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As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds., Comment: 18 pages. Comments are welcome

Nowadays, there is a strong need to deploy the target speaker separation (TSS) model on mobile devices with a limitation of the model size and computational complexity. To better perform TSS for mobile voice communication, we first make a dual-channel dataset based on a specific scenario, LibriPhone. Specifically, to better mimic the real-case scenario, instead of simulating from the single-channel dataset, LibriPhone is made by simultaneously replaying pairs of utterances from LibriSpeech by two professional artificial heads and recording by two built-in microphones of the mobile. Then, we propose a lightweight time-frequency domain separation model, LSTM-Former, which is based on the LSTM framework with source-to-noise ratio (SI-SNR) loss. For the experiments on Libri-Phone, we explore the dual-channel LSTMFormer model and a single-channel version by a random single channel of Libri-Phone. Experimental result shows that the dual-channel LSTM-Former outperforms the single-channel LSTMFormer with relative 25% improvement. This work provides a feasible solution for the TSS task on mobile devices, playing back and recording multiple data sources in real application scenarios for getting dual-channel real data can assist the lightweight model to achieve higher performance.

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at $t=1$ gives the weighted number of the spanning trees of the graph., Comment: 13 pages. Comments are welcome

Recently, Convolutional Neural Network (CNN) and Long short-term memory (LSTM) based models have been introduced to deep learning-based target speaker separation. In this paper, we propose an Attention-based neural network (Atss-Net) in the spectrogram domain for the task. It allows the network to compute the correlation between each feature parallelly, and using shallower layers to extract more features, compared with the CNN-LSTM architecture. Experimental results show that our Atss-Net yields better performance than the VoiceFilter, although it only contains half of the parameters. Furthermore, our proposed model also demonstrates promising performance in speech enhancement., Comment: Submitted to Interspeech 2020

This is a sequel to [arXiv:1708.09092v2]. For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1| 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2]. As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$ studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta}(G, c)$ for a positive coloring $c$ of $G$, where $\Delta_{(G, c)}(t)$ is constructed from certain regular covering space of the complement of $G$ in $S^3$ and it is the Euler characteristic of the Heegaard Floer homology of $G$ that we studied before. When $G$ is a plane graph, we provide a topological interpretation to the vertex state sum of $\underline{\Delta}(G, c)$ by considering a special Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface., Comment: 26 pages. Some notations may vary slightly from published version

We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph $\mathbb{G}$, we refine the construction and obtain a framed ambient isotopy invariant $\Delta_{(\mathbb{G},c)}(t)$. The invariant $\Delta_{(\mathbb{G}, c)}(t)$ satisfies a series of relations, which we call MOY-type relations, and conversely these relations determine $\Delta_{(\mathbb{G}, c)}(t)$. Using them we provide a graphical definition of the Alexander polynomial of a link. Finally, we discuss some properties and applications of our invariants., Comment: This version is accepted for publication in Selecta Mathematica. We thank the referee for careful reading and many helpful comments

For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched covers of the knot, obstruct the smooth sliceness of the knot. These invariants bear some structural resemblances to Casson-Gordon invariants, which obstruct the topological sliceness of a knot. Se-Goo Kim found a polynomial splitting property for Casson-Gordon invariants. In this paper, we show a similar result for Ozsvath and Szabo's d-invariants. We give an application of the result., Comment: 12 pages, 1 figure. This paper was published one year ago

A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative unknotting number, and discuss its application to two classes of knots., Comment: 14 pages, 5 figures. This paper was published two years ago

We generalize the construction of the Heegaard Floer homology for a singular knot to that for a balanced bipartite graph. For a given graph, we provide a combinatorial description of the Euler characteristic of its Heegaard Floer homology by using the "Kauffman states" on a graph diagram., Comment: The paper is rewritten. We study balanced bipartite graphs, which generalize both trivalent graphs and singular knots. We show how to apply Alishahi and Eftekhary's refinement of sutured Floer homology to extract interesting invariants for a graph. In the latter half, we study the Alexander invariant of a graph, and calculate it for Kinoshita theta-graph as an example

This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The second part concerns an explicit form of composite links with H(2)-unknotting number one. In the last part, we develop a method of studying the H(2)-unknotting numbers of some tangle unknotting number one knots via 2-bridge knots., Comment: 14 pages, 9 figures

We give an obstruction to unknotting a knot by adding a twisted band, derived from Heegaard Floer homology., Comment: 6 pages, 4 figures

Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots., Comment: 27 pages, 16 figures.