Josh Krienke

Currently based both in the St. Paul/Minneapolis area, Josh is a mathematician interested primarily in low-dimensional topology and its connections with other fields of math. He just graduated from Bard College with a BA in Math, and will be attending University of Nebraska-Lincoln in the Fall as a graduate teaching assistant and PhD student. He has participated in a variety of research projects, including through the Polymath Jr. program in 2023, through the CSU Chico REU in 2024, and through the Bard Summer Research Institute in 2025. Additionally Josh recently completed his year-long Bard Senior Thesis “The Front Multicrossing Number of Links” under the advisement of Dr. Caitlin Leverson.

Josh also has a deep love and commitment for teaching math. He has spent the last three years as a math tutor in various capacities, including as a Bard course tutor, Bard one-on-one tutor, Bard math study room tutor, and a private tutor. He prides himself on being a patient teacher, and for leaving space for questions and curiosity. His goal is for students to leave sessions with just as much intuition for solving further problems as they have clarity on how to apply the tools necessary to solve the problems at hand.

Research

The Front Multicrossing Number of links

In the 2024-25 school year I completed my Bard Senior Thesis under the advisement of Dr. Caitlin Leverson, who is a knot theorist and Assistant Professor at Bard College. In the project, I defined a particular projection of knots, called the “front multicrossing projection”, a front projection which allows multicrossings. By considering the smooth knot type of a knot with such a projection, I then defined the front multicrossing number of a knot (roughly: the minimal number of multicrossings in any front multicrossing projection of a smooth knot). Building off of work from Amit Kumar, Jake Murphy, and Brian Naff, who showed that any smooth knot with a front multicrossing number of 1 must be an unknot, I classified all links with a front multicrossing number of two (of which there are many infinite families).

See the contact form below if you have any further questions about our results. The final senior project will be up in the Bard digital commons, and is currently being revised so that it can be submitted for publication.

CSU, Chico REU 2024

In the Summer of 2024 I worked with John Lind on exploring the relationships between knots and graph Laplacians. By taking the dual graph of an orientable, planar surface with a knot as its boundary (assigning regions to vertices of the graph and crosses to edges of the graph), one can take the graph Laplacian of the dual graph, and by deleting a number of rows and columns equal to 1 plus the number of tracer circuits used to obtain a planar surface, recover a Seifert Matrix of the knot. We discovered this relationship shortly before finding a suite of papers by Daniel Silver and Susan Williams which proved these results in 2018. From there, we pivoted and worked on proving the result at the level of homology and applying the relationship between graph Laplacians and Seifert Matrices to reframe results about the order of the first homology group of the n-fold cyclic branched cover of the knot complement in terms of graph Laplacians.

See the contact form below if you have any further questions about our results.

Polymath Jr. 2023

In the Summer of 2023 I worked with Alex Zupan, Jeffrey Meier, and Evan Scott on computing the ribbon numbers of 12 crossing ribbon knots. Of the 108 ribbon knots we were given, we ultimately found the ribbon number for 74 of them (along with finding close upper and lower bounds for the remaining 34 knots). We used a variety of known tools to make initial bounds on the ribbon numbers for the knots we were given, but ended up spending most of our time developing new tools to bound the ribbon numbers. Most importantly, we calculated the set ℜ₄ which is the set of all Alexander Polynomials for ribbon knots of ribbon number less than or equal to 4. As part of this process I created a maximal table of over 350 ribbon codes with ribbon number 4 or less. After organizing the list by Alexander Polynomial, I found another unique move on ribbon codes which we defined as a leaf isotopy. Through the leaf isotopy and other isotopies between ribbon codes, we reduced the list down to at most 118 inequivalent, indecomposable, irreducible ribbon codes. As further research, we are interested in finding a complete set of moves on ribbon codes to show when two are the same.

See the contact form below if you have any further questions about our results. Our article has been posted here, and will be submitted for publication soon!

Teaching

Tutoring Experience

I’ve been tutoring since the Fall of 2022 in a number of different roles. I started as a tutor in Bard’s Math Study Room, on call twice a week to help students with any number of classes from Calculus to Abstract Algebra to Real Analysis, etc. I also worked with students one-on-one through Bard during the 2023-2024 academic year, meeting with students individually to help with classes such as Calculus 1 & 2, Proofs and Fundamentals, Elementary Linear Algebra, and Real Analysis. In the Fall of 2023 I was a dedicated course tutor for Calculus 1 & 2, and in the Spring of 2024 I was a dedicated course tutor for Proofs and Fundamentals and Elementary Linear Algebra. Additionally, I worked as a private tutor weekly online with a student in high school algebra through the 2023-2024 academic year. Between my last two years at Bard I tutored 98 different people official a total of 499 times, which means students returned for an average of over 5 times to work with me.

