Sam Nelson -- My Research Students

Since 2004 I have been publishing papers cowritten with with undergraduate students. I didn't set out to become an undergraduate research mentor; it just sortof happened. The facts are simply that (1) I'm perfectly willing to work with motivated students with any level of preparation and (2) nearly all my students have been undergrads. It follows that I do a lot of projects that involve undergraduate research students.

Gabe It all started during my first year as a VAP at the University of California at Riverside. One afternoon, a former student from my Vector Calc II class from the previous quarter dropped by my office hours to say "hi" and ask me about undergraduate research. He wanted to know, was it really possible for an undergraduate to contribute to research?

Having been a McNair Scholar as an undergrad myself, I replied that undergrad students can definitely contribute to research. The student, Gabriel Murillo, asked if I had any project ideas he could work with me on. I said I'd see what I could come up with, and within a few days we had a project idea -- continuing the computation of isomorphism classes of finite Alexander quandles using a theorem from my dissertation. We worked on the project over the summer, completed the required computations, and wrote up a paper which we posted to arxiv.org and submitted for publication. It has now appeared the Journal of Knot Theory and its Ramifications.

Benita Toward the end of that summer, I ran into a former student from my topology class, Benita Ho, at the gym. When I mentioned that I was finishing a paper I was writing with a student, she asked if I had any other project ideas. I said I'd see what I could come up with, and after a few days I contacted her with a project idea involving representing finite quandles symbolically as matrices. We met informally about once per week during the fall quarter of my second year at UCR, and we posted our paper to arxiv.org on my 30th birthday. It has appeared in the electronic journal Homology, Homotopy and Applications.

Anthony The spring quarter of my second year at UCR, Gabe was taking my topology class and wanted to do a second project with me; we settled on trying to find a method for determining whether a finite quandle is isomorphic to an Alexander quandle. One day Gabe was waiting outside my office working on the project when Anthony Thompson, a classmate from my topology class, stopped to ask what he was doing. Gabe explained the project and Anthony decided to stick around for our meeting. Indeed, Anthony later told me he stayed up all night thinking about the problem, even missing a few classes the next day as a result. We devised and implemented an algorithm which finds all Alexander presentations of a finite quandle from its matrix representation. The resulting paper has now appeared in J. Knot Theory Ramifications.

Todd That spring quarter, another former student, Todd Macedo, asked me about doing a project. Todd was a computer science major, so I put him to work on a distributed algorithm version of the finite quandle computer search program from my paper with Benita. We enlisted the aid of my friend Richard Henderson of Red Hat Software, who offered to run our search on his personal network. After an initial run of several hours on several processors, Richard modified our algorithm such that the n=6 case completed in under 2 seconds on a single processor, and was able to get the n=7 and 8 cases, while the n=9 case is still out of range even with a large network. Our paper has appeared in the Journal of Symbolic Computation.

Natasha Meanwhile, a freshman from my calculus class that spring, Natasha Harrell, started showing up to my research meetings with Anthony and Gabe. Soon enough, she asked me for a project idea. Around the same time, another former student from a vector calc class, John Vo, asked me about doing a project. By this time I had started keeping a running list of project ideas I thought were suitable for joint work with undergrads. Anthony wanted to do another project, as did Rohit Jain, another former calculus student. I sent the four of them a list of four project ideas to choose from.

Natasha chose the project idea of using quandle difference invariants to detect nonclassicality in virtual knots and links. We wrote some Maple programs using the method for computing the quandle counting invariant symbolically from a quandle matrix developed in my paper with Richard and Todd together with a program for generating all 4-crossing Gauss codes, and we found that some 85% or so of non-evenly intersticed 4-crossing Gauss codes with nontrivial counting invariant values for the six smallest connected quandles have nonclassicality detected by quandle difference invariants. Our paper appeared in Topology Proceedings following our presentation of our work at the 2006 Spring Topology and Dynamics conference in North Carolina.

John John chose the project of extending the symbolic matrix representation of quandles from my paper with Benita to biquandles, a generalization of quandles. We were able to classify all biquandles with two, three and four elements using our method as well as write Maple software for computing the biquandle counting invariants for any knot or link given its Gauss code. One of the four element biquandles we found detects the non-triviality of all of the Kishino knots, not an easy thing to do. Our paper has appeared in Homology, Homotopy and Applications.

