Minimal degree sequence for 2-bridge knots

Abstract

Abstract. We discuss polynomial representations for 2-bridge knots and determine the minimal degree sequence for all such knots. We apply the connection between rational tangles and 2-bridge knots.1. Introduction. It is known that, up to ambient isotopy, every noncompact knot is equivalent to some polynomial knot [10]. In fact, one can say that the set of all non-compact knot types is the same as the set of all polynomial knot types because the isotopy between two equivalent polynomial knots can be given by a one-parameter family of polynomial embeddings. Thus, for a given knot type K (always non-compact for us) there exist three real polynomials f (t), g(t) and h(t) such that the map t → (f (t), g(t), h(t)) represents K. If deg f (t) = l, deg g(t) = m and deg h(t) = n, then we say that the triple (l, m, n) is a degree sequence of K. We define (l, m, n) to be the minimal degree sequence for the knot type K if it is minimal amongst all degree sequences of K with respect to the lexicographic ordering in N 3 . A knot can be well understood if its minimal degree sequence is known. In fact a polynomial knot with the minimal degree sequence has a diagram that cannot be reduced further. Thus, to determine the minimal degree sequence of a given knot type is an interesting problem.In our earlier papers we have found a degree sequence for all torus knots ([6], [7]) and also the minimal degree sequence for torus knots of type (2, 2n + 1) ([7]). The minimal degree sequence for a general torus knot of type (p, q) is still not known, except for the special case when q = 2p − 1 ([5]). For convenience we state some known partial results:

Polynomial representation for long knots

“…Theorem 14 [11] The minimal degree sequence for a 2-bridge knot having minimal crossing number N is given by…”

“…Hence, for a given knot-type estimating a degree sequence and eventually the minimal degree sequence is an important aspect for polynomial knots. In our earlier papers ( [14], [7], [8], [10], [11], [12]) we have estimated a degree sequence and the minimal degree sequence for torus knots and 2-bridge knots.…”

Polynomial Unknotting and Singularity Index

“…We see that the equations (10), (11) and (14) do not have any real solutions. Also equation (8) is identical with equation (12) and equation (9) is identical with equation (15) and equation (13) have real solutions.…”

Constructing real rational knots by gluing

“…The image of such an embedding is a long knot which upon one point compactification gives our knot in S 3 . In this connection the question of obtaining the polynomials of minimal degree [9,5] for a given knot type was explored. Polynomial parametrizations have certain disadvantages.…”