Started () articles should be accessible to advanced undergraduates.

Joint with Robert Ghrist and Phil Holmes. Springer-Verlag, 1997. 208 pages.

More on Knots in Robinson's Attractor, Preprint. Joint with Ghazwan AL-Hashimi.

**Abstract.** In an earlier paper the second author made a study
of the knotted periodic orbits in a strange attractor for a set of
differential equations in a paper by Clark Robinson. The attractor
is modeled by a Lorenz-like template. It was shown that the knots
and links are positive but need not be positive braids. Here we
show that they are fibered, have positive signature, and that each
knot-type appears infinitely often. We then construct a zeta type
function that counts periodic orbits by the twisting of the local
stable manifolds.

Knots in Flows, Preprint.

**Abstract.**
This is a survey article submitted by invitation for inclusion in the *Encyclopedia of Knot Theory*, CRC Press. Published on December 2, 2020.

Further study of simple Smale flows using four band templates,

**Abstract.**
In this paper, we discuss how to realize a non singular
Smale flow with a four band template on 3-sphere. This extends
the work done by the second author on Lorenz Smale flows, Bin Yu
on realizing Lorenz Like Smale flows on 3-manifold and continues
the work of Elizabeth Haynes and the second author on realizing
simple Smale flows with a different four band template on 3-sphere.

Realizing Full

**Abstract.**
Smale flows on 3-manifolds can have invariant saddle sets that are
suspensions of shifts of finite type. We look at Smale flows with
chain recurrent sets consisting of an attracting closed orbit *a*,
a repelling closed orbit *r* and a saddle set that is a suspension
of a full *n*-shift and draw some conclusions about the knotting
and linking of *a* and *r*. For example, we show for all values of
*n* it is possible for *a* and *r* to be unknots. For any even
value of *n* it is possible for *a* and *r* to be the Hopf link,
a trefoil and meridian, or a figure-8 knot and meridian.

Trefoil Surgery.

**Abstract.**
This is an expository account of a theorem of Louise
Moser that describes the types of manifolds that can be constructed
via Dehn surgery along a trefoil in the 3-sphere. These include
lens spaces, connected sums of two lens spaces, and certain Seifert
fibered spaces with three exceptional fibers. Various concepts from
the topological theory of three dimensional manifolds are developed
as needed.

Simple Smale Flows with a Four Band Template, with Elizabeth Haynes,

**Abstract.**
We study simple Smale flows on S^{3} and other 3-manifolds modeled by the Lorenz template and another template with four bands but that still has cross section a full 2-shift.

Nonsingular Smale flows in the 3-sphere with one attractor and one repeller.

The aim of this paper is to show that any two knots can be realized as an attractor and repeller
pair for some nonsingular Smale flow on S^{3} with any linking number. We view this as
progress, albeit limited, to the conjecture that all two component links can be realized as
an attractor-repeller pair in a nonsingular Smale flow on S^{3} with just one other basic
set of saddle type.

Linear Vector Fields. PS, PDF. [4 pages] (Unpublished)

This paper is aimed at undergraduate readers. It explores some connections between vector calculus and linear algebra.

Transverse Foliations to nonsingular Morse-Smale flows on the 3-sphere and Bott-integrable Hamiltonian systems.

Qualitative Theory of Dynamical Systems, Vol. 7, No. 2, December 2008. PS, PDF. [5 pages]

In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse-Smale flows on the 3-sphere have transverse foliations. We then observe that there is a connection to flows arising from certain Hamiltonian systems and from certain contact structures.

Factoring Families of Positive Knots on Lorenz-like Templates.

Journal of Knot Theory and Its Ramifications, Vol. 17, No. 10, October 2008

PS, PDF. [15 pages]

We show that for *m* and *n* positive, composite closed orbits
realized on the Lorenz-like template *L(m,n)* have two prime
factors, each a torus knot; and that composite closed orbits on
*L(-1,-1)* have either two for three prime factors, two of which
are torus knots.

