Question

2. Prove that 2-bridge knots are alternating in two steps (a) (Adams, Exercise 2.17) Prove that every rational knot is alternating (by finding an alternating diagram) (b) Prove that every 2-bridge knot is a rational knot.

Answer 1

2 - bridge knots also called rotational knots these are called 2-bridge knots because they are presenting with two local maximum at the top of figure (a)_we have to prove that every rotational knot is alternating let us see, first positive and negative twist therefore positive twist over are passes from SW to NE and under are passes from NW to SE formation of rotational tangles \r\n therefore an alternating projection is a knot projection if we are traveling along knot we alternate between over crossing and undercrossing (b) therefore l very 2-bridge has two local maximum or bridges at top of figure they are rotational knot only because connection with real numbers. Every rotational knot has projection in which Conway notation consists positive integer let k(p/q) denote 2-bridge knot associated to p/q therefore 0 lessthanorequalto p/q < 1 as p/q = 0 - unknot every boundary slope continued fraction has [+ - + ...]pattern