ? Some important things to keep in mind as we study knots: To show that two knots are equivalent we have to produce a deformation from one to the other.
To show that two knots are not equivalent, we would have to show that there can be no such deformation.
How can we make sure that we have thought of all possible deformations? Physical knot theory Some knot theorists are also interested in understanding knots made out of "real stuff," in which physical rigidity plays a significant role in determining topological equivalence. photo by David J. Fred Knots made from "real stuff" Alain Goriely (2005) Knots made from "real stuff" Knots made from "real stuff" Knots made from "real stuff" Dietler, Pieranski, Kasas, Stasiak (2002) Valle, Favre, Roca, Dietler (2005) Dietrich-Buchecker and Sauvage (1989) An octagonal 8 knot 19 An octagonal 8 ? 18 No! This is an optical illusion! To construct this octagon, the "straight edges" must bend to avoid each other.
(Proof by linear algebra.) ? Some interesting facts about polygonal knots... Every triangle, quadrilateral, and pentagon is unknotted.
A polygonal trefoil knot requires at least 6 edges.
A polygonal figure-eight knot requires at least 7 edges.
There are nine additional polygonal knot types that can
be constructed with 8 edges. A deformation of a 7-edge figure-eight knot Calvo and Morstad (2002) Open Question: Can this knot equivalence be established without stretching and shrinking of the edges? Polygonal isotopes There are two types of hexagonal trefoil knots that are not geometrically equivalent. Similarly, there are two types of 7-edge figure-eight knots that are not geometrically equivalent. Open Question: Are there polygonal isotopes of other knots? Of the unknot? Open Question: How many polygonal knot types can be constructed using n edges when n is greater than 8? Regular knots Open Question: What about polygons with equal edges and equal angles? An 11-edge regular trefoil knot with 109.4 angles Comar et al (2006) o Lattice knots Diao (1993) A 24-edge lattice trefoil knot Open Question: What if the unit-length edges are always parallel to the coordinate axes? Thank You!