Zachary Bryhtan

Iowa ยท zacha-remove-ry-bryhtan@uiowa.edu

Hey! My name is Zach and I am a fifth year Mathematics PhD student at the University of Iowa. My interests in many areas of math including commutative algebra, combinatorics, and knot theory. I am currently working with my advisor, Prof. Isabel Darcy, to complete a database of 2- string tangles with up to 10 crossings (think KnotInfo).

Socials

Education

University of Iowa

Doctor of Philosophy
Mathematics

Expected May 2024

August 2019-Current

University of Iowa

Master of Science
Mathematics

August 2019-May 2021

University of Wisconsin Parkside

Bachelor of Science
Mathematics and Physics

Completed Minor in French

August 2014-May 2019

Publications

A collection of articles, presentations or talks.

Talk - Tangle Tabulation

The classification of knots, links, and tangles has applications to fields including molecular biology where proteins are modeled by tangle equations. Online databases contain millions of knots and links including diagrams, notations, and properties. Tangles have not been classified to the same extent despite their close relation to knots and links and usefulness in applications. To build a database of 2-string tangles, we use a Conway-like notation to count and generate tangles using known canonical forms for rational, Montesinos, and generalized Montesinos tangles as building blocks.

October 2023

Poster - Classifying Non-Rational 2-String Tangles

Classifying knots, links, and tangles is an ongoing process with applications to fields including molecular biology where protein actions may be modelled by tangle equations. For knots and links, there are databases which list thousands of these shapes including diagrams, notations, and properties. Tangles have not been published to the same level so, while properties can be found for a simple category of tangles (rational tangles), information is still needed for the more complex tangles. To help build a database of 2-string tangles, we use a matrix-like notation to describe any tangle which is more easily interpreted and manipulated by a computer then translated into a more descriptive and common notation (Conway notation).

November 2022

Conferences

American Mathematical Society’s Fall Central Sectional Meeting

2023

Special Session on Applied Knot Theory. See more about the event here. Slides presented here.

8th Mexican Workshop on Applied Geometry and Topology

2022

Workshop over several days with talks covering topics including applications of topology to biology and algebraic geometry to statistics. See more about the event here. Poster presented here

Topological Data Visualization Workshop

2022

Workshop over several days with talks covering applications of topological data analysis in many disciplines. See more here.

University of Wisconsin System Research in the Rotunda

2019

Undergraduate students gather to present research they are doing in many areas of science. See more here.

Teaching

MATH:1005 College Algebra

‘‘Algebraic techniques, equations and inequalities, functions and graphs, exponential and logarithmic functions, systems of equations and inequalities.’’

MATH:1350 Quantitative Reasoning for Business

‘‘Algebraic techniques and modeling; quantitative methods for treating problems that arise in management and economic sciences; topics include algebra techniques, functions and functional models, exponential and logarithmic functions and models, and a thorough introduction to differential calculus; examples and applications from management, economic sciences, and related areas.’’

MATH:1380 Calculus and Matrix Algebra for Business

‘‘Quantitative methods for treating problems arising in management, economic sciences, related areas; introduction to differential and integral calculus, systems of linear equations and matrix operations.’’

MATH:1440 Mathematics for the Biological Sciences

‘‘Relations, functions, coordinate systems, graphing, polynomials, trigonometric functions, logarithmic and exponential functions; discrete mathematics, probability; examples and applications from biological sciences.’’

MATH:1460 Calculus for the Biological Sciences

‘‘One-semester survey of calculus for students in biological or life sciences; nontheoretical treatment of differential and integral calculus; brief introduction to differential equations and probability with calculus, with applications to the life sciences.’’

MATH:1560 Engineering Mathematics II: Multivariable Calculus

‘‘Vector geometry; functions of several variables; polar coordinates; partial derivatives, gradients, directional derivatives; tangent lines and planes; max/min/parametric curves, curvilinear motion; multiple integrals; vector fields, flows; integration on curves, work; divergence, flux, Green’s theorem.’’

MATH:1860 Calculus II

‘‘Techniques of integration including by-parts, trigonometric Integrals, trigonometric substitutions, partial fractions, improper integrals; applications (i.e., arclength), area surfaces of revolutions, application to physics; introduction to differential equations; parametric equations and polar coordinates; infinite sequences and series, convergence tests, power series, Taylor polynomials and series.’’

PSQF:1075 Educational Psychology and Measurement

Guest lecture covering approaches to classroom management including benefits of positive learning environments and how to conduct yourself, establish rules, handle discipline to create and maintain a positive learning environment.

Blog

Here you will find a spattering of things that I have done, seen, or thought about which may or may not be math related.

Need Helpful Math Papers?

I was introduced to Keith Conrad when taking my first course in Abstract Algebra. His expository papers in Group and Ring theory were incredibly useful to me, and beyond that I was able to investigate so much more beyond course content in a digestible way. If you have not had the pleasure of perusing his expository papers, I encourage you to do so (they are all posted here )!
April 19, 2023

A Guide to Conway Notation

I first encountered J. Conway’s notation for knots the way I imagine most people do, by reading his original article “An enumeration of knots and links, and some of their algebraic properties.” While this was sufficient for a basic understanding, I found that the notation had evolved somewhat since 1970. I could not find a single convenient source covering common conventions, or the extensions discussed by A. Caudron in his paper “Classification des noeuds et des enlacements” (written completely in French).
March 29, 2023