Josh Krienke

Currently based both in Minnesota’s Twin Cities and New York’s Hudson Valley, Josh is a mathematician interested in a broad range of geometric fields of study. His primary interest lies in topology, and more specifically in knot theory. He spent the Summer of 2023 studying ribbon knots with Alex Zupan and Jeffery Meier through the Polymath Jr. Research Program. Then, in the Summer of 2024 Josh participated in the CSU Chico REU studying connections between knot invariants and graph Laplacians with John Lind. In the Fall of 2024, Josh will begin his senior project at Bard College under the advisement of Caitlin Leverson in a topic related to knot theory. Additionally, Josh has done research in finite projective geometry with Lauren Rose and on connections between algebraic curves and knots with Charles Doran.

Josh also has a deep love and commitment for teaching math. He has spent the last two 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

Bard Senior Thesis

In the 2024-25 school year I will be working on my Senior Thesis as part of the requirements for completing the math major at Bard College. I will work under the advisement of Caitlin Leverson, who is a knot theorist and Assistant Professor at Bard College. I am currently in the early stages of researching multicrossing front projections of Legendrian knots. In particular, I’m working on classifying Legendrian knots with front projections that have 2 multicrossings.

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 Real Analysis to Algebraic Curves (in the Spring of 2024 I tutored 12 unique classes, for reference). I’ve also been a one-on-one tutor through Bard since the Spring of 2023, 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. Between all of my tutoring responsibilities at Bard, I tutored 29 unique students on 100 different occasions in the Fall of 2023, and 36 unique students on 102 different occasions in the Spring of 2024. Additionally, I worked as a private tutor weekly online with a student in high school algebra through the 2023-2024 academic year. I plan to continue working privately with students and in the Bard Math Study Room through the 2024-2025 academic year.

Classes I’ve Taken

Below is a list of math classes I’ve taken.

• 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

I am currently taking Complex Analysis and working on my Senior thesis (which counts as a 4 credit course). In addition, I am auditing Advanced Ordinary Differential Equations.

Math Talks

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

• 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.

 Resume

Resume available upon request. An updated resume is coming soon.

Creative Writing

To view my compositions, prose, and album reviews, click below