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In groundbreaking work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex. Lemma 3 of that paper provides a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. This lemma is false. We present counterexamples, a correction with detailed proof, and an analysis of the consequences of this error on subsequent works that rely on this lemma., Comment: 18 pages, 14 figures

An extensive body of research has demonstrated that the use in a K-12 classroom of technology, such as the Internet, computers, and software programs, enhances the learning of mathematics (Cheung & Slavin, 2013; Cohen & Hollebrands, 2011). In particular, growing empirical evidence supports that certain types of technology, such as intelligent tutoring systems and adaptive learning systems, have a positive impact on students' academic achievement in math and their attitudes toward math (Arroyo, Burleson, Tai, Muldner, & Woolf, 2013; Ma, Adesope, Nesbit, & Liu, 2014; Pane, Griffin, McCaffrey, & Karam, 2013; Steenbergen-Hu & Cooper, 2013). These kinds of learning systems yield positive effects by providing students with personalized instruction tailored to "the pace, order, location, and content of a lesson uniquely for each student" (Enyedy, 2014, p. 3). However, despite the recognized benefits, many teachers still struggle with successfully integrating technology into their instruction. Through funding from the state legislature, over 200,000 K-12 students were given access to 11 mathematics education technology products. The authors surveyed teachers to understand the implementation successes and challenges. The authors' review of prior research and the TPACK framework informed the research as the authors analyzed open-ended survey data on teachers' perceptions of the education technology implementation over one school year. At the end of the year, researchers collected data from the state office of education on student assessment and demographic characteristics for use in a quasi-experiment to understand the impact of the mathematics technology products. The following research questions were addressed: (1) Is there any significant effect of using mathematics education technology through the statewide grant program on student state achievement; (2) Is there any significant effect of using the technology for students who met the fidelity of implementation benchmark; (3) How were the education technology products being used; (4) With what features of the products or experiences are teachers most satisfied; (5) What concerns or challenges have teachers experienced with use of the products; (6) What barriers limit teachers from using the products to their desired level; and (7) How have teachers used the performance management features of the products? While the state assessment data is very important, the authors provide a detailed overview of teacher feedback because it sheds light on their experiences implementing the products/programs and opportunities to learn lessons from implementation that can inform future years of implementation. Tables and figures are appended. [SREE documents are structured abstracts of SREE conference symposium, panel, and paper or poster submissions.]

We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{\mathcal{F}^m_n\}$ of the string link concordance group $\mathcal{C}^m$. We utilize the Thom-Pontryagin construction to show that the Sato-Levine invariants $\bar{\mu}_{(iijj)}$ must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence $0\rightarrow \mathcal{F}^m_0/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_{0.5} \rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_0 \rightarrow 0$ does not split for links of two or more components, in contrast to the fact that it splits for knots. Considering lower terms of the filtration $\{\mathcal{F}^m_n\}$ in the short exact sequence $0\rightarrow \mathcal{F}^m_{-0.5}/\mathcal{F}^m_{0} \rightarrow \mathcal{C}^m/\mathcal{F}^m_{0} \rightarrow \mathcal{C}^m/\mathcal{F}^m_{-0.5} \rightarrow 0$, we show that while the sequence does not split for $m\ge 3$, it does indeed split for $m=2$. We conclude that the quotient $\mathcal{C}^2/\mathcal{F}^2_0 \cong \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_2 \oplus \mathbb{Z}$., Comment: 10 pages, 4 figures

Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by $\mathcal{F}_n$. It has been shown that $\mathcal{F}_n/\mathcal{F}_{n.5}$ is a very large group for $n\ge 0$. For a generalization to the setting of links the third author showed that $\mathcal{F}_{n.5}/\mathcal{F}_{n+1}$ is non-trivial. In this paper we provide evidence that for knots $\mathcal{F}_{0.5}=\mathcal{F}_1$. In particular we prove that every genus 1 algebraically slice knot is 1-solvable., Comment: 19 pages, 10 figures, to appear in Transactions of the American Mathematical Society

The $n$-solvable filtration of the $m$-component smooth (string) link concordance group, $$\dots \subset \mathcal{F}^m_{n+1} \subset \mathcal{F}^m_{n.5} \subset \mathcal{F}^m_n \dots \subset \mathcal{F}^m_1 \subset \mathcal{F}^m_{0.5} \subset \mathcal{F}^m_0 \subset \mathcal{F}^m_{-0.5} \subset \mathcal{C}^m,$$ as defined by Cochran, Orr, and Teichner, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. The focus of this paper is to give a characterization of the set of 0-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric classification of $\mathbb{L}_0^m$, the set of links up to 0-solve equivalence. We show that $\mathbb{L}_0^m$ has a group structure isomorphic to the quotient $\mathcal{F}_{-0.5}/\mathcal{F}_0$ of concordance classes of string links and classify this group, showing that $$\mathbb{L}_0^m \cong \mathcal{F}_{-0.5}^m/\mathcal{F}_0^m \cong \mathbb{Z}_2^m \oplus \mathbb{Z}^{m \choose 3} \oplus \mathbb{Z}_2^{m \choose 2}.$$ Finally, using results of Conant, Schneiderman, and Teichner, we show that 0-solvable links are precisely the links that bound class 2 gropes and support order 2 Whitney towers in the 4-ball., Comment: 34 pages