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Background Knot concordance Fractals Satellite operations and knot concordance Arunima Ray Brandeis University San Francisco State University February 18, 2016 Arunima Ray (Brandeis) Satellite operations and knot concordance February 18,

SLIDE 1

Background Knot concordance Fractals

Satellite operations and knot concordance

Arunima Ray

Brandeis University

San Francisco State University

February 18, 2016

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 1 / 13

SLIDE 2

Background Knot concordance Fractals

Knots

Definition

A knot is an embedding S1 ֒ → R3.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 2 / 13

SLIDE 3

Background Knot concordance Fractals

Knots

Definition

Two knots are said to be isotopic if one can be deformed into another through embeddings in R3. Isotopy is a 3–dimensional equivalence relation.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 3 / 13

SLIDE 4

Background Knot concordance Fractals

Knots

Definition

Two knots are said to be isotopic if one can be deformed into another through embeddings in R3. Isotopy is a 3–dimensional equivalence relation.

Figure: These are all pictures of the same knot, called the unknot.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 3 / 13

SLIDE 5

Background Knot concordance Fractals

Knots

Theorem (Lickorish–Wallace, 1960s)

Any closed, connected, orientable manifold can be obtained from R3 by performing an operation called ‘surgery’ on a collection of knots.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 4 / 13

SLIDE 6

Background Knot concordance Fractals

Knots

Theorem (Lickorish–Wallace, 1960s)

Any closed, connected, orientable manifold can be obtained from R3 by performing an operation called ‘surgery’ on a collection of knots. Knot theory also has applications to algebraic geometry, statistical mechanics, DNA topology, quantum computing, . . . .

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 4 / 13

SLIDE 7

Background Knot concordance Fractals

Knot concordance

Definition

Knots K0, K1 are concordant if they cobound a smoothly embedded annulus in S3 × [0, 1]. Knots modulo concordance form the knot concordance group C. K0 S3 × {0} S3 × [0, 1] K1

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 5 / 13

SLIDE 8

Background Knot concordance Fractals

Topological knot concordance

Definition

Knots K0, K1 are topologically concordant if they cobound a locally flat, topologically embedded annulus in S3 × [0, 1]. Knots modulo topological concordance form the topological knot concordance group Ctop. K0 S3 × {0} S3 × [0, 1] K1

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 6 / 13

SLIDE 9

Background Knot concordance Fractals

Exotic knot concordance

Definition

Knots K0, K1 are exotically concordant if they cobound a smoothly embedded annulus in a smooth manifold M homeomorphic to S3 × [0, 1], i.e. a possibly exotic S3 × [0, 1]. Knots modulo exotic concordance form the exotic knot concordance group Cex. K0 S3 M K1 If the smooth 4–dimensional Poincar´ e Conjecture holds, then C = Cex.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 7 / 13

SLIDE 10

Background Knot concordance Fractals

Proposition

A knot is the unknot if and only if it is the boundary of a disk. That is, K is the unknot if and only if g(K) = 0.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 8 / 13

SLIDE 11

Background Knot concordance Fractals

Proposition

A knot is the unknot if and only if it is the boundary of a disk. That is, K is the unknot if and only if g(K) = 0. If T is the trefoil knot, g(T) = 1. Therefore, the trefoil is not equivalent to the unknot.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 8 / 13

SLIDE 12

Background Knot concordance Fractals

Connected sum of knots

Figure: The connected sum of two trefoil knots, T#T

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 9 / 13

SLIDE 13

Background Knot concordance Fractals

Connected sum of knots

Figure: The connected sum of two trefoil knots, T#T

Proposition

Given two knots K and J, g(K#J) = g(K) + g(J).

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 9 / 13

SLIDE 14

Background Knot concordance Fractals

Connected sum of knots

Figure: The connected sum of two trefoil knots, T#T

Proposition

Given two knots K and J, g(K#J) = g(K) + g(J). Therefore, g(T# · · · #T

n copies

) = n

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 9 / 13

SLIDE 15

Background Knot concordance Fractals

Connected sum of knots

Figure: The connected sum of two trefoil knots, T#T

Proposition

Given two knots K and J, g(K#J) = g(K) + g(J). Therefore, g(T# · · · #T

n copies

) = n Corollary: There exist infinitely many distinct knots!

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 9 / 13

SLIDE 16

Background Knot concordance Fractals

Connected sum of knots

Figure: The connected sum of two trefoil knots, T#T

Proposition

Given two knots K and J, g(K#J) = g(K) + g(J). Therefore, g(T# · · · #T

n copies

) = n Corollary: There exist infinitely many distinct knots! Corollary: We can never add together non-trivial knots to get a trivial knot.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 9 / 13

SLIDE 17

Background Knot concordance Fractals

Slice knots

Recall that a knot is equivalent to the unknot if and only if it is the boundary of a disk in R3.

