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Mathematics major Kevin Penderghest’05 (Lindenwold, N.J.) is no stranger to geometry.
In fact, earlier in his college career, he found a solution to a geometry problem that was superior to any answer that had ever come before it.
Solving that problem, which asked for the most effective way to cover one circle, say a quarter, with a number of other circles, such as dimes, earned Penderghest the chance to present his solution at the Regional Student Mathematical Conference hosted by Moravian College.
In January, he will present his recent research on knot theory in Atlanta at the joint national meeting of the American Mathematical Society and Mathematics Association of America. He co-authored a paper on his findings and submitted it for publication to The Journal of Knot Theory.
Captain of the cross country team, Penderghest worked this summer with John E. Meier, associate professor of mathematics at Lafayette, and three other students from colleges across the nation in the National Science Foundation’s Research Experience for Undergraduates program at Lafayette, applying what was already known about knots towards a three-dimensional version.
Penderghest’s participation was made possible by Lafayette’s distinctive EXCEL Scholars program, in which students conduct research with faculty while earning a stipend. The program has helped to make Lafayette a national leader in undergraduate research. Many of the more than 160 students who participate each year share their work through articles in academic journals and/or conference presentations.
“The scope of the project extended beyond simply graphing a knot,” says Meier, who chose Penderghest in part for his ability to solve the difficult geometry problem involving circles.
“Imagine a computer network in which wires from all of the components become tangled between the back of the machines and the wall outlet. We paid attention to how tangled those wires were from getting [the user] to one place from another,” he explains. “And what we really tried to figure out was if someone was handed one of these things with all those tangled wires, could anything be done to untangle it.”
All prior theories assumed that nothing could be done to untangle the wires unless they are unplugged from the backs of the machines. Meier hoped that the students could develop an easy test to determine if that three-dimensional knot could be undone without becoming unplugged. Their results were encouraging.
“We showed that when a graph is alternating, the algebra of the group is very complicated, but not completely unsolvable,” Penderghest says.
“What I decided to do this summer was to figure out if research is something I want to do the rest of my life,” he adds.
More immediately, it augmented his education at Lafayette.
“As I take more classes with stuff that I’ve never seen before, it helps me be able to handle it better,” he says. “This year I’m doing a lot of stuff with these theories. [My summer research] was good practice for that.”
In addition to competing in cross country, Penderghest has been a member of the track team for four years, volunteers as a lector at the Roman Catholic Mass on campus, is a member of the Calculus Calvary, and works as a mathematics lab proctor.
He is a graduate of Paul VI High School.
As a national leader in undergraduate research, Lafayette sends one of the largest contingents to the National Conference on Undergraduate Research each year. Forty-two students were accepted to present their work at the last annual conference in April.