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A knitted surface is a surface with boundary properly embedded in $D^2 \times B^2$, which is a generalization of a braided surface. A knitted surface is called a 2-dimensional knit if its boundary is the closure of a trivial braid. From a 2-dimensional knit $S$, we obtain a surface-link in $\mathbb{R}^4$ by taking the closure of $S$. We show that any surface-link is ambiently isotopic to the closure of some 2-dimensional knit. Further, we consider another type of the closure of a knitted surface, called the plat closure. It is known that any trivial surface-knot is ambiently isotopic to the plat closure of a knitted surface of degree 2. We show that the plat closure of any knitted surface of degree 2 is a trivial surface-link, and any trivial surface-link is ambiently isotopic to the plat closure of a knitted surface of degree $2$. We also show the same result for the closure of 2-dimensional knits of degree 2., Comment: 20 pages, 17 figures

Boundaries in gauge theory and gravity give rise to symmetries and charges at both finite and asymptotic distance. Due to their structural similarities, it is often held that soft modes are some kind of asymptotic limit of edge modes. Here, we show in Maxwell theory that there is an arguably more interesting relationship between the \emph{asymptotic} symmetries and their charges, on one hand, and their \emph{finite-distance} counterparts, on the other, without the need of a limit. Key to this observation is to embed the finite region in the global spacetime and identify edge modes as dynamical $\rm{U}(1)$-reference frames for dressing subregion variables. Distinguishing \emph{intrinsic} and \emph{extrinsic} frames, according to whether they are built from field content in- or outside the region, we find that non-trivial corner symmetries arise only for extrinsic frames. Further, the asymptotic-to-finite relation requires asymptotically charged ones (like Wilson lines). Such frames, called \emph{soft edges}, extend to asymptotia and realize the corner charge algebra by ``pulling in'' the asymptotic one from infinity. Realizing an infinite-dimensional algebra requires a new set of \emph{soft boundary conditions}, relying on the distinction between extrinsic and intrinsic data. We identify the subregion Goldstone mode as the relational observable between extrinsic and intrinsic frames and clarify the meaning of vacuum degeneracy. We also connect the asymptotic memory effect with a more operational \emph{quasi-local} one. A main conclusion is that the relationship between asymptotia and finite distance is \emph{frame-dependent}; each choice of soft edge mode probes distinct cross-boundary data of the global theory. Our work combines the study of boundary symmetries with the program of dynamical reference frames and we anticipate that core insights extend to Yang-Mills theory and gravity., Comment: 69 pages, 13 figures

Quantum link models (QLMs) are generalizations of Wilson's lattice gauge theory formulated with finite-dimensional link Hilbert spaces. In certain cases, the non-Abelian Gauss Law constraint can be exactly solved, and the gauge invariant subspace embedded onto local spin Hamiltonians for efficient quantum simulation. In $(1+1)d$ previous studies of the $SO(3)$ QLM coupled to adjoint fermionic matter have been shown to reflect key properties of QCD and nuclear physics, including distinct confining/deconfining phases and hadronic bound states. We extend the model to $(2+1)d$ dimensions for the first time, and report on our initial results. We review the construction of gauge-invariant state space for the proposed models, and study the single-plaquette ground state via exact-diagonalisation. We provide indications of a rich phase diagram which shows both spontaneous and explicit chiral symmetry breaking, confinement, and distinct magnetic phases characterised by different plaquette expectation values., Comment: 9 pages, 7 figures, proceedings to talk presented at the 41st International Symposium on Lattice Field Theory, July 28th - August 3rd 2024

This is an English translation of the expository article written by the author in Japanese for publication in {\em Sugaku}. The author will explain Milnor invariants from the viewpoint of his research., Comment: 30 pages many figures

The Vol-Det Conjecture, formulated by Champanerkar, Kofman and Purcell, states that there exists a specific inequality connecting the hyperbolic volume of an alternating link and its determinant. Among the classes of links for which this conjecture holds are all alternating hyperbolic knots with at most 16 crossings, 2-bridge links, and links that are closures of 3-strand braids. In the present paper, Burton's bound on the number of crossings for which the Vol-Det Conjecture holds is improved for links with more than eight twists. In addition, Stoimenow's inequalities between hyperbolic volumes and determinants are improved for alternating and alternating arborescent links with more than eight twists., Comment: 10 pages, 2 figures