Rohit opted to return to his native India and pursue a career in medicine, a loss for mathematics but a gain for medicine. Anthony chose to work on an open-ended kind of project about studying Latin quandles, i.e. finite quandles whose operation matrices form Latin squares. Such quandles are also distributive quasigroups, and we read a few papers from the quasigroup literature but have not yet been able to prove our main conjecture, namely that all Latin quandles are Alexander. We know from our computer searches that the conjecture is true at least up to order 8. Unfortunately, we didn't feel that we had enough new material to justify a paper, and when Anthony graduated, we put the project on hold.

The winter quarter of my third and final year at UCR saw several more requests from students for project ideas. John and Natasha both wanted to do a second project, and two new students, Conrad Creel and Daisy Lam, asked me for project ideas. Conrad was the first student to ask me for a project without having taken a class with me, though not the last. Daisy had taken my linear algebra class the previous summer.

John and I enlisted the help of Jim Dolan, a member of UCR's occasional quandles group (other members included John Baez, Xiao-Song Lin, Alissa Crans and Derek Wise) who was studying a connection between the roots of the Jones polynomial and the fundamental Alexander quandle of a knot. We embarked on a project to define an extension of the Kauffman bracket polynomial using a virtual crossing as a kind of smoothing. We were able to define the invariant but unable to determine whether the new invariant was really new, and we ultimately decided to table the paper.

Daisy Daisy and I tried out several research ideas before finally settling on the idea of trying to extend my classification theorem for finite Alexander quandles to finite Alexander biquandles. We were able to show that two Alexander biquandles are isomorphic iff their (1-st) submodules are isomorphic and they have sets of coset representatives satisfying certain extra criteria. These extra criteria are always satisfied if the Alexander biquandles are actually Alexander quandles, but need not be for general Alexander biquandles. The resulting paper has appeared in the International Journal of Mathematics.

Conrad Conrad Creel, who was working as a software developer as well as a student of mathematics, proposed working with me on a symbolic computation project relating to knot theory. This time I suggested we write software to compute Yang-Baxter cocycles of finite biquandles and the knot and link invariants they define, using the symbolic matrix representation of finite biquandles from my paper with John Vo. The cool thing about this is that the software works in an algebra-agnostic way -- you don't actually have to know formulas for the biquandle operations, just the biquandle matrix. Our paper has appeared in J. Symbolic Computation.

For my second paper with Natasha, we decided to look at the counting invariants associated to quandles with trivial orbits. We found that we could use the counting invariants associated to a specific family of quandles to recover the linking number of a two-component link. This is of interest since, as a complete invariant of knots and unsplit links in S³ up to reflection, the knot quandle should determine nearly all of the other invariants. In particular, understanding how various knot and link invariants arise from the knot quandle has the potential to tell us a lot about how these invariants are related to each other. Our paper has appeared in the Journal of Knot Theory and its Ramifications.

Adam For the final summer that I spent at UCR, I had one more student who asked for a project, Esteban Adam Navas. Like Conrad, Adam had not taken any classes from me, though he did sit in one one of my calculus lectures after being invited by a friend who was in my class. For this project we decided to study a type of finite quandle defined in terms of a symplectic form on a finite vector space, which we called "symplectic quandles" (though we've since learned that these are also called "quandles of transvections"). We were able to prove some results about these quandles and use their extra structure to enhance the quandle counting invariant. Our paper has appeared in the Osaka Journal of Mathematics.

After reaching the three year limit as a VAP at UCR, I returned to Whittier College for a year. There I met two more students who wanted to do undergraduate research.

Jacquie Jacquelyn Rische was a student from my abstract algebra II class who had some previous research experience -- she had done an REU project on number theory and error-correcting codes the previous summer, for which she won a prize at the Joint Meetings in 2007. While discussing my recent work with Adam, we decided to try to generalize the symplectic quandle definition to finite biquandles. After an initial computer search to find the appropriate form for the operations, we were able to prove a number of results about these bilinear biquandles as we call them (since the form in general is bilinear and not always symplectic) and we were able to define biquandle versions of the symplectic quandle invariants from my paper with Adam. Our paper has appeared in Colloquium Mathematicum.

Jose Jose Ceniceros was a student from my combinatorics class at Whittier with excellent taste in music. We originally set out to extend my classification theorem for finite Alexander quandles to finite Alexander virtual quandles, but when this turned out to be easy, we decided to try my backup plan of extending Yang-Baxter cocycle invariants to virtual biquandles. We successfully defined an infinite family of invariants of virtual knots and links using pairs of compatible cocycles from the Yang-Baxter cohomology and a new "S-cohomology" theory. These invariants reduce to the ordinary Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. Our paper has appeared in Transactions of the American Mathematical Society.

