Research

We introduce and study the notion of equivariant Q-sliceness for strongly invertible knots. On the constructive side, we prove that every Klein amphichiral knot, which is a strongly invertible knot admitting a compatible negative amphichiral involution, is equivariant Q-slice in a single Q-homology 4-ball, by refining Kawauchi's construction and generalizing Levine's uniqueness result. On the obstructive side, we show that the equivariant version of the classical Fox-Milnor condition, proved recently by the first author, also obstructs equivariant Q-sliceness. We then introduce the equivariant Q-concordance group and study the natural maps between concordance groups as an application. We also list some open problems for future study.

We collect and discuss various results on an important family of knots and links called Turk's head knots and links Th(p,q). In the mathematical literature, they also appear under different names such as rosette knots and links or weaving knots and links. Unless being the unknot or the alternating torus links T(2,q), the Turk's head links Th(p,q) are all known to be alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime. The Turk's head links Th(p,q) are also both positive and negative amphichiral if p is chosen to be odd. Moreover, we highlight and present several more results, focusing on Turk's head knots Th(3,q). We finally list several open problems and conjectures for Turk's head knots and links. We conclude with a short appendix on torus knots and links, which might be of independent interest. 

Using Milnor invariants, we prove that the concordance group of 2-string links is not solvable. As a consequence, we prove that the equivariant concordance group of strongly invertible knots is also not solvable, and we answer a conjecture posed by Kuzbary.

By considering a particular type of invariant Seifert surfaces we define a homomorphism from the (topological) equivariant concordance group of directed strongly invertible knots to a new equivariant algebraic concordance group. We prove that this map lifts both Miller and Powell's equivariant algebraic concordance homomorphism and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial result on the isomorphism type of the equivariant algebraic concordance group and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that our construction can obstruct equivariant sliceness for knots with Alexander polynomial one.

We study the equivariant concordance classes of 2-bridge knots, providing an easy formula to compute their butterfly polynomial, and we prove that no 2-bridge knot is equivariantly slice. Finally, we introduce a new invariant of equivariant concordance for strongly invertible knots. Using this invariant as an obstruction we strengthen the result on 2-bridge knots, proving that every 2-bridge knot has infinite order in the equivariant concordance group. 

We prove that the equivariant concordance group of strongly invertible knots is not abelian by exhibiting an infinite family of nontrivial commutators.