Given a tied link $L$, the invariant $\langle\langle\cdot\rangle\rangle$ generalizes the Kauffman bracket of classical links. However, the analogues of Kauffman states and their relationship to this invariant are not immediately clear. We address this question by defining the Aicardi-Juyumaya states, and show that the contribution of each AJ-state to $\langle\langle\cdot\rangle\rangle$ does not depend on the chosen resolution tree. We also present an algorithm to compute the double bracket of a tied link diagram, and use it to find pairs of examples of (oriented) tied links sharing the same Homflypt polynomial but different tied Jones polynomial., Comment: 17 pages, 13 figures, 1 tables

I investigated the capability of medial quandle, quandle whose operation satisfying that $(a_1*b_1)*(a_2*b_2)=(a_1*a_2)*(b_1*b_2)$, to detect causality in (2+1)-dimensional globally hyperbolic spacetime by determining if they can distinguished the connected sum of two Hopf links from an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. I show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles is an example that this result can apply to. Inspired by this result, I also derived a generalized theorem about the coloring of medial quandles on a specific type of tangles, which help to determine whether this quandles can distinguish between a wider range of knots and links or not.

In this paper we investigate the relationship between links in bridge position and plat presentations. We will show that the Hilden double coset classes of plat presentations of a link are equivalent to bridge positions of the link up to bridge isotopy. This correspondence allows us to reframe algebraic questions about plat presentations in terms of bridge positions. We demonstrate some results about both plat presentations and links in bridge position using this correspondence. For instance, we reprove that there is only one Hilden double coset class of the n-bridge unknot in $\mathbb{S}^3.$ We also show that there is only a single double coset class for torus knots in plat position. Finally, we discuss how this correspondence may be used to investigate plat closures of knots, which is the subject of ongoing research., Comment: 21 pages, 18 figures, Comments Welcome

The theory of motivic superpolynomials is developed, including its extension to algebraic links, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis and DAHA recurrences for iterated torus links. The DAHA construction is provided. Applications are considered to affine Springer fibers and compactified Jacobians of type $A_n$ in the most general case (for arbitrary characteristic polynomials), rho-invariants of algebraic knots, motivic formulas for the DAHA-Jones polynomials, including the case of $A_1$. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. The 2nd connection conjecture links the superpolynomials to the $L$-functions. DAHA theory is adjusted to the definition and main properties of superpolynomials. The motivic tools are systematically exposed, a certain unification of Schubert Calculus with the theory of plane curve singularities., Comment: 95 pages

This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with signatures that deviate by a fixed amount from their crossing numbers. The main results include the identification of all links for which the sum of the signature and crossing number equals 2, which are shown to be closures of positive 3-braids. Additionally, we explore the implications of these findings in the context of band surgeries and their applications to vortex knots and DNA topology.

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., Comment: 35 pages, 10 figures and 1 table. Minor typos are fixed and more references are added

The set of real-analytic Legendrian links with respect to the standard contact structure on the 3-sphere $S^3$ corresponds both to the set of totally tangential $\mathbb{C}$-links as defined by Rudolph and to the set of stable knotted field lines in Bateman electromagnetic fields of Hopf type. It is known that every isotopy class has a real-analytic Legendrian representative, so that every link type $L$ admits a holomorphic function $G:\mathbb{C}^2\to\mathbb{C}$ whose zeros intersect $S^3$ tangentially in $L$ and there is a Bateman electromagnetic field $\mathbf{F}$ with closed field lines in the shape of $L$. However, so far the family of torus links are the only examples where explicit expressions of $G$ and $\mathbf{F}$ have been found. In this paper, we present an algorithm that finds for every given link type $L$ a real-analytic Legendrian representative, parametrised in terms of trigonometric polynomials. We then prove that (good candidates for) examples of $G$ and $\mathbf{F}$ can be obtained by solving a system of linear equations, which is homogeneous in the case of $G$ and inhomogeneous in the case of $\mathbf{F}$. We also use the real-analytic Legendrian parametrisations to study the dynamics of knots in Bateman electromagnetic fields of Hopf type. In particular, we show that no compact subset of $\mathbb{R}^3$ can contain an electromagnetic knot indefinitely., Comment: 26 pages, 7 figures

We show that there are infinitely many triples of non-isotopic hyperbolic links in the lens space $L(4,1)$ such that the three lifts of each triple in $S^{3}$ are isotopic. They are obtained as the lifts of links in $S^{3} / Q_{8}$ by double covers, where $Q_{8}$ is the quaternion group. To construct specific examples, we introduce a diagram of a link in $S^{3} / Q_{8}$ obtained by projecting to a square. The diagrams of isotopic links are connected by Reidemeister-type moves., Comment: 15 pages, 16 figures

