On the Alexander polynomial and the signature invariant of two-bridge knots

Abstract

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.

Knot Floer homology of satellite knots with (1,1)-patterns

“…Let H=(Σ, α a , β, w, z) be a genus-one bordered diagram for (S 1 × D 2 , P ) corresponding to the standard meridian-longitude parametrization of the torus boundary. We first place α(K) and (β, w, z) in a specific position on the torus T 2 : Identify T 2 as the obvious quotient space of the squre [0, 1]× [0, 1] and divide the square into four quadrants by the segments { 1 2 } × [0, 1] and [0, 1] × { 1 2 }. Include α(K) into the first quadrant and extend it horizontally/vertically.…”

“…Finally, if the lower row is of type (c) in Figure 32, then the upper row must be of type (1) in Figure 31, and the corresponding pairing diagram is shown in Figure 36. Note in this case, w(P ) ≥ −σ − 2 and τ = A(x 0 ) = τ = w(P )+σ 2 + 1.…”