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An Introduction to Knot Theory

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An Introduction to Knot Theory Compiled from presentations and work by: David Freund, Sarah Smith-Polderman, Danielle Shepherd, Joseph Smith, Michael Bush, and Katelyn French Advised by: Dr. John Ramsay and Dr. Jennifer Bowen Take a piece of string Loop it around and through itself Connect the ends A mathematical knot [I]. Links are composed of interwoven knots (components). A link with one component is a knot. Links are commonly represented as two- dimensional images called projections. Knot projections are different representations of a given knot Three projections of the trefoil knot [II]. The three Reidemeister moves [III]. In the mirror image of a link, every overcrossing is switched an undercrossing and vice versa Most knots, like the trefoil knot, are not amphichiral (equivalent to their mirror image) The left handed trefoil (left) and its mirror image, the right handed trefoil (right). There is no combination of Reidemeister moves that can transform one into the other [IV]. Knots and links can be classified through their invariants An invariant is an inherent characteristic of a link that is the same for any projection The Perko Pair, they were thought to be different knots for 75 years [V]! Crossing number P-colorability Linking number Knot groups Genus Bridge number Polynomials Braid Index The crossing number of a knot or link A, denoted c(A), is the fewest number of crossings that occurs in any projection of the knot or link. The crossing number of the trefoil is 3. The two projections on the left are at the trefoil’s crossing number, while the one on the right has 4 crossings [2]. An asterisk denotes the 2* mirror image of a link Crossing number 41 Number of components Index (as catalogued by Rolfsen) [4] 2 41 There is only one knot with no crossings: the unknot Millions of links have been catalogued Links organized by their Alexander-Briggs notations [VI]. Add the values of all the crossings between different components Divide the absolute value by 2 +1 ­ 1 A strand is a portion of a knot or link between two undercrossings. A link is tricolorable if: . The three strands coming together at a crossing are all different colors or all the same color . At least two colors must be used to color the link A tricolored trefoil knot [VII]. p-colorability generalizes tricolorability In a torus link, denoted T(m,n), the link wraps around the longitude m times and the meridian n times. A link on a trefoil [VIII]. Rectangular representation of a torus link A Klein bottle is a non-orientable surface We use a three-dimensional once-punctured Klein bottle A Klein bottle [IX]. A glass Klein bottle. Rectangular representation of a K(5,4) . Klein link K(m,n) labeling n=4 m=5 K(5,4) on the Klein bottle Layer 1 2 3 4 Link after Klein bottle has been removed Untangle the resulting link and classify Digital Catalogue 3* 87 Braid generators: Braid word: −1 −1 휎푖휎푖 = 1 = 휎푖 휎푖 휎푖휎푖+1휎1 = 휎푖+1휎푖휎푖+1 휎푖휎푗 = 휎푗휎푖 if 푖 − 푗 ≥ 2 Conjugation Stabilization −1 −1 퐵 = 휎푖퐵휎푖 = 휎푖 퐵휎푖, For an n-string braid B, −1 where 1 ≤ 푖 ≤ 푛. 퐵 = 퐵휎푛 = 퐵휎푛 . A general braid word for a torus link is [1]: m (1 2 n1) A general braid word for a Klein link adds a half-twist to the torus braid word: n1 m 1 1 1 ( 1 2 n1) ( n1 n2 i ) i1 In a full twist (denoted Δ2), each string returns to its original position in the braid In a half twist (denoted Δ), strings reverse their position in the braid A full twist on a 5-string braid. A half twist on a 5-string braid. When m ≥ n the general braid word of a K(7,4) Klein link can be simplified to the braid word w: Half Twist (Δ) I. http://en.wikipedia.org/wiki/Figure- eight_knot_(mathematics) II. http://vismath7.tripod.com/nat/#f4 III. http://mathworld.wolfram.com/ReidemeisterMoves.ht ml IV. http://en.wikipedia.org/wiki/Trefoil_knot V. http://www.math.toronto.edu/drorbn/classes/0102/Feyn manDiagrams/NonObvious/Perko_640.jpg VI. http://www.math.buffalo.edu/~menasco/knot- theory.html VII. http://en.wikipedia.org/wiki/Tricolorability VIII. http://www.cgtp.duke.edu/~psa/cls/261/knot53.html IX. http://en.wikipedia.org/wiki/Klein_bottle X. http://www.wallpaperpin.com/wallpaper/1600x1200/rop e-macro-knot-node-desktop-free-wallpaper-2458.html 1) Colin C. Adams. The Knot Book. American Mathematical Society, 2004. 2) W.B. Raymond Lickorish. An Introduction to Knot Theory. Springer, New York, 1997. 3) Kunio Murasugi. Knot Theory & Its Applications. Birkhäuser, 2008. 4) Dale Rolfsen. Knots and Links. Publish or Perish, Houston, 1976. [X] .
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