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Introduction to Geometric Knot Theory Introduction to Geometric Knot Theory Elizabeth Denne Smith College CCSU Math Colloquium, October 9, 2009 What is a knot? Definition A knot K is an embedding of a circle in R3. (Intuition: a smooth or polygonal closed curve without self-intersections.) A link is an embedding of a disjoint union of circles in R3. Trefoil knot Figure 8 knot Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 2 / 24 When are two knots the same? K1 and K2 are equivalent if K1 can be continuously moved to K2. (Technically, they are ambient isotopic.) A knot is trivial or unknotted if it is equivalent to a circle. A knot is tame if it is equivalent to a smooth knot. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 3 / 24 When are two knots the same? K1 and K2 are equivalent if K1 can be continuously moved to K2. (Technically, they are ambient isotopic.) A knot is trivial or unknotted if it is equivalent to a circle. A knot is tame if it is equivalent to a smooth knot. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 3 / 24 When are two knots the same? K1 and K2 are equivalent if K1 can be continuously moved to K2. (Technically, they are ambient isotopic.) A knot is trivial or unknotted if it is equivalent to a circle. A knot is tame if it is equivalent to a smooth knot. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 3 / 24 Tame knots A wild knot p :: Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 4 / 24 How can you tell if two knots are equivalent? Idea Use topological invariants denoted by I. If K1 is equivalent to K2 then I(K1) = I(K2), Crossing number (knot tables) cr(K ) = min ( min (# crossings of K )) K 2[K ] directions Alexander or Jones polynomial 3 Knot group π1(R n K ) Principle If I(K1) 6= I(K2) then K1 is not equivalent to K2. Knots are distinguished using invariants. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 5 / 24 How can you tell if two knots are equivalent? Idea Use topological invariants denoted by I. If K1 is equivalent to K2 then I(K1) = I(K2), Crossing number (knot tables) cr(K ) = min ( min (# crossings of K )) K 2[K ] directions Alexander or Jones polynomial 3 Knot group π1(R n K ) Principle If I(K1) 6= I(K2) then K1 is not equivalent to K2. Knots are distinguished using invariants. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 5 / 24 How can you tell if two knots are equivalent? Idea Use topological invariants denoted by I. If K1 is equivalent to K2 then I(K1) = I(K2), Crossing number (knot tables) cr(K ) = min ( min (# crossings of K )) K 2[K ] directions Alexander or Jones polynomial 3 Knot group π1(R n K ) Principle If I(K1) 6= I(K2) then K1 is not equivalent to K2. Knots are distinguished using invariants. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 5 / 24 How can you tell if two knots are equivalent? Idea Use topological invariants denoted by I. If K1 is equivalent to K2 then I(K1) = I(K2), Crossing number (knot tables) cr(K ) = min ( min (# crossings of K )) K 2[K ] directions Alexander or Jones polynomial 3 Knot group π1(R n K ) Principle If I(K1) 6= I(K2) then K1 is not equivalent to K2. Knots are distinguished using invariants. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 5 / 24 Knot and Link Table Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 6 / 24 Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 7 / 24 Total curvature κ(γ) of a curve γ For a polygon P, κ(P) is the sum of exterior angles. For a curve γ, take polygons P with vertices on γ, then κ(γ) = max κ(P) P Equivalently for a smooth knot K , κ(K ) is the total angle through which the unit tangent vector turns through. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 8 / 24 Total curvature κ(γ) of a curve γ For a polygon P, κ(P) is the sum of exterior angles. For a curve γ, take polygons P with vertices on γ, then κ(γ) = max κ(P) P Equivalently for a smooth knot K , κ(K ) is the total angle through which the unit tangent vector turns through. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 8 / 24 Some history ... 1929 Fenchel proved that for a closed curve γ in R3, the total curvature κ(γ) ≥ 2π with equality iff γ is a convex planar curve. 1947 Borsuk proved same for γ in Rn and conjectured that the total curvature of a nontrivial knot ≥ 4π. Theorem (1950 Fáry-Milnor) For any nontrivial tame knot K , the total curvature κ(K ) ≥ 4π. