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.
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.
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.
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.
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.
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 4crossing Gauss codes, and we found that
some 85% or so of nonevenly intersticed 4crossing 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 nontriviality
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 openended 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, XiaoSong 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 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 (1st) 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 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 YangBaxter 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
algebraagnostic 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 twocomponent link. This is of interest
since, as a complete invariant of knots and unsplit links in S^{3}
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.
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.
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 errorcorrecting 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 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 YangBaxter 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
YangBaxter cohomology and a new "Scohomology" theory. These invariants
reduce to the ordinary YangBaxter 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.
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 tenuretrack 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.
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 warmup 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.
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 semiregularly. After our initial idea started
to fizzle, we hit on the idea of using shadow colorings to extend
our previous work on virtual YangBaxter cohomology.
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.
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 4space into
semisheets 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 tenuretrack position. I could not
have been happier to accept.
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.
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 nonquandle racks. Our
paper, which we call the
HHNYZ paper, has appeared in the Osaka Journal of Mathematics.
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 XiaoSong 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 wellreceived talk on my work with Sinan at Knots in Washington
in the spring of 2011.
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
Ndegeneracy 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 Ibundles"), if we allow nonorientable 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.
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.
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 20102011 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.
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.
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!
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 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 biracklabeled links.
Our paper has appeared in
Topology and its Applications.
During the 20112012 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 tenuretrack 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 ReshetikhinTuraev invariants customized for
biracklabeled 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.
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 "antiabelian" and
"Latin Alexander" and has appeared in the
New York Journal of Mathematics.
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 4space, 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.
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 handlebodyknots using the idea of symmetric quandles, quandles
with an involution satisfying certain conditions designed to allow coloring of
knots in nonorientable 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 biquandlecolored
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 singleauthored
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 4space with diagrams
called marked vertex diagrams or chdiagrams, 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 nonorientable knotted surfaces in 4space, as well as
virtual knotted surfaces in other 4manifolds. 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 codiscoverers 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 bruteforce 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 SL_{2}(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.
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.
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 KOOKTAPU 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.
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.
In the fall of 2016 a student from my discrete math class a few years back,
Julien Chien, was finishing an offcampus 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.
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 handlebodylinks,
which can be understood as knotted handlebodies embedded in thickened
possibly nonorientable 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.
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.
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 handlebodylink
Reidemeister moves, which can be used to color oriented trivalent spatial
graphs and handlebodylinks. Our
paper has appeared in the
International Journal of Mathematics.
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
Rmodule. Our
paper has appeared in the
International Journal of Mathematics.
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.
In 20182019 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 singlecomponent crossings and multicomponent crossings, but realized
we could generalize this to a much more versatile stucture we call
multitribrackets 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.
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 tribracketcolored 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 twoweek
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 threeweek
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.
During the summer of 2019
I coauthored 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.
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
TAPUKOOK and to visit my collaborators in Pusan and in Japan, but the
COVID19 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!
Want to add your name to this list? Drop by my office!
Copyright © 20072020 Sam Nelson.
