They can be described by Jones and Alexander polynomials, which can be generalized to the HOMFLY polynomial.
Two distinct ones of these cannot have the same exterior by the Gordon-Luecke theorem. Dowker notation can describe their projection. Twist, poke, and slide make up the Reidemeister moves on diagrams of these. They can be described by Jones and Alexander polynomials, which can be generalized to the HOMFLY polynomial. A simple loop is known as the un- one of these; a simple, non-trivial one is the trefoil. Slightly more general structures permitting open ends are known as braids. For 10 points, name these closed non-intersecting curves that cannot be untangled.
Two distinct ones of these cannot have the same exterior by the Gordon-Luecke theorem.
{knots}