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Fun with Knots: Invariants and Applications

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Fun with Knots: Invariants and Applications Fun with knots: Invariants and applications Mark Ioppolo School of Mathematics and Statistics University of Western Australia August 18, 2011 Mark Ioppolo Fun with knots: Invariants and applications 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Figure: Vortices in the air around aeroplane wings A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = fx 2 R : jjxjj = 1g A link is a disjoint union of knots Knots are covered more formally in Baez and Munian's Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book. What is a knot? Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together. A link is a disjoint union of knots Knots are covered more formally in Baez and Munian's Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book. What is a knot? Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together. A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = fx 2 R : jjxjj = 1g Knots are covered more formally in Baez and Munian's Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book. What is a knot? Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together. A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = fx 2 R : jjxjj = 1g A link is a disjoint union of knots What is a knot? Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together. A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = fx 2 R : jjxjj = 1g A link is a disjoint union of knots Knots are covered more formally in Baez and Munian's Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book. Example 1: The figure 8 Figure: A figure 8 knot in 3-space Example 2: The Borromean rings Figure: The Borromean rings Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1! Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing. Isotopy: When are two knots the same? Figure: Topology in a nutshell - artist unknown Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing. Isotopy: When are two knots the same? Figure: Topology in a nutshell - artist unknown Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1! Isotopy: When are two knots the same? Figure: Topology in a nutshell - artist unknown Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1! Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing. Knot diagrams A knot diagram is a projection onto the plane where the neighbourhood of every point is one of the following: Figure: Allowed neighbourhoods in a knot diagram Theorem Every knot is isotopic to a knot whose projection contains no catastrophes. Knot diagrams cont. Sometimes a projection will generate a knot diagram with a neighbourhood of the form: Figure: Uh oh, try again This is called a catastrophe. Knot diagrams cont. Sometimes a projection will generate a knot diagram with a neighbourhood of the form: Figure: Uh oh, try again This is called a catastrophe. Theorem Every knot is isotopic to a knot whose projection contains no catastrophes. Example 3: Some knot diagrams Figure: The unknot (i), Haken's knot (ii) and Goeritz's knot (iii) are isotopic to each other Note that for any knot K we have K ? U = U ? K = K where U denotes the unknot. Prime knots: If K = K1 ? K2 for knots K1; K2 then K1 = K and K2 = U. Connected sums Notation: K1 ? K2 1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands Prime knots: If K = K1 ? K2 for knots K1; K2 then K1 = K and K2 = U. Connected sums Notation: K1 ? K2 1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands Note that for any knot K we have K ? U = U ? K = K where U denotes the unknot. Connected sums Notation: K1 ? K2 1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands Note that for any knot K we have K ? U = U ? K = K where U denotes the unknot. Prime knots: If K = K1 ? K2 for knots K1; K2 then K1 = K and K2 = U. Example 4: Prime knots Figure: Prime knot diagrams with 7 crossings or less Theorem Two knots are isotopic if and only if the corresponding knot diagrams can be connected by a finite sequence of Reidemeister moves Reidemeister moves Figure: The three kinds of Reidemeister moves Reidemeister moves Figure: The three kinds of Reidemeister moves Theorem Two knots are isotopic if and only if the corresponding knot diagrams can be connected by a finite sequence of Reidemeister moves There is an algorithm that will test whether or not a knot with n crossings is isotopic to an unknot I would not recommend trying this. Unknot recognition Theorem (Hass,Lagarias) There is a positive constant c, such that for each n ≥ 1, any unknotted knot diagram n crossings can be transformed to the trivial knot diagram using at most k = 2cn Reidemeister moves Unknot recognition Theorem (Hass,Lagarias) There is a positive constant c, such that for each n ≥ 1, any unknotted knot diagram n crossings can be transformed to the trivial knot diagram using at most k = 2cn Reidemeister moves There is an algorithm that will test whether or not a knot with n crossings is isotopic to an unknot I would not recommend trying this. Side note: Hass and Lagarias built their 3-sphere by gluing tetrahedra. A knot is represented by a sequence of edges. Unknot recognition cont. Hass and Lagarias' least upper bound was 1 2 2dt 6 · 840n + 2dt + 1 2 where d = 107 and t is the number of tetrahedra. Taking c = 1011 is sufficient. Unknot recognition cont. Hass and Lagarias' least upper bound was 1 2 2dt 6 · 840n + 2dt + 1 2 where d = 107 and t is the number of tetrahedra. Taking c = 1011 is sufficient. Side note: Hass and Lagarias built their 3-sphere by gluing tetrahedra. A knot is represented by a sequence of edges. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic. Figure: The Perko pair: Perko discovered an error in Little's tables in 1974 Knot tabulation Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Figure: The Perko pair: Perko discovered an error in Little's tables in 1974 Knot tabulation Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic. Knot tabulation Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic. Figure: The Perko pair: Perko discovered an error in Little's tables in 1974 Knot tabulation cont. Figure: Number of prime alternating links per crossing number and number of components.
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