Since 2004 I have written 12 papers with students from my classes or other undergrads. I didn't set out to become an undergraduate research mentor; it just sortof happened.

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 was accepted by the Journal of Knot Theory and its Ramifications and is currently waiting to appear.

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.

Another student from my topology class, Chau-Yim (Jason) Wong, dropped by my office to say hello early that fall quarter, and when I mentioned the paper I had just completed and the project I was working on, Jason asked if I had any other ideas. Indeed I did -- the matrix representation of finite quandles Benita and I had developed made it clear how quandles can be broken down into subquandles, and we decided to study this type of decomposition further. Ultimately, we wrote a paper which appeared in J. Knot Theory Ramifications.

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 of Gabe 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.

Another former student, Todd Macedo, asked me about doing a project that spring quarter. 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.

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 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 with 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 table the paper.

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 Alexandeer quandles, but need not be for general Alexander biquandles. The resulting paper is currently in peer review.

Conrad, who is 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 been accepted and will appear 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 is currently in peer review.

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 I'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 is currently in peer review.

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

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 is currently in peer review.

My most recent research student is Jose Ceniceros, 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 cocycles from the Yang-Baxter cohomology and a new S-cohomology theory which reduce to the ordinary Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. Our paper is currently in peer review.