Ryan Following my second year at Whittier College, I spent one year as a visiting assistant professor at Pomona College. As a regular member of the Claremont Topology seminar since spring 2003, I was no stranger to Pomona College. After inviting students from my classes to come to my seminar talk, I started working with Ryan Wieghard, a freshman in my linear algebra class, on a project on finite Coxeter racks and their enhancements of the rack counting invariants. Our paper has appeared in J. Knot Theory Ramifications.

That spring, when the tenure-track position I had accepted at California State University, Dominguez Hills was revoked due to budget issues, I was quickly offered a VAP position for one year at neighboring Claremont McKenna College, which I was happy to accept.

Tim Toward the end of my second and final semester at Pomona College, a student from my Topology class, Tim Carrell, asked me to be his senior thesis advisor, a role I found myself informally playing for Jose as well. I agreed and we set out to do some summer research as a warm-up for Tim's upcoming thesis research that fall. We were able to settle a number of conjectures as well as define a new family of link invariants using generalized rack polynomials. Our paper has appeared in the Journal of Algebra and its Applications.

Johanna Early in my visiting year at CMC, a student from neighboring Scripps College, Johanna Hennig, dropped by my office to ask me to be her senior thesis advisor. We decided to look at an enhancement of the birack counting counting invariants using finite groups we call "column groups". Our paper has appeared in the Journal of Knot Theory and its Ramifications.

While taking a year off before graduate school, Jose suggested we start work on a second paper. We selected a couple of papers to read and started meeting semi-regularly. After our initial idea started to fizzle, we hit on the idea of using shadow colorings to extend our previous work on virtual Yang-Baxter cohomology.

Jessica That fall, my honors calculus III course included CMC student Jessica Ceniceros, who had taken my calculus I class at Pomona the previous year and who had attended a few of my meetings with Jose, who happens to be her brother. Jessica and I decided to start a project of our own that fall; alas, like my work with Daisy, we ended up going through two complete project ideas before hitting on one that looked likely to work out in a practical way -- the first turned out to be equivalent to previous work, while the second invariant gave us trivial values on the knots smalls enough for our software to compute quickly. The third idea involves enhancements of the rack counting invariants using (t,s)-racks. Our paper has appeared in the International Journal of Mathematics.

Wesley Another student from my honors calculus III class, Wesley Chang, asked to do a project. I knew Wesley was a high school student, but it wasn't until our third research meeting that I realized he was still a junior in high school! Starting in the spring of 2009, we considered a few ideas before deciding to apply the shadow coloring idea to quandle and rack based counting invariants. Before Wesley finished at Claremont High, I was delighted to be able to inform him that our paper has appeared in J. Knot Theory Ramifications.

Tim's senior thesis project involved surface biquandles, the algebraic structure determined by dividing knotted surfaces in 4-space into semi-sheets at the singular set and getting axioms determined by the Roseman moves. Tim was able to show that, as is the case with quandles, surface biquandles are just biquandles. Tim's work was sufficiently independent that we felt Tim should adapt his senior thesis to a solo paper; meanwhile I started receiving emails from knot theorist friends in England and Japan asking for copies of Tim's thesis.

Tim and Johanna both gave excellent talks at the 2009 Pacific Coast Undergraduate Mathematics Conference at the University of California at Riverside. Wesley, Jose, Jessica and I came along to lend support as well as to see the other talks.

As the spring semester wore on and I hadn't had a single job interview, I started to prepare for an eighth year as a visiting professor. Then one morning while preparing my lectures, my department chair stopped by to tell me I should be hearing from the Dean. Much to my surprise, Claremont McKenna College had decided to create a new tenure line in order to offer me a tenure-track position. I could not have been happier to accept.

Scott Not long after that happy day, I received a letter from a student at the University of Wisconsin, Madison. Scott Pellicane had read several of my papers (as well as this very webpage!) and wanted to know if I would do a project with him if came to Claremont for the summer. I said "sure" and sent a list of possible project ideas. We decided to study the structure of a biquandle from my paper with Daisy Lam, which I had since noticed was an example of what Allison Henrich and I had decided to call semiquandles. Ultimately Scott uncovered a connection between Latin semiquandles, finite linear switches and finite Weyl algebras, drawing on the work of Roger Fenn and Vladimir Turaev. Our paper is in preparation.

