Untwisting information from Heegaard Floer homology

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several 10 and 11-crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsv\'ath-Szab\'o tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p,q > 1.Comment: 21 pages, 11 figures; final version, accepted for publication in Algebraic & Geometric Topolog

Just Mathematics: Getting Started Teaching Postsecondary Math for Social Justice

Following the summer 2020 civil rights movement and increasing attention to the intersections of mathematics with politics and power, many math educators have reported a desire to implement an antiracist pedagogy and to examine the intersections of their subject with issues of equity, inclusion, and social justice. Many resources exist for K-12 math educators interested in incorporating social justice into their curricula, but resources are comparatively scarce for college and university instructors (though this is changing quickly!). We discuss why one may want to teach mathematics for social justice, how to begin to implement issues of social justice into postsecondary math courses, and publicly available social justice materials for postsecondary math courses

Unknotting via null-homologous twists and multi-twists

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1.Comment: 14 pages, 6 figure

Comparing demographics of signatories to public letters on diversity in the mathematical sciences

In its December 2019 edition, the \textit{Notices of the American Mathematical Society} published an essay critical of the use of diversity statements in academic hiring. The publication of this essay prompted many responses, including three public letters circulated within the mathematical sciences community. Each letter was signed by hundreds of people and was published online, also by the American Mathematical Society. We report on a study of the signatories' demographics, which we infer using a crowdsourcing approach. Letter A highlights diversity and social justice. The pool of signatories contains relatively more individuals inferred to be women and/or members of underrepresented ethnic groups. Moreover, this pool is diverse with respect to the levels of professional security and types of academic institutions represented. Letter B does not comment on diversity, but rather, asks for discussion and debate. This letter was signed by a strong majority of individuals inferred to be white men in professionally secure positions at highly research intensive universities. Letter C speaks out specifically against diversity statements, calling them "a mistake," and claiming that their usage during early stages of faculty hiring "diminishes mathematical achievement." Individuals who signed both Letters B and C, that is, signatories who both privilege debate and oppose diversity statements, are overwhelmingly inferred to be tenured white men at highly research intensive universities. Our empirical results are consistent with theories of power drawn from the social sciences.Comment: 21 pages, 2 tables, 2 figures; minor textual edits made to previous versio

In a real-life setting, direct-acting antivirals to people who inject drugs with chronic hepatitis c in Turkey

Background: People who inject drugs (PWID) should be treated in order to eliminate hepatitis C virus in the world. The aim of this study was to compare direct-acting antivirals treatment of hepatitis C virus for PWID and non-PWID in a real-life setting. Methods: We performed a prospective, non-randomized, observational multicenter cohort study in 37 centers. All patients treated with direct-acting antivirals between April 1, 2017, and February 28, 2019, were included. In total, 2713 patients were included in the study among which 250 were PWID and 2463 were non-PWID. Besides patient characteristics, treatment response, follow-up, and side effects of treatment were also analyzed. Results: Genotype 1a and 3 were more prevalent in PWID-infected patients (20.4% vs 9.9% and 46.8% vs 5.3%). The number of naïve patients was higher in PWID (90.7% vs 60.0%), while the number of patients with cirrhosis was higher in non-PWID (14.1% vs 3.7%). The loss of follow-up was higher in PWID (29.6% vs 13.6%). There was no difference in the sustained virologic response at 12 weeks after treatment (98.3% vs 98.4%), but the end of treatment response was lower in PWID (96.2% vs 99.0%). In addition, the rate of treatment completion was lower in PWID (74% vs 94.4%). Conclusion: Direct-acting antivirals were safe and effective in PWID. Primary measures should be taken to prevent the loss of follow-up and poor adherence in PWID patients in order to achieve World Health Organization’s objective of eliminating viral hepatitis

The untwisting number of a knot

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. First, we show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer. Second, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to show that several 10-and 11-crossing knots cannot be unknotted by a single positive or negative generalized crossing change. We also use the Ozsváth-Szabó tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p and q