State sum invariants of knots from idempotents in quandle rings

Abstract: We use idempotents in quandle rings in combination with the state sum invariants of knots to distinguish all of the 12965 prime oriented knots up to 13 crossings using only 21 connected quandles and three quandles made of idempotents in quandle rings. We also distinguish all knots up to 13 crossings from their mirror images using the same 24 quandles. Furthermore, we distinguish all of the 2977 prime oriented knots up to 12 crossings using only 10 connected quandles and three quandles made of idempotents in quandle rings. This improves a result in [Quandle colorings of knots and applications, J. Knot Theory Ramifications 23 (2014), no. 6, 1450035]. Our computations are achieved with the help of Python and Maple softwares.