Classes I’ve Taken

Below is a list of math classes I’ve taken and tutored

• Senior Project (2 semesters)

• Abstract Algebra II Tutorial

• Knot Theory (Audit) (*)

• Complex Analysis (*)

• Advanced Ordinary Differential Equations (Audit)

• Point-Set Topology Tutorial

• Algebraic Curves (*)

• Research in Finite Geometry Tutorial

• Real Analysis (*)

• Abstract Algebra (*)

• Vector Calculus (*)

• Arithmetic of Listening (*)

• Proofs and Fundamentals (*)

• Elementary Linear Algebra (*)

• Calculus II (*)

• AP Calculus AB (*) (**)

• AP Statistics (**)

• Precalculus (*) (**)

• Algebra 2 Advanced (*) (**)

• Geometry Advanced (*) (**)

(*) indicates whether I’ve tutored students for the class before (note that I’ve also tutored for classes I haven’t taken so this isn’t comprehensive)

(**) indicates a high school class

Math Talks

I’ve given a number of math talks, listed below:

• Constructing Sidon Sets in Finite Fields Using Conics (Pt 1 & 2) - two 30 min presentations in collaboration with Sebastian Sargenti - BSRI - 5+ in-person - June 13th & 20th, 2025

• On the Correspondence Between Sidon Sets and Linear Codes - 30 min presentation - BSRI - 5+ in-person - June 6th, 2025

• The Front Multicrossing Number of Links - 20 min presentation - Senior Project Capstone - 10+ in-person - May 20th, 2025

• A Theory of Modality - 50 min presentation - Bard Spectra Math Seminar (Math Mondays) - 15+ in-person - March 3rd, 2025

• Bridges Between Laplacian Matrices and Knot Invariants - 10 min presentation - JMM 2025 - AMS Contributed Paper Session on Topology, I - 15+ in-person - January 8th, 2025

• Avoiding Triplets in the Card Game Spot-It! - 20 min presentation in collaboration with Oliver Vanderploeg - AMS Fall Eastern Sectional Meeting in Albany - AMS Special Session on Topics in Recreational Math and Finite Geometry, I - 20+ in person - October 19th, 2024

• CSU Chico REU Final Presentation - 20 min presentation - CSU Chico REU - 15+ in-person - August 7th, 2024

• Ribbon Knots, Numbers, and Codes - 60 min presentation - CSU Chico REU - 15+ in-person - June 18th, 2024

• Characterizing the Topology of Singularities - 10 minute presentation - Algebraic Curves capstone - 10+ in-person - May 20th, 2024

• Ribbon Knots, Numbers, and Codes - 20 min presentation - Bard Math Seminar - 20+ in-person - September 20th, 2023

• Exploring Ribbon Knots - 20 min presentation in collaboration with Minyi Liang, Samuel Lowry, and Ege Malkoc - Polymath Jr. capstone - Audience of 100+ online - August 12th, 2023

For information about any of the topics or to request my slides, see the contact form below.

Bard Spectra Math Club

In the Spring of 2024, I took on the responsibility of co-club head of the Bard Spectra Math Club. We hosted community building events such as game nights or the Pi Day events at Bard. In addition, when the Bard Math Seminar stopped running, we began a lecture series in the Fall of 2024 with two invited speakers that semester.

With the proof of concept run of the series having been successful, we turned the lecture series into a weekly “Math Monday” seminar in the Spring of 2025. We had four student speakers and three invited speakers, some of whom gave talks over Zoom and some of whom came to Bard in-person. Attendance averaged to about ten students per seminar, all with a wide range of backgrounds. We did our best to make the talks accessible, and were relatively successful in priming speakers to give thoughtful talks. It was also a wonderful chance for students to practice long-form lectures (up to an hour long) on topics that weren’t necessarily related to their classes. The intention was to construct a space where students can feel comfortable sharing their interests, and have an outlet to remind each other why we love math—particularly in the trenches of finals or otherwise inevitable periods of stress. I tended to host the talks, while my co-club heads helped with scheduling and finances (and will hopefully be continuing the seminar next year!).

 Resume

(Updated last on December 5th, 2024)

Creative Writing

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