The Topology and Dynamics of Flows

Open Problems in Topology II

Elliott Pearl Editor

Elsever Press, 2007. PDF [13 pages]

After a brief survey of various types of flows (Morse-Smale, Smale, Anosov and partially hyperbolic) we focus on Smale flows on the 3-sphere. However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms.

Knots on a Positive Template have a Bounded Number of Prime Factors.

Algebraic and Geometric Topology 5 (2005), paper no. 24, pages 563-576.

Preprint:PostScript, PDF.

Templates are branched 2-manifolds with
semi-flows used to model ``chaotic''hyperbolic invariant sets of
flows on 3-manifolds. Knotted orbits on a template correspond to
those in the original flow. Birman and Williams conjectured that
for any given template the number of prime factors of the knots
realized would be bounded. We prove a special case when the
template is *positive*; the general case is now known
to be false.

Factoring Positive Braids via Branched Manifolds

Topology Proceedings, Volume 30 Number 1 (2006), pp. 403-416.

[418K, postscript], [203K, PDF], Published version.

We show that a positive braid is composite if and only if the
factorization is ``visually obvious'' by placing the braid

Twistwise Flow Equivalence and Beyond... (Appendix joint with Mike Boyle)

The Proceedings of the Max Planck Institute Workshop on Algebraic and Topological Dynamics, 171--186,

Edited by S. Kolyada, Y. Manin, & T. Ward, Contemporary Mathematics, Vol. 385, American Mathematical Society, 2005.

PostScript, PDF.

An expository account of recent progress on twistwise flow equivalence. There is a new result in the appendix.

Book review of

SIAM Reviews, Volume 47, Number 2 (2005) 397--400. PostScript, PDF.

Equivariant Flow Equivalence of Shifts of Finite Type by Matrix Equivalence over group Rings.

Joint with Mike Boyle.

Proceedings of the London Mathematical Society,

Volume 91 Part 1 (July 2005). Post Script, PDF.

Let G be a finite group. We classify G-equivariant flow
equivalence of nontrivial irreducible shifts of finite
type in terms of (i) elementary equivalence of matrices over the integral
group ring ZG and (ii) the conjugacy class in ZG of the group of G-weights
of cycles based at a fixed vertex. In the case G=Z/2, we have the classification for
twistwise flow equivalence. We include some algebraic results and examples related to
the determination of elementary equivalence over ZG.
This involves the algebraic K-theory group K_{1}(ZG).

Periodic Prime Knots and Topologically transitive Flows on 3-Manifolds

Joint with Bill Basener.

Topology and Its Applications,

Volume 153, Issue 8, 1 February 2006, Pages 1236-1240.

In postscript, In PDF, Published version

Suppose that *p* is a nonsingular (fixed point free)
C^{1} flow on a smooth closed 3-dimensional manifold
*M* with *H _{2}(M)=0*. Suppose that

The Linking Homomorphism of One-Dimensional Minimal Sets

Joint with Alex Clark.

Topology and Its Applications, Volume 141, (2004) Pages 125-145.

In postscript, and PDF. Outline with link to software.

We introduce a way of characterizing the linking of one-dimensional minimal sets in three-dimensional flows and carry out the characterization for some minimal sets within flows modeled by templates, with an emphasis on the linking of Denjoy continua. We also show that any aperiodic minimal subshift of minimal block growth has a suspension which is homeomorphic to a Denjoy continuum.

Weapons with Depleted Uranium: Public Risks and Perceptions

This is an unpublished somewhat informal report. It is not a math paper. April 2003. MSWORD

**Abstract:**
Does battlefield residual depleted uranium pose a significant
public health risk? How should activists respond?

Quantum Invariants for Templates

Joint with L. Kauffman and M. Saito.

Journal of Knot Theory and Its Ramifications, Vol. 12, No. 5 (2003) 653-681.

PostScript, PDF.

We define quantum-type invariants for templates that appear in certain dynamical systems. Such invariants are derived from bialgebras and their quantizations called braided Hopf algebras that are defined by Majid. Diagrammatic relations between projections of templates and the algebraic structures are used to define invariants.

Flows with knotted closed orbits

Joint with John Franks.

The Handbook of Geometric Topology, pages 471--497, North-Holland, Amsterdam, 2002.

