# Biquandle Knot Invariants

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## Papers and Presentations

**The Structure of Biquandle Brackets (with Adu Vengal and Will Hoffer)**

Paper (published in the Journal of Knot Theory and its Ramifications), JKTR DOI, on the arXiv.

Poster (presented at the Young Mathematicians Conference, 2018).

**Categorifying Biquandle Brackets (with Adu Vengal)**

Paper (published in the Journal of Knot Theory and its Ramifications), JKTR DOI, on the arXiv.

My honors thesis at OSU (simplified version of above paper, includes Mathematica package documentation).

Slides (presented at the Young Mathematicians Conference, 2019), on Sergei Chmutov's webiste.

Poster (presented at the Joint Mathematics Meetings, 2020).
## Mathematica Packages

Biquandles.wl

BiquandleBrackets.wl

Biquandle2Cocycles.wl

BiquandleBracketCanonical2Cocycle.wl

You can also download this .ZIP file, which contains all of the above packages (and a bonus one).

Then, you can extract it into your "Applications" folder in your Mathematica install folder, and you can import all of the packages by typing "<<BiquandleInvariants`" (without the quotes) in Mathematica.
## Conjectures and Questions

**1:** If \(X\) is a biquandle and \(\beta\) is a biquandle bracket over \(X\), then \(\beta\) is still a biquandle bracket over the trivial biquandle on the underlying set of \(X\).

**2:** How does the biquandle homology invariant \(\operatorname{Bh}_\beta\) compare to the biquandle bracket invariant \(\Phi_X^\beta\)?

**3:** Does a true Khovanov homology-style categorification of the biquandle bracket exist?