In July of 2009, I gave an invited address at the UnKnot conference at Dennison Univeristy in Ohio, accompanied by Jose, Jessica and Scott. All three students gave outstanding talks about our various projects.

Jessica gave poster presentations about our work on (t,s)-racks in both the Claremont Colleges fall poster session in 2009 and at the AMS/MAA Joint Meetings in New Orleans in January 2010. Jessica also accompanied me to the spring 2010 Knots in Washington conference at George Washington University in Washington, D.C., where she gave a talk on our work on (t,s)-racks. While there, she found an idea for her senior thesis project while watching a talk on twisted virtual knots by my friend Naoko Kamada.

Garret
and Quingcheng Jonah and Aaron During the summer of 2010, I participated for the first time in a formal REU (Research Experiences for Undergraduates) summer program. The Claremont Colleges were running several REU projects with a grant from the NSF, and for the first time ever the Claremont Center for the Mathematical Sciences was also running summer REU projects with funding from the Fletcher Jones Foundation (No, it's not related to the SoCal car dealership :) During the first eight weeks of the summer, I worked with CGU student Garret Heckel, Claremont McKenna students Aaron Haas and Jonah Yuen, and Pomona College student Quingcheng Zhang on a project about extending quandle module knot invariants to the case of non-quandle racks. Our paper, which we call the HHNYZ paper, has appeared in the Osaka Journal of Mathematics.

Sinan During the summer of 2010, I received an email from University of Chicago undergrad Sinan Aksoy, a friend of my previous research student Wesley Chang from Claremont High, who also happens to be the son of my CMC Math colleague Asuman Askoy. Sinan would be in Claremont for rest of the summer after the REU and wanted to work with me on a project. We decided to look at BiKei and involutory biracks, in the process answering a question asked of me by Xiao-Song Lin a few years earlier: can biquandle counting invariants detect invertibility (also known as reversibility) in oriented knots and links? The answer is yes, as we were able to show. Our paper has appeared in J. Knot Theory Ramifications. I gave a well-received talk on my work with Sinan at Knots in Washington in the spring of 2011.

Mike Back in the fall of 2009, Mike Grier, a Pomona College student from my linear algebra class back in fall 2007, contacted me to ask about doing a project which would eventually become his senior thesis. A Math and Psychology double major, Mike was writing two senior thesis, both involving original research. We started meeting regularly to work on the project during the spring of 2010 and continued through the fall of 2011. After first looking at some ideas involving skein invariants, we settled on a project on kei algebras and the invariants of unoriented links we can get from them. Our paper has appeared in Homology, Homotopy and Applications.

As Jose was finishing his Master's degree at Cal State Los Angeles and applying for PhD programs, my friend Mohamed Elhamdadi at the University of South Florida in Tampa suggested we invite Jose to join in a project we were working on involving extending N-degeneracy in rack homology to the case of birack homology. Jose was interested, and the two of us flew out to Tampa during fall break for a few days of research. The conversations there ultimately resulted in a joint paper with Mohamed and his student Matt Green on augemnted birack homology in 2013; it has appeared in the International Journal of Mathematics.

Meanwhile, Jessica and I started our work on twisted virtual biracks for her CMC senior thesis. Where virtual knots and links can be understood in terms of simple closed curves in thickened orientable surfaces (also known as "trivial I-bundles"), if we allow non-orientable surfaces such as the real projective plane or the klein bottle, we get twisted virtual knots and links. Jessica gave a talk on twisted virtual biracks at the Pacific Coast conference in the spring of 2011. Our paper based on portions of her senior thesis has appeared in the journal Topology and its Applications.

Gina Late in the spring of 2010, Gina Bauernschmidt, another Pomona College student from the same linear algebra class where I met Ryan Weighard and Mike Grier, contacted me to ask about doing a project which would eventually become her senior thesis. We started discussing project ideas during the spring and summer of 2010, eventually deciding to extend the rack module idea from the HHNYZ paper to the case of biracks. Our paper has appeared in the journal Communications in Contemporary Mathematics.

Katie During the summer of 2010 I was contacted by Katie Pelland, another rising Pomona College senior, about advising her senior thesis. Pomona College's Mathematics department imposes a limit of three senior thesis students per faculty per year, and Katie would be my fourth senior thesis student for the 2010-2011 academic year...but fortunately, I work for CMC, not Pomona, and thus was not bound by the Pomona limit :) Katie and I ultimately decided to extend the birack module idea from Gina's project to include shadow labelings, thus defining what we call the Shadow Algebra and Shadow Modules. Our paper has appeared in J. Knot Theory Ramifications.