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 9 / 24 Bridge number Definition The bridge number br(K ) of a knot K is br(K ) = min ( min (# maxima of K )) K 2[K ] directions br(Unknot)=1, br(Trefoil)= 2 = br(Figure 8) br(810) = 3 Theorem (1949 Milnor) Given a tame knot K , κ(K ) > 2πbr(K ) and br(K ) ≥ 2 for nontrivial knots. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 10 / 24 The ropelength problem Goal Minimize ropelength — the length of a knot or link which has a tube of fixed diameter around it. Tight knots are those knots which minimize ropelength. Question What do tight knots look like? What is the ropelength of a tight knot? Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 11 / 24 Definition The ropelength R(K ) of a knot K is the quotient of its length over thickness R(K ) = Len(K )/τ(K ). Definition The thickness τ(K ) is the diameter of the largest embedded normal tube around the knot K . Consider local and global conditions here. 1 K g 2 g 1 2 K Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 12 / 24 Theorem (2002 CKS and GMSvdM) There is a ropelength minimizer in any (tame) link type; any minimizer is C1;1, with bounded curvature. CKS also described a family of tight links. Alas, there are many links for which the minimum ropelength is not known. For example a keychain with ≥ 7 keys. The minimum ropelength is not known for any knot type. Numerical simulations have shown that the trefoil is the shortest possible knot with minimum ropelength about 16.372. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 13 / 24 Theorem (2002 CKS and GMSvdM) There is a ropelength minimizer in any (tame) link type; any minimizer is C1;1, with bounded curvature. CKS also described a family of tight links. Alas, there are many links for which the minimum ropelength is not known. For example a keychain with ≥ 7 keys. The minimum ropelength is not known for any knot type. Numerical simulations have shown that the trefoil is the shortest possible knot with minimum ropelength about 16.372. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 13 / 24 Theorem (2002 CKS and GMSvdM) There is a ropelength minimizer in any (tame) link type; any minimizer is C1;1, with bounded curvature. CKS also described a family of tight links. Alas, there are many links for which the minimum ropelength is not known. For example a keychain with ≥ 7 keys. The minimum ropelength is not known for any knot type. Numerical simulations have shown that the trefoil is the shortest possible knot with minimum ropelength about 16.372. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 13 / 24 Difficult Problem Relate the ropelength to link type using topological invariants. Example: Let L be a link type with minimum crossing number n and minimum ropelength R(L). Find constants c1; c2; p; q such that p q c1n ≤ R(L) ≤ c2n 3=4 There are families of torus links for which c1n ≤ R(L). 3=2 Diao, Ernst and Yu (2004) proved R(L) ≤ c2n . 1 It is conjectured that R(L) ≤ c3n . Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 14 / 24 Difficult Problem Relate the ropelength to link type using topological invariants. Example: Let L be a link type with minimum crossing number n and minimum ropelength R(L). Find constants c1; c2; p; q such that p q c1n ≤ R(L) ≤ c2n 3=4 There are families of torus links for which c1n ≤ R(L). 3=2 Diao, Ernst and Yu (2004) proved R(L) ≤ c2n . 1 It is conjectured that R(L) ≤ c3n . Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 14 / 24 Quadrisecants secant quadrisecant (alternating) Secant = 2-secant trisecant Trisecant = 3-secant Quadrisecant = 4-secant quadrisecant (simple) Definition An n-secant line is an oriented line in R3 which intersects K in at least n places. An n-secant is an ordered n-tuple of points in K which lie on an n-secant line. Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 15 / 24 Quadrisecants Three types of quadrisecants: simple, flipped and alternating. Determined by comparing the order of abcd along the line and along the unoriented knot — there are jS4=D4j = 3 dihedral orderings. ab c d a b c d a b c d Order:abcd abdc acbd simple flipped alternating Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 16 / 24 Theorem (D-, 2004) Every nontrivial tame knot in R3 has at least one alternating quadrisecant.
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