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We prove that the SL(2, C) character variety of a hyperbolic, freely 2-periodic knot has two canonical components. We also prove that the hyperbolic torsion polynomial of such a knot satisfies a factorization condition which seems to be particularly effective at identifying freely 2-periodic knots.

We investigate a polynomial factorization problem which naturally arises in Hartley's factorization condition on the Alexander polynomial of freely periodic knots. We give a number-theoretic interpretation of this factorization condition, which allows for efficient computation. As an application, we prove that any polynomial which is not a product of cyclotomic polynomials can be the Alexander polynomial of a freely $p$-periodic knot for only finitely many $p$. As a demonstration of the computational efficiency of these methods, we also show that the Alexander polynomial of any freely-periodic L-space knot with genus at most 16 must be a product of cyclotomic polynomials. We conjecture that any periodic or freely periodic L-space knot must be an iterated torus knot., Comment: 12 pages

We classify all finite group actions on knots in the 3-sphere. By geometrization, all such actions are conjugate to actions by isometries, and so we may use orthogonal representation theory to describe three cyclic and seven dihedral families of symmetries. By constructing examples, we prove that all of the cyclic and four of the dihedral families arise as symmetries of prime knots. The remaining three dihedral families apply only to composite knots. We also explain how to distinguish different types of symmetries of knots using diagrammatic or topological data. Our classification immediately implies a previous result of Sakuma: a hyperbolic knot cannot be both freely periodic and amphichiral. In passing, we establish two technical results: one concerning finite cyclic or dihedral subgroups of isometries of the 3-sphere and another concerning linking numbers of symmetric knots.

We show that any strongly negative amphichiral knot with a trivial Alexander polynomial is equivariantly topologically slice.

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize the factor $f(z)$ to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in $S^3$, answering a problem of Hartley-Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group., Comment: Updated to match the version accepted for publication in Revista Matem\'atica Iberoamericana. The new version includes a complete proof of the equivariant Reidemeister moves for strongly negative amphichiral knots, and a table of strongly negative amphichiral knots through 12 crossings

We consider free symmetries on cobordisms between knots. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. A key tool is the homology cobordism classification of lens spaces using d-invariants. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.

We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature., Comment: 14 pages, 9 figures

We prove obstructions to a strongly negative amphichiral knot bounding an equivariant slice disk in the 4-ball using the determinant, Spinc-structures and Donaldson's theorem. Of the 16 slice strongly negative amphichiral knots with 12 or fewer crossings, our obstructions show that 8 are not equivariantly slice, we exhibit equivariant ribbon diagrams for 5 others, and the remaining 3 are unknown. Finally, we give an obstruction to a knot being strongly negative amphichiral in terms of Heegaard Floer correction terms., Comment: 20 pages, 6 figures

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new strongly invertible concordance group invariants, Donaldson's theorem, and the g-signature. We find many new examples where the equivariant 4-genus is larger than the 4-genus., Comment: 41 pages, 14 figures. Comments are very welcome!

Let $L$ be an alternating prime non-split link in $S^3$. We use the category of flypes between reduced alternating diagrams for $L$ to classify involutions on $L$. As consequences, we show that the quotient of an alternating periodic link is alternating, and that all freely 2-periodic alternating links have an even number of components., Comment: Updated to agree with the version accepted by Proceedings of the AMS. 15 pages, 9 figures. arXiv admin note: text overlap with arXiv:1906.04308

Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype. This implies that for any odd prime order action on K there is either a reduced alternating periodic diagram or a reduced alternating free periodic diagram. Finally, we deduce that the quotient of an odd periodic alternating knot is also alternating.

Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we give computational evidence, and produce applications to the Alexander polynomials of $\widetilde{K}$ and $K$., Comment: Updated to match the version accepted for publication in Bulletin of the London Mathematical Society

A knot $K$ is definite if $|\sigma(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces., Comment: 2 pages

Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a hyperbolic knot, we provide partial results in the direction of the conjecture that M never covers M'., Comment: Corrected Theorem 2 (and its proof) by adding a necessary condition that was missing. Thanks to Tye Lidman for pointing out the error