Aparna During the summer of 2011, I worked with Pomona college student Aparna Sarkar on a project with funding from CMC. Initially we set out to generalize the Lie rack idea from Fenn and Rourke to the case of biracks. We enlisted the help of my frequent partner in conference session organizing, Alissa Crans, who knows much more about Lie Algebras than I do. Alissa's location at LMU gave us a great excuse to have research meetings in places like the Santa Monica pier and Mel's Diner in Hollywood. The Lie Algebra idea evolved into something we called 'rack magmas', which we then found were a special case of things called 'dynamical cocycles.' Our paper has appeared in the New York Journal of Mathematics.

Catherine,
Susan, Jackson and Nikki During the summer of 2011 I directed a project with the Claremont Colleges Mathematics REU (Research Experiences for Undergraduates) site with funding from the NSF (National Science Foundation). My student partners for this project were Jackson Blankstein, Catherine Lepel, Susan Kim and Nicole Sanderson. Together we extended the ideas from Katie and Jessica's senior thesis projects to define virtual shadow and twisted virtual shadow module invariants. Our paper has appeared in the International Journal of Mathematics. Catherine, Jackson and Nikki each reported giving talks or posters about our project (Susan had graduated), and Nikki's poster even won a prize at the AMS/MAA Joint Meetings in 2012. Congrats!

Emily In 2012 I directed senior thesis projects for two students, one from Scripps and one from Pomona. Emily Watterberg was a student from my Modern Geometry course in the fall of 2010 who asked me about doing a senior thesis. We decided to extend the idea from my paper with Alissa and Aparna on rack dynamical cocycles to the case of biracks. Along the way, we found a way to dramatically simplify the idea by thinking in terms of birack homorphisms, effectively turning any birack homomorphism into an enhancement of the counting invariant. Our paper has appeared in J. Algebra Applications.

Evan Evan Cody was a Pomona College student who approached me about doing a senior thesis back in 2010 before spending a semester in France, "The Land of Galois". We decided to try the idea of looking at birack modules with polynomial entries, which lend themselves naturally to defining customized Alexander polynomials of birack-labeled links. Our paper has appeared in Topology and its Applications.

During the 2011-2012 academic year I came up for early tenure at CMC. Normally the tenure decision happens at the end of the sixth year of one's assistant professor appointment, but my department decided that my nearly twelve years of teaching full courses and my large number of publications merited an early tenure decision. Moreover, while it was technically only my third year of tenure-track at CMC, I had been teaching at CMC for four years and at the Claremont Colleges for five, so it wasn't really a big stretch. I was delighted one afternoon to receive a text message from the Dean of the Faculty letting me know that the committee had unanimously voted to grant my tenure and promote me to Associate Professor.

In the fall of 2011 I received an email from my friend and collaborator Rena Levitt about a student in her Calculus class at Pomona, Veronica Rivera. Veronica was a student at Claremont High who was taking math classes at Pomona and was interested in doing some research. It turned out that Veronica also knew of my previous Claremont High student Wesley Chang, so Rena sent her over to talk to me. We decided to look at a similar but more general idea to Evan's thesis project, defining Reshetikhin-Turaev invariants customized for birack-labeled links. Since these invariants are related to the representation theory of objects called "quantum groups", we called these invariants "quantum enhancements." Opting to start with the simplest case (unoriented tangles and involutory biracks, from my paper with Sinan), we finished a the first paper in what we expect to be a series of papers on this topic late in the summer of 2012. The paper has appeared in the Journal of Knot Theory and its Ramifications.

Sheri In the spring of 2013 I had my first sabbatical -- the first semester since I was four years old in which I neither taught nor took any course -- and thus took a year off from writing papers with senior thesis students. I spent the sabbatical and the summer travelling to Florida, Louisiana, and Korea, giving colloquia, semianr and conference talks. At one conference at Caltech that spring I had a long conversation with Jim Hoste, my colleague at Pitzer College (and one of the discoverers of the HOMFLYPT polynomial), about which knots have finite involutory quandles. After a lengthy series of discussions with Jim and Rena on the topic, I became interested in the question of which other quotients of the fundamental quandle of a knot are finite. Thus, when Sherilyn Tamagawa of Scripps College asked me about doing a senior thesis, I suggested we look at quotients of knot quandles. Our paper includes results about quotient quandles satisfying conditions we call "anti-abelian" and "Latin Alexander" and has appeared in the New York Journal of Mathematics.