[663K, postscript], [318K, pdf].

We survey results concerning dynamics of flows on the 3-sphere with special attention to the relationship between dynamical invariants and invariants of geometric topology.

Visually building Smale flows in S

Topology & Its Applications, 106 (2000), no. 1, 1--19.

[1317K, postscript], [620K, pdf].

A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, Smale flows in the 3-sphere.

Positive Knots and Robinson's attractor

Journal of Knot Theory and its Ramifications, Vol. 7, No. 1 (1998) 115--121.

[345K, postscript], [161K, pdf].

We study knotted periodic orbits which are realized in an attractor of a certain ODE first described by Clark Robinson. These knots can be presented so as to have all positive crossings, but may not be restricted to positive braids.

Invariants of twist-wise flow equivalence

Electronic Research Announcements, AMS, Vol. 3 (1997), pp. 126-130.

[254K, postscript], [148K, pdf].

See below.

Knot Factoring

Mathematics Monthly, April 2000.

[590K, postscript], [282K, PDF].

Knots can be factored uniquely into primes, up to order. We hope that our presentation of this classic result will be accessible to advanced undergraduates.

Invariants of Twist-wise Flow Equivalence

Discrete and Continuous Dynamical Systems, Vol. 4, No. 3, July 1998, 475--484.

[435K, postscript], [191K, pdf].

Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.

An invariant for basic sets of Smale flows

Ergodic Theory and Dynamical Systems, Vol 17, 1997, pp. 1437-1448.

[150K, postscript], [151K, pdf]. Also see the Errata, [155K, postscript], [85K, pdf]

We consider one dimensional flows which might arise as a hyperbolic invariant set a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

Positive braids with a half twist are prime

Journal of Knot Theory and its Ramifications, 6 (1997), no. 3, 405--415.

[320K, postscript], [126K, pdf].

We prove that a knot which can be represented by a positive braid with a half twist is prime. This is done by associating to each such braid a smooth branched 2-manifold with boundary and studying its intersection with a would-be cutting sphere.

A Mathematician Reads "Social Text"

Notices of the AMS, October 1996, 1127--1131.

HTML.

An analysis of the issue of the postmodern journal Social Text in which physicist Alan Sokal published his spoof article.

Knots about Stokes' theorem

Journal of College Mathematics, March 1996.

[486K, postscript], [225K, PDF] .

Connections between knot theory and Stokes' Theorem as presented to students in a sophomore level vector calculus class.

A zeta function for flows with positive template

Topology & Its Applications, 66 (1995) 199-213.

[232K, postscript], [214K, pdf].

A zeta function for a map f:M ---> M is a device for counting periodic
orbits. For a flow however, there is not a clear meaning to the period
of a closed orbit. We circumvent this for hyperbolic 3-flows which have

Educating Dilbert

Undergraduate Mathematics Education Trends, March 1995.

HTML.

A commentary on calculus reform efforts.

The prime decomposition of knotted periodic obits in dynamical systems

The Journal of Knot Theory and its Ramifications , Vol. 3 No. 1 (1994) 83-120.

[479K, postscript], [317K, pdf].

Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.

Composite knots in the figure-8 knot complement can have any number of prime factors

Topology and Its Applications, 55 (1994) 261-272.

[199K, postscript], [143K, pdf],

We study an Anosov flow in the complement of the figure-8 knot in the 3-sphere. In 1983 Joan Birman and Bob Williams conjectured that the knot types of the periodic orbits of this flow could have at most two prime factors. Below, we give a geometric method for constructing knots in this flow with any number of prime factors.

Prime decomposition of knots in Lorenz-like templates

Journal of Knot Theory and its Ramifications , Vol. 2, No. 4 (1993) 453-462.

[189K, postscript], [158K, pdf] (some figures missing).

R. F. Williams showed that all knots in the Lorenz template are prime. His proof included the cases where any number of positive twists were added to one of the template's branches. However, Williams does give an example of a composite knot in a template with a single negative twist. We will show that in all the negative cases composite knots do exist, and give a mechanism for producing many examples.