The new generation of domain-specific AI accelerators is characterized by rapidly increasing demands for bulk data transfers, as opposed to small, latency-critical cache line transfers typical of traditional cache-coherent systems. In this paper, we address this critical need by introducing the FlooNoC Network-on-Chip (NoC), featuring very wide, fully Advanced eXtensible Interface (AXI4) compliant links designed to meet the massive bandwidth needs at high energy efficiency. At the transport level, non-blocking transactions are supported for latency tolerance. Additionally, a novel end-to-end ordering approach for AXI4, enabled by a multi-stream capable Direct Memory Access (DMA) engine simplifies network interfaces and eliminates inter-stream dependencies. Furthermore, dedicated physical links are instantiated for short, latency-critical messages. A complete end-to-end reference implementation in 12nm FinFET technology demonstrates the physical feasibility and power performance area (PPA) benefits of our approach. Utilizing wide links on high levels of metal, we achieve a bandwidth of 645 Gbps per link and a total aggregate bandwidth of 103 Tbps for an 8x4 mesh of processors cluster tiles, with a total of 288 RISC-V cores. The NoC imposes a minimal area overhead of only 3.5% per compute tile and achieves a leading-edge energy efficiency of 0.15 pJ/B/hop at 0.8 V. Compared to state-of-the-art NoCs, our system offers three times the energy efficiency and more than double the link bandwidth. Furthermore, compared to a traditional AXI4-based multi-layer interconnect, our NoC achieves a 30% reduction in area, corresponding to a 47% increase in GFLOPSDP within the same floorplan.

In this paper, we present a novel method to compute the determinant of a link using Fourier-Hadamard transforms of Boolean functions. We also investigate the determinant of centrally symmetric links (a special class of strong achiral links). In particular, we show that the determinant of a centrally symmetric link with an even number of components is equals zero.

We introduce "book links" as a generalization of braids in open book decompositions; this new class of objects includes both braids and plats as special cases. We then prove a version of Markov's theorem in this general setting by extending the theory of open book foliations., Comment: 11 pages, 9 figures

In this paper, we establish simple $k$-fold summation expressions for the Quot and motivic Cohen--Lenstra zeta functions associated with the $(2,2k)$ torus links. Such expressions lead us to some multiple Rogers--Ramanujan type identities and their finitizations, thereby confirming a conjecture of Huang and Jiang. Several other properties of the two zeta functions will be examined as well.

Target selection is crucial in pharmaceutical drug discovery, directly influencing clinical trial success. Despite its importance, drug development remains resource-intensive, often taking over a decade with significant financial costs. High failure rates highlight the need for better early-stage target selection. We present GATher, a graph attention network designed to predict therapeutic gene-disease links by integrating data from diverse biomedical sources into a graph with over 4.4 million edges. GATher incorporates GATv3, a novel graph attention convolution layer, and GATv3HeteroConv, which aggregates transformations for each edge type, enhancing its ability to manage complex interactions within this extensive dataset. Utilizing hard negative sampling and multi-task pre-training, GATher addresses topological imbalances and improves specificity. Trained on data up to 2018 and evaluated through 2024, our results show GATher predicts clinical trial outcomes with a ROC AUC of 0.69 for unmet efficacy failures and 0.79 for positive efficacy. Feature attribution methods, using Captum, highlight key nodes and relationships, enhancing model interpretability. By 2024, GATher improved precision in prioritizing the top 200 clinical trial targets to 14.1%, an absolute increase of over 3.5% compared to other methods. GATher outperforms existing models like GAT, GATv2, and HGT in predicting clinical trial outcomes, demonstrating its potential in enhancing target validation and predicting clinical efficacy and safety.