Gill Not long after the Fall semester of 2014 started, Pomona student Gillian Grindstaff asked me about doing a senior thesis. We initially spent some time looking at computing quotients of quandles associated to knotted surfaces in 4-space, but got distracted thinking about Lie algebras. Our paper on a type of enhancement of the quandle counting invariant known as Lie Ideal Enhancements (or LIE invariants, a recursive acronym) has appeared in the Osaka Journal of Mathematics.

Melinda In the Spring of 2014 I started work with Scripps College student Melinda Ho on a project for her senior thesis. I had recently started work with a colleague from Japan, Atushi Ishii, on biquandle invariants for spatial graphs and handlebody-knots using the idea of symmetric quandles, quandles with an involution satisfying certain conditions designed to allow coloring of knots in non-orientable spaces. Our paper uses the structure of symmetric quandles and biracks to enhance the birack counting invariant. The paper has appearing in J. Knot Theory Ramifications.

Late in the spring of 2014, a Pomona student from my discrete mathematics class, S***** P*******, asked me about doing a project over the summer. We decided on an idea I had been thinking about for a while but hadn't yet seen how to make work, namely finite type invariants of biquandle-colored knots. We met somewhat infrequently over the summer but managed to solve the problem, and did so in a way that would lead to solutions to similar problems in my later research -- the key idea is to associate the colors not with the arcs of the knot but with the crossings. When we wrote the paper, S**** felt she hadn't contributed enough and refused to let me list her as a coauthor, despite my protestations. With a heavy heart, I posted the paper as a single-authored paper but still mentioned her nontrivial contributions.

In the spring of 2014, Veronica mentioned to me that her sister Patricia wanted to do a project as well, and we started talking about some ideas. I had seen some exciting talks at conference in Korea in the summer of 2013 involving a way of representing knotted surfaces in 4-space with diagrams called marked vertex diagrams or ch-diagrams, so we decided to see if we could say anything about these. After an initial false start using something we called ribbon biquandles, the final version of our paper uses the involutory biquandles defined in my paper with Sinan to define counting invariants for both orientable and non-orientable knotted surfaces in 4-space, as well as virtual knotted surfaces in other 4-manifolds. The paper will appear in the Journal of Knot Theory and its Ramifications in a special issue for the 60th birthday of Jozef Przytycki, one of the co-discoverers of the HOMFLYPT polynomial. This brought to two the number of sets of siblings with whom I've written papers :)

In the Fall of 2014, I was selected to receive a Collaboration Grant from the Simons Foundation which for five years provides funding for travel to give talks at conferences and colloquia and to, well, collaborate with colleagues around the world.

In the spring of 2015, Veronica (then a sophomore at fellow Claremont Consortium member Harvey Mudd College) suggested doing another project. Our first paper had a very promising idea, but brute-force searching for quantum enhancements was very inefficient, so we wanted to find a better approach. I had recently seen an amazing talk by Charlie Frohman on the relationship between the representations of the special linear group SL₂(C) and the Kauffman bracket polynomial, so we selected some of Charlie's papers to read. Neither of us were experts on representation theory, so we turned to HMC professor Michael Orrison, who often teaches a course on representation theory at HMC. The three of us started meeting and reading though Charlie's papers, hoping to find some inspiration for new quantum enhancements...

And then over spring break, I realized that the key idea from the finite type enhancements paper could be applied to skein invariants to yield new quantum enhancements. We wrote computer code to compute the new invariants with exciting results -- this class of invariants includes both classical quantum invariants like the Jones, Alexander, HOMFLYPT and Kauffman polynomials and biquandle cocycles invariants as special cases, while also including new invariants. Our paper, called Quantum Enhancements via Biquandle Brackets, has appeared in J. Knot Theory Ramifications.

Leo In the spring of 2015 I met Pomona student Leo Selker, who was looking for a research project for the summer. We discussed a few ideas and decided to look at parity biquandles, the topic of my friend Aaron Kaestner's PhD dissertation. In classical knot theory, there are always an even number of crossing points between the over and under instance of any crossing, but for virtual knots this number can be even or odd. In a parity biquandle, we have different operations at even vs. odd crossings. We decided to look at the counting invariants for these parity biquandles, which we completed in about one week. We then decided to include cocycles and developed a theory of parity biquandle cocycle invariants reminiscent of my first paper with Jose. After discussions with Aaron, we decided to invite Aaron to join us as a coauthor; the paper has appeared in the journal Topology and its Applications. Leo gave a talk about the paper at the Knots in Washington XLI conference in the fall of 2015.

