Quandle colorings of knots and applications.
We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that ifQis a simple quandle of prime power order thenQand the dual quandleQ* ofQhave the same number of colorings for all knots and conjecture that this holds for all Alexander quandlesQ. We study a knot invariant based on a quandle homomorphismf:Q1→Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.
Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
W Edwin
WE
Clark
Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
Mohamed
M
Elhamdadi
Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
Masahico
M
Saito
Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
Timothy
T
Yeatman
Quandles; knot colorings; knot invariants; mirror images of knots