A graph $G$ of constant link $L$ is a graph in which the neighborhood of any vertex induces a graph isomorphic to $L$. Given two different graphs, $H$ and $G$, the induced Tur\'an number ${\rm ex}(n; H, G{\rm -ind})$ is defined as the maximum number of edges in an $n$-vertex graph having no subgraph isomorphic to $H$ and no copy from $G$ as an induced subgraph. Our main motivation in this paper is to establish a bridge between graphs with constant link and induced Tur\'an numbers via the class of $t$-regular, $k$-uniform (linear) hypergraphs of girth at least $4$, as well as to present several methods of constructing connected graphs with constant link. We show that, for integers $t \geq 3$ and $k \geq 3$, ${\rm ex}(n; C_k, K_{1,t}{\rm -ind}) \leq (k - 2)(t - 1)n/2$ and that equality holds for infinitely many values of $n$. This result is built upon the existence of graphs with constant link $tL$ with restricted cycle length, which we prove in another theorem. More precisely, we show that, given a graph $F$ with constant link $L$ and circumference $c$, then, for all integers $t \geq 2$ and $g > c$, there exists a graph with constant link $tL$ which is free of cycles of length $l$, for all $c < l < g$. We provide two proofs of this result using distinct approaches. We further present constructions of graphs with constant links $tL$, $t \geq 2$, and restricted cycle length based on Steiner Systems. Finally, starting from a connected graph of constant link $tL$, for $t \geq 2$, having order $n$ and restricted cycle lengths, we provide a method to construct an infinite collection of connected graphs of constant link $tL$ that preserves the cycle length restriction, and whose orders form an arithmetic progression $qn$, $q \geq 1$., Comment: 23 pages, 5 figures

We have investigated the evolutionary connections of the isolated neutron star (NS) populations including radio pulsars (RPs), anomalous X-ray pulsars (AXPs), soft gamma repeaters (SGRs), dim isolated NSs (XDINs), ``high-magnetic-field'' RPs (``HBRPs''), central compact objects (CCOs), rotating radio transients (RRATs), and long-period pulsars (LPPs) in the fallback disc model. The model can reproduce these NS families as a natural outcome of different initial conditions (initial period, disc mass, and dipole moment, $\mu$) with a continuous $\mu$ distribution in the $\sim 10^{27} - 5 \times 10^{30}$ G cm$^3$ range. Results of our simulations can be summarised as follows: (1) A fraction of ``HBRPs'' with relatively high $\mu$ evolve into the persistent AXP/SGR properties, and subsequently become LPPs. (2) Persistent AXP/SGRs do not have evolutionary links with CCOs, XDINs, and RRATs. (3) For a wide range of $\mu$, most RRATs evolve passing through RP or ``HBRP'' properties during their early evolutionary phases. (4) A fraction of RRATs which have the highest estimated birth rate seem to be the progenitors of XDINs. (5) LPPs, whose existence was predicted by the fallback disc model, are the sources evolving in the late stage of evolution before the discs become inactive. These results provide concrete support to the ideas proposing evolutionary connections between the NS families to account for the ``birth-rate problem'', the discrepancy between the cumulative birth rate estimated for these systems and the core-collapse supernova rate., Comment: 9 pages, 3 figures, accepted for publication in MNRAS

Growing traffic over the high-frequency (HF) band poses significant challenges to establishing robust communication links. While existing spread-spectrum HF transceivers are, to some degree, robust against harsh HF channel conditions, their performance significantly degrades in the presence of strong co-channel interference. To improve performance in congested channel conditions, we propose a filter-bank based multicarrier spread-spectrum waveform with noncontiguous subcarrier bands. The use of noncontiguous subcarriers allows the system to at once leverage the robustness of a wideband system while retaining the frequency agility of a narrowband system. In this study, we explore differences between contiguous and noncontiguous systems by considering their respective peak-to-average power ratios (PAPRs) and matched-filter responses. Additionally, we develop a modified filter-bank receiver structure to facilitate both efficient signal processing and noncontiguous channel estimation. We conclude by presenting simulated and over-the-air results of the noncontiguous waveform, demonstrating both its robustness in harsh HF channels and its enhanced performance in congested spectral conditions.

In this paper, we calculate the annihilating polynomial of the colored Jones polynomial for the Hopf link and the Whitehead link and consider the link version of the AJ conjecture. We also give another annihilating polynomial of the colored Jones polynomial for those links., Comment: 23 pages, 0 figures

The precision of link-level theoretical performance analysis for emerging wireless communication paradigms is critical. Recent studies have demonstrated the excellent fitting capabilities of the mixture gamma (MG) distribution in representing small-scale fading in outdoor terahertz (THz)-band scenarios. Our study establishes an in-depth performance analysis for outdoor point-to-point THz links under realistic configurations, incorporating MG small-scale fading combined with the misalignment effect. We derive closed-form expressions for the bit-error probability, outage probability, and ergodic capacity. Furthermore, we conduct an asymptotic analysis of these metrics at high signal-to-noise ratios and derive the necessary convergence conditions. Simulation results, leveraging precise measurement-based channel parameters in various configurations, closely align with the derived analytical equations.