Jake In the Spring of 2016 my student Jake Rosenfield asked me about doing a senior thesis, with a bit of a catch -- we had only one week to essentially find an idea and complete the original research side of the project. Fortunately Jake had just taken my algebraic topology class and was familiar with homology, so we were quickly able to introduce a notion of bikei homology and compute some examples. Our paper has appeared in the journal Homology, Homotopy and Applications.

I gave talks about biquandle brackets at conferences around the world including the 2016 Knots in Hellas conference at Ancient Olympia in Greece celebrating the 70th birthday of Louis Kauffman, the KOOK-TAPU seminar in Pusan, South Korea and a colloquium talk delivered via Skype in Moscow, Russia, in addition to the usual venues such as Knots in Washington and various AMS meetings. Several other papers have so far been written involving biquandle brackets, including two collaborations between myself and colleagues in Japan as well as independent projects by mathematicians in China and Russia.

Harry In the spring of 2015 my colleague Deanna Needell and I wrote a paper introducing a new algebraic structure we called biquasiles which we used to define invariants of knots and links. In the Fall of 2016, Deanna and I started working with Pomona College senior WonHyuk "Harry" Choi. The idea was to define Boltzmann weights for biquandle colorings analogous to those used in quandle and biquandle cocycle invariants. Our paper has been published in the Journal of Knot Theory and its Ramifications.

Julien In the fall of 2016 a student from my discrete math class a few years back, Julien Chien, was finishing an off-campus CS major at Harvey Mudd College and asked me about doing a senior thesis project. (Here in Claremont, students can not only take classes at other colleges in the consortium and have senior thesis advisors at other colleges in the consortium, they can even in some cases have majors at other campuses!) We decided to look at algorithms for computing the operation tables of medial abelian bikei for virtual knots. In particular, we were curious about which virtual knots have finite fundamental medial bikei. Julien gave a talk on our work at CMC's Athenaeum as part of a showcase of CMC senior theses in the Spring of 2017. Our paper has appeared in the Journal of Symbolic Computation.

Yuqi In the Fall semester of 2017 I completed a paper with a Yuqi Zhao, a student from my Modern Geometry class in 2016 and my senior thesis advisee. We introduced Reidemeister moves and a category of coloring algebras for a new type of knotted object known as twisted virtual handlebody-links, which can be understood as knotted handlebodies embedded in thickened possibly non-orientable surfaces up to stabilization. I was invited to give a talk specifically about this paper by my collaborator Atushi Ishii at the University of Tsukuba near Tokyo at a local conference in October 2017. Unlike the previous conferences I had attended in Japan, at this conference all but three of the talks were in Japanese! Fortunately, I had been studying Japanese fairly seriously for a bit over a year at the time and was able to start my talk with "すみません、私の日本語がうまくないから、英語で話します" or "I'm sorry, but because my Japanese is not good, I'll give my talk in English". Our paper is currently in peer review.

Shane In the spring of 2018 another former Modern Geometry student and senior thesis advisee of mine, Shane Pico, and I completed a paper together. In this one we adapted the biquasile idea to the case of virtual knots and links using a new "tribracket" notation inspired by Maciej Niebrzydowski. Unlike the cases of quandles and biquandles, with biquasiles and tribrackets one cannot extend colorings to the virtual case by simply ignore virtual crossings because from a diagram, it is impossible to tell whether two regions in a virtual knot's surface complement are the same; thus we had to take the virtual crossings into account by defining compatible classical and virtual tribrackets. Our paper has appeared in the Journal of Knot Theory and its Ramifications.

Paige In the fall of 2018, my former Scripps senior thesis student Sherri Tamagawa, now a PhD student at UCSB, came back to Claremont to teach a course at Scripps. As we chatted at a Claremont Mathematics Colloquium reception, Sherri mentioned that she has a friend at the nearby University of La Verne, Paige Graves, who had done some undergrad research with Sherri at an REU over the summer and wanted to do more research. As I seem to have more project ideas I want to do than I could possibly hope to complete in a lifetime at this point, I was able to make a few suggestions. The three of us decided to work together on a project defining an algebraic structure we call "Niebrzydowski Algebras", essentially a set with a tribracket and a multiplication satisfying compatibility conditions coming from the spatial graph and handlebody-link Reidemeister moves, which can be used to color oriented trivalent spatial graphs and handlebody-links. Our paper has appeared in the International Journal of Mathematics.

