Booth Id:
MATH032
Category:
Mathematics
Year:
2024
Finalist Names:
Kim, Aaron (School: Bronx High School of Science)
Abstract:
Knot theory, counting a knot invariant for a knot L has been long studied. Using an algebraic structure called quandle, we calculate the knot invariant, in other words coloring of the knot. Schlank and Davis showed that by converting the knots to the closed braid σ ̄ of n strings through Alexander’s theorem, it is possible to compute the expected number of the coloring of the quandle Q through the Hilbert Polynomial PQ(x). We first identify the unknown quandle of size 4, J2, and compute the associated Hilbert Polynomial. Furthermore, we introduce two new methods to decompose a quandle, which are fiber product and twisted disjoint union. These 2 categorical acts allow a unique mapping so that it may be used in multiple mapping for the same two quandles. Finally, by these methods and known results by Schlank and Davis, we classified all the quandles of size 6 and computed the Hilbert Polynomial of a selected few.