We provide a purely combinatorial proof of a skein exact sequence obeyed by double-point enhanced grid homology. We also extend the theory to coefficients over $\mathbb {Z},$ and discuss alternatives to the Ozsváth–Szabó $\tau $ invariant.

We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Anstee, Przytycki, and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants that can be distinguished by Khovanov homology, but not by the Jones polynomial.

How does DNA, the molecule containing genetic information, change its three-dimensionalshape during the complex cellular processes of replication, recombination and repair? Thisis one of the core questions in molecular biology which cannot be answered without helpfrom mathematical modeling. Basic concepts of topology and geometry can be introduced inundergraduate teaching to help students understand counterintuitive complex structuraltransformations that occur in every living cell. Topoisomerases, a fascinating class ofenzymes involved in replication, recombination and repair, catalyze a change in DNAtopology through a series of highly coordinated mechanistic steps. Undergraduate biologyand mathematics students can visualize and explore the principles of topoisomerase actionby using easily available materials such as Velcro, ribbons, telephone cords, zippers andtubing. These simple toys can be used as powerful teaching tools to engage students inhands-on exploration with the goal of learning about both the mathematics and the biologyof DNA structure.