Edward In the spring of 2018 a student from my Algebraic Topology class, Yingqui (Edward) Shi, asked to do a research project with me. Together with Deanna Needell (now at UCLA), we applied the quandle module idea to the case of tribrackets to define tribracket modules, structures defined from a tribracket and a commutative ring with identity R such that each tribracket coloring of a knot or link diagram gets an associated invariant R-module. Our paper has appeared in the International Journal of Mathematics.

Karina In the summer of 2018, another Algebraic Topology student, Karina Cho, applied for and won a competitve summer research stipend from Harvey Mudd College to do research with me over the summer. We developed the idea of quandle cocycle quivers, directed graphs which reveal extra structure in the set of quandle colorings of a knot. Our first paper was completed over the summer of 2018 and appeared in J. Knot Theory Ramifications in early 2019.Our second paper, Quandle Cocycle Quivers, formed the basis for Karina's senior thesis (my first HMC senior thesis student!) and has appeared in the journal Topology and its Applications. Karina scored a highly competitve NSF Graduate School Fellowship and left Claremont to pursue graduate studies at Stony Brook.

Evan In 2018-2019 my advisee Evan Paultich, who started with my calculus class and stuck with me for linear algebra, multivariable calculus, algebraic topology and modern geometry, asked to do a research project and senior thesis. We started with the idea of tribrackets with different operations at single-component crossings and multi-component crossings, but realized we could generalize this to a much more versatile stucture we call multi-tribrackets which include virtual tribrackets as a special case, as well as other structures like parity virtual tribrackets. Our paper has appeared in the Journal of Knot Theory and its Ramifications.

Laira In the spring of 2019, Patricia Rivera joined me and another of my former students, Laira Aggarwal, to finish up a paper we had been working on for several semesters. In this paper we introduced tribracket brackets, the tribracket version of biquandle brackets, identifying the conditions needed on the coefficients in a skein expansion to give invariants of tribracket-colored knot and link diagrams. Our paper has been accepted for publication in the Mediterranean Journal of Mathematics.

In the summer of 2019 I travelled to east Asia twice. First, I had a two-week visit to Pusan National University (including an afternoon visit to Kyungpook National University in Daegu to give a talk), during which my collaborator Jieon Kim and I started projects with two graduate students at PSU, Suhyeon Joung and Minju Seo. Later in the summer, I came to Japan for a three-week tour, giving talks and starting projects with collaborators in Nagoya, Nara, Osaka and Tokyo. My Japanese speaking ability had improved to the point that I was able to comfortably order food in restaurants in Japanese and read train schedules in kanji, including my for the first time spending a complete day alone without seeing anyone I already knew.

Forest During the summer of 2019 I co-authored a paper with another HMC student, Forest Kobayashi. Forest had previously done an independent study with me on category theory that veered into algebraic topology and knot theory, and like Karina he competed for and won HMC's Borelli Fellowship to fund his summer work with me. In this paper we defined an analog of biquandle brackets for virtual knots and links with coloring by parity biquandles. The obvious name for these structures, namely "parity biquandle brackets", has already been used by Ilyutko and Manturov, so after briefly considering making a distinction between "(parity biquandle) brackets" and "parity (biquandle brackets)", we opted to name these structures "Kaestner brackets" after Aaron Kaestner, who introduced parity biquandles in his dissertation. Our paper is in peer review.

Karma In the Fall of 2020 a student from my Linear Algebra class a few semesters prior, Karma Istanbouli, asked me to be her senior thesis advisor. We decided to continue the quandle coloring quiver work I had started with Karina by adding quandle modules to the quivers. Given a finite quandle and a commutative ring with identity, we select a subset of quandle endomorphisms and a quandle module over our ring; then in the resulting quandle coloring quiver, we assign the space of quandle module colorings of the knot diagram to the vertices in the quiver. The resulting invariants generalize the quandle module polynomials we had defined in previous papers as well as the quandle quiver invariants. Our paper is in peer review.

In the summer of 2020 I had planned to return to Daegu to speak at TAPU-KOOK and to visit my collaborators in Pusan and in Japan, but the COVID-19 global pandemic had other ideas. Since my Simons Foundation Collaboration grant had run its course, funding for this travel would have been harder to come by, and my new duties as Chair of the CMC Department of Mathematical Sciences were making it harder to find the time to focus on research. Happily, at the start of the summer I found out my next Simons grant had been approved!

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