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?

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 ∈[K ] directions

Alexander or Jones polynomial 3 Knot group π1(R \ 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 ∈[K ] directions

Alexander or Jones polynomial 3 Knot group π1(R \ 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 ∈[K ] directions

Alexander or Jones polynomial 3 Knot group π1(R \ 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 ∈[K ] directions

Alexander or Jones polynomial 3 Knot group π1(R \ 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

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 ∈[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 } 2 } 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.

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 |S4/D4| = 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.

Quadrisecant history: 1933 Pannwitz: Every nontrivial generic polygonal knot has at least 2u2 quadrisecants. 1994 Kuperberg: Every nontrivial tame knot has at least one quadrisecant. 2004 Budney, Conant, Scannell, Sinha: The coefﬁcient of z2 in the Conway polynomial may be computed by counting alternating quadrisecants with an appropriate multiplicity. , 2004 D- : Every nontrivial C1 1 knot in R3 has at least one essential alternating quadrisecant.

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 17 / 24 Theorem (D-, 2004) Every nontrivial tame knot in R3 has at least one alternating quadrisecant.

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 17 / 24 Strategy Assume the knot K has unit thickness, then ropelength is just the length of K (the core curve). Use quadrisecants and extra geometry to get a lower bound for ropelength. Extra geometry due to unit thickness of K :

I Curvature bounded above by 2. I Given p ∈ K , let B(p) be the unit ball centered at p. Then B(p) contains a single unknotted arc of K with length at most π. I Other arcs of the knot stay outside this unit ball. 0 b b a0 1 ∠apb = θ β s α Length arc(ab) is a r p √ √ r 2 − 1 + θ − (α + β) + s2 − 1

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 18 / 24 Strategy Assume the knot K has unit thickness, then ropelength is just the length of K (the core curve). Use quadrisecants and extra geometry to get a lower bound for ropelength. Extra geometry due to unit thickness of K :

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 18 / 24 Computations Given quadrisecant abcd, deﬁne r := |b − a|, s := |c − b|, t := |d − c|.

Find Len(K√) in terms of r, s and t. 2 1 Let f (r) := r − 1 + arcsin( r ). Len(γac) ≥ f (r) + f (s), Len(γda) ≥ f (r) + s + f (t).

f(r) f (r) f((ss)) r s t 2 1 r s t r s aa b b c d d r s

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 19 / 24 Computations Given quadrisecant abcd, deﬁne r := |b − a|, s := |c − b|, t := |d − c|.

Find Len(K√) in terms of r, s and t. 2 1 Let f (r) := r − 1 + arcsin( r ). Len(γac) ≥ f (r) + f (s), Len(γda) ≥ f (r) + s + f (t).

f(r) f (r) f((ss)) r s t 2 1 r s t r s aa b b c d d r s

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 19 / 24 Computations Given quadrisecant abcd, deﬁne r := |b − a|, s := |c − b|, t := |d − c|.

Find Len(K√) in terms of r, s and t. 2 1 Let f (r) := r − 1 + arcsin( r ). Len(γac) ≥ f (r) + f (s), Len(γda) ≥ f (r) + s + f (t).

f(r) f (r) f((ss)) r s t 2 1 r s t r s aa b b c d d r s

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 19 / 24 The results

Theorem (D-, Diao, Sullivan 2006) Any nontrivial knot has ropelength at least 15.66.

Recall: the tight trefoil knot has ropelength ≈ 16.372, so we were pretty close.

Note the ropelength of a knot with simple essential quadrisecants is at least 15.94, ﬂipped essential quadrisecants is at least 13.936.

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 20 / 24 Thank you!

Acknowledgements: My fantastic coauthors: Jason Cantarella, Yuanan Diao, John Sullivan, Nancy Wrinkle.

Further reading: J. Cantarella, R.B. Kusner, J.M. Sullivan, On the minimum ropelength of knots and links. Invent. Math 150:2 (2002) pp 257–286. E. Denne, Y. Diao, J.M. Sullivan, Quadrisecants give new bounds for ropelength. Geom. Topol. 10 (2006) pp 1–26. Y. Diao, C. Ernst, X. Yu, Hamiltonian knot projections and lengths of thick knots. Topology Appl. 136 (2004), no. 1-3, pp 7–36.

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 21 / 24 Deﬁnition Let α, β, γ be disjoint arcs from a to b. Let X := R3 r (α ∪ γ) and let h(α) be a parallel curve to α ∪ β, chosen so that α ∪ γ has linking number zero with δ. Then (α, β) is inessential if δ is also null homotopic in X. We say (α, β) is essential if it is not inessential.

h(α)

a b β

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 22 / 24 Lemma (DDS 2006)

If γab is an essential arc in a unit-thickness knot then ( 2π − 2 arcsin(|a − b|/2) if |a − b| < 2, Len(γab) ≥ π if |a − b| ≥ 2.

a b

Πa Πb

Πq

Elizabeth Denne (Smith College) Geometric Knot Theory 9th October 2009 23 / 24 Notes

If K is the unknot then arc γab is inessential for a, b ∈ K . If both arcs γab and γba for a, b ∈ K are inessential then K is the unknot.

Deﬁnition

Secant ab is essential if both γab and γba are essential. An essential alternating quadrisecant abcd is an alternating quadrisecant which is essential in the second segment bc

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 24 / 24