Concordance of Bing doubles and boundary genus

Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature $\sigma$, then the n-iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than $2^{n-1}\sigma$. The same result holds with $\sigma$ replaced by $2\tau$, twice the Ozsvath-Szabo knot concordance invariant.Comment: 13 pages, 7 figure

On the nonorientable 4-genus of double twist knots

We investigate the nonorientable 4-genus $\gamma_4$ of a special family of 2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that $\gamma_4(C(m,n)) \le 3$. By using explicit constructions to obtain upper bounds on $\gamma_4$ and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on $\gamma_4$, we produce infinite subfamilies of $C(m,n)$ where $\gamma_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that $\gamma_4$ lies in one of the sets $\{1,2\}, \{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $|m|$ and $|n|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.Comment: Some exposition is revised, a figure is added, and typos are corrected, following comments from the refere

Ozsváth-Szabó and Rasmussen Invariants of Cable Knots

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants τ and s on Km,n, the (m,n)–cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on Km,n differ from their value on the torus knot Tm,n by fixed constants for all but finitely many n\u3e0. Combining this result together with Hedden’s extensive work on the behavior of τ on (m,mr+1)–cables yields bounds on the value of τ on any (m,n)–cable of K. In addition, several of Hedden’s obstructions for cables bounding complex curves are extended