Definition

A knot K is slice if it is the boundary of a disk in R3 × [0, ∞).

y, z x w

Figure: Schematic picture of the unknot

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 10 / 13

SLIDE 18

Background Knot concordance Fractals

Slice knots

Recall that a knot is equivalent to the unknot if and only if it is the boundary of a disk in R3.

Definition

A knot K is slice if it is the boundary of a disk in R3 × [0, ∞).

y, z x w

Figure: Schematic picture of the unknot

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 10 / 13

SLIDE 19

Background Knot concordance Fractals

Slice knots

Recall that a knot is equivalent to the unknot if and only if it is the boundary of a disk in R3.

Definition

A knot K is slice if it is the boundary of a disk in R3 × [0, ∞).

y, z x w y, z x w

Figure: Schematic picture of the unknot and a slice knot

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 10 / 13

SLIDE 20

Background Knot concordance Fractals

Examples of slice knots

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 11 / 13

SLIDE 21

Background Knot concordance Fractals

Examples of slice knots

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 11 / 13

SLIDE 22

Background Knot concordance Fractals

Examples of slice knots

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 11 / 13

SLIDE 23

Background Knot concordance Fractals

Examples of slice knots

Knots of this form are called ribbon knots.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 11 / 13

SLIDE 24

Background Knot concordance Fractals

Examples of slice knots

Knots of this form are called ribbon knots. Knots, modulo slice knots, form a group called the knot concordance group, denoted C.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 11 / 13

SLIDE 25

Background Knot concordance Fractals

Fractals

Fractals are objects that exhibit ‘self-similarity’ at arbitrarily small scales. i.e. there exist families of injective functions from the set to smaller and smaller subsets (in particular, the functions are non-surjective).

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 12 / 13

SLIDE 26

Background Knot concordance Fractals

Fractals

Fractals are objects that exhibit ‘self-similarity’ at arbitrarily small scales. i.e. there exist families of injective functions from the set to smaller and smaller subsets (in particular, the functions are non-surjective).

Conjecture (Cochran–Harvey–Leidy, 2011)

The knot concordance group C is a fractal.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 12 / 13

SLIDE 27

Background Knot concordance Fractals

Satellite operations on knots

P K P(K)

Figure: The satellite operation on knots

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 13 / 13

SLIDE 28

Background Knot concordance Fractals

Satellite operations on knots

P K P(K)

Figure: The satellite operation on knots

Any knot P in a solid torus gives a function on the knot concordance group, P : C → C K → P(K) These functions are called satellite operators.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 13 / 13

SLIDE 29

Background Knot concordance Fractals

The knot concordance group has fractal properties

Theorem (Cochran–Davis–R., 2012)

Large (infinite) classes of satellite operators P : C → C are injective.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 14 / 13

SLIDE 30

Background Knot concordance Fractals

The knot concordance group has fractal properties

Theorem (Cochran–Davis–R., 2012)

Large (infinite) classes of satellite operators P : C → C are injective.

Theorem (R., 2013)

There are infinitely many satellite operators P and a large class of knots K such that P i(K) = P j(K) for all i = j.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 14 / 13

SLIDE 31

Background Knot concordance Fractals

The knot concordance group has fractal properties

Theorem (Cochran–Davis–R., 2012)

Large (infinite) classes of satellite operators P : C → C are injective.

Theorem (R., 2013)

There are infinitely many satellite operators P and a large class of knots K such that P i(K) = P j(K) for all i = j.

Theorem (Davis–R., 2013)

There exist satellite operators that are bijective on C.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 14 / 13

SLIDE 32

Background Knot concordance Fractals

The knot concordance group has fractal properties

Theorem (Cochran–Davis–R., 2012)

Large (infinite) classes of satellite operators P : C → C are injective.

Theorem (R., 2013)

There are infinitely many satellite operators P and a large class of knots K such that P i(K) = P j(K) for all i = j.

Theorem (Davis–R., 2013)

There exist satellite operators that are bijective on C.

Theorem (A. Levine, 2014)

There exist satellite operators that are injective but not surjective.

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 14 / 13

SLIDE 33

Background Knot concordance Fractals

Fractals

What is left to show? In order for C to be a fractal, we need some notion of distance or size, to see that we have smaller and smaller embeddings of C within itself. One way to do this is to exhibit a metric space structure on C. There are several natural metrics on C, but we have not yet found one that works well with the current results on satellite operators. The search is on!

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 15 / 13

SLIDE 34

The origins of mathematical knot theory

1880s: Kelvin (1824–1907) hypothesized that atoms were ‘knotted vortices’ in æther. This led Tait (1831–1901) to start tabulating knots. Tait thought he was making a periodic table!

Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 1 / 2

SLIDE 35

Examples of knots

Figure: Knots in circular DNA.

(Images from Cozzarelli, Sumners, Cozzarelli, respectively.) Arunima Ray (Brandeis) Satellite operations and knot concordance February 18, 2016 2 / 2