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Our university is one campus of the larger, 23-campus California State University system. In 2017, the Chancellor of the system discontinued the developmental mathematics programs at all 23 campuses. This caught us by surprise and compelled us to act quickly. This article gives an overview of our response, highlighting the change our department underwent to improve our pre-calculus stream and general education courses, making them more interactive, supportive, and student-centered.

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 1 1 -stick knots in the sh-lattice are the trefoil knot ( 3 1 ) and the figure-eight knot ( 4 1 ). [ABSTRACT FROM AUTHOR]

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan's lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator. [ABSTRACT FROM AUTHOR]

We adapt work of Kirby–Thompson and Zupan to define an integer invariant L(T)$\mathcal {L}(\mathcal {T})$ of a bridge trisection T$\mathcal {T}$ of a smooth surface S$S$ in S4$S^4$ or B4$B^4$. We show that when L(T)=0$\mathcal {L}(\mathcal {T})=0$, then the surface S$S$ is unknotted. We also show that for a trisection T$\mathcal {T}$ of an irreducible surface, bridge number produces a lower bound for L(T)$\mathcal {L}(\mathcal {T})$. Consequently L$\mathcal {L}$ can be arbitrarily large. [ABSTRACT FROM AUTHOR]

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the ℓ 1 -norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion. [ABSTRACT FROM AUTHOR]

We show that the disk complex of a genus g > 1 Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of (2 g − 2) -dimensional spheres. This implies that genus g > 1 Heegaard surfaces for the 3-sphere are topologically minimal with index 2 g − 1. [ABSTRACT FROM AUTHOR]

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in R³ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant. [ABSTRACT FROM AUTHOR]

The Gordian graph is the graph with vertex set the set of knot types and edge set consisting of pairs of knots that have a diagram wherein they differ at a single crossing. The bridge number is a classical knot invariant that is a measure of the complexity of a knot. It can be refined by another, recently discovered, knot invariant known as "bridge distance." We show, using arguments that are almost entirely elementary, that each vertex of the Gordian graph is adjacent to a vertex having arbitrarily high bridge number and bridge distance. [ABSTRACT FROM AUTHOR]

In this paper, we characterize all links in $$S^3$$ with bridge number at least three that have a bridge sphere of distance two. We show that if a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: [ABSTRACT FROM AUTHOR]

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus ≥2, then the coarse mapping class group of (S, V, W) is isomorphic to the mapping class group of (S, V, W). [ABSTRACT FROM AUTHOR]