David Freund and Sarah SmithPolderman Advised By Dr Jennifer Bowen and Dr John Ramsay Introduction to Knot Theory Knot theory is a branch of topology that explores the properties of knots Figure 1 The unknot trefoil knot and the figureeight knot ID: 203189
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Klein LinksDavid Freund and Sarah Smith-PoldermanAdvised By: Dr. Jennifer Bowen and Dr. John Ramsay
Introduction to Knot Theory
Knot theory is a branch of topology that explores the properties of knots.
Figure 1. The unknot, trefoil knot, and the figure-eight knot.A mathematical knot is formed by taking a piece of string and wrapping it around and through itself before gluing the ends of the string.ApplicationsApplications for knot theory can be found in various academic fields, including physics, chemistry, and molecular biology. For example, the wrapping of circular DNA can be modeled using knots (Adams, 2004).BraidsIt is known that every knot can be represented as a braid (Adams, 2004). A braid can be thought of pieces of string being dangled from a bar which are then looped around each other and then connected to another bar.Figure 2. A braid and its closure.Figure 3. Braid generators.
Torus KnotsTorus knots are formed by wrapping a string around a torus without letting the string cross over itself and then gluing the ends of the string. rwgrayprojects.com/Lynn/DoubleTorus/dt01.htmlFigure 4. A torus knot.Klein LinksKlein links are formed using an oriented rectangular diagram for a Klein bottle. The vertical edges are connected, forming a cylinder. Then, the horizontal edges are subsequently connected by deforming the bottom of the cylinder to pass through itself and connect with the top of the cylinder.Figure 5. Rectangular diagram and the Klein bottle.We create our Klein links by placing strings on the rectangle and then examining the form of the knot on the Klein bottle that is formed by connecting the edges.Figure 6. (0,4)-Klein link.
ResultsThe (n,n)-Klein link is equivalent to the (n,n-1)-Klein link.The (n,n)-Klein link is the mirror image of the (0,n)-Klein link.The (1,n)-Klein link is the disjoint union of the unknot and the (0,n-1)-Klein link.The (m,2)-Klein link is equivalent to the (m-1,2)-torus knot.The (m,2m)-Klein link is the disjoint union of the (m,m)-Klein link and the (0,m)-Klein link.If m is even then the (m,n)-Klein link is composed of ceiling(n/2) components.If m is odd then the (m,n)-Klein link is composed of ceiling([n+1]/2) components.The (n,n)-Klein link can be written as the braid word (σ1σ2 …σq-1)(σ1σ2…σq-2)…σ1.ReferencesColin C. Adams. The Knot Book. American Mathematical Society, 2004.Louisa Catalano, David Freund, Rutendo Ruzvidzo, Jennifer Bowen, and John Ramsay. A preliminary study of Klein knots. In Proc. Midstates Conference for Undergraduate Research in Computer Science and Mathematics, pages 10-17, Spring Field, OH, 2010.Jeffery Weeks. The Shape of Space. Taylor & Francis Group, 2002.