In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity we use these predictions to identify braid representatives that are likely to be quasipositive, which we then subject to further testing to verify. Using these techniques we identify 84 new quasipositive 11 and 12crossing knots. Furthermore, we show that neural networks are also able to predict and help compute the slice genus and Ozsváth-Szabó τ -invariant of knots.

We construct new families of closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature that are interesting with respect to the geography problem. Note that e(M) and σ (M) are in turn completely determined by χ h (M) and c 2 1 (M), that is, e(M) = 12χ h (M) − c 2 1 (M) and σ (M) = c 2 1 (M) − 8χ h (M). When M is a complex surface, χ h (M) is the holomorphic Euler characteristic of M while c 2 1 (M) is the square of the first Chern class of M. The classical "geography problem" in algebraic geometry, originally posed by Persson [1981], asks which ordered pairs of positive integers can be realized as the pair (χ h (M), c 2 1 (M)) for some minimal complex surface M of general type. The related "botany problem", which is a lot more difficult, asks for the classification of all minimal complex surfaces with a given pair of invariants (χ h , c 2 1). The symplectic geography problem, first posed in [McCarthy and Wolfson 1994], asks which ordered pairs of integers can be realized as (χ h (M), c 2 1 (M)) for some minimal symplectic 4-manifold M. There has been steady progress on the symplectic geography problem in recent years and the problem has been completely solved for simply connected minimal symplectic 4-manifolds with negative signature

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group S 2n+1 . We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that any two permutations which represent isotopic knots can be related by a sequence of these moves and their inverses.

No abstract

Abstract. We define the notion of a braided link cobordism in S 3 × [0, 1], which generalizes Viro's closed surface braids in R 4 . We prove that any properly embedded oriented surface W ⊂ S 3 × [0, 1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when ∂W already consists of closed braids. These surfaces are closely related to another notion of surface braiding in D 2 × D 2 , called braided surfaces with caps, which are a generalization of Rudolph's braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4-space, as well as constructing singular fibrations on smooth 4-manifolds from a given handle decomposition.