New link detection results using closures of 3-braids.
problem Link detection using homology theories.
method Closure operations on 3-braids and homology theories.
result Detection of specific links using link Floer homology, Khovanov homology, and annular Khovanov homology.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
Paper extends link Floer homology detection to almost braided links.
problem Detecting links that are nearly braided using link Floer homology.
method Inspired by nearly fibered knots, uses clasp-braids and topological characterisation.
result Khovanov homology detects Whitehead link and infinite families of links.
Khovanov homology detects specific links.
problem Detecting specific links using Khovanov homology.
method Using Z/2-coefficients, proving detection of specific links.
result Khovanov homology detects L7n1 and a trefoil with meridian union.
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexand…
Prime homology detects split links in prime characteristic.
problem Detecting split links in prime characteristic.
method Uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients.
result Reduced sl(P) link homology detects split links in Z/P. Empirical evidence suggests link polynomials can detect causality in spacetimes.
problem Detect causality in (2+1)-dimensional globally hyperbolic spacetimes. method Introduced a new invariant of certain tangles related to the Conway polynomial.
result The Conway polynomial does not detect causality in certain spacetime scenarios.
New methods detect and create Brunnian links efficiently.
problem Detecting and constructing Brunnian links when each component is known to be an unknot.
method Two practical methods based on observation and handwork.
result Established a general way to construct new Brunnian links in bulk.
Symplectic quandles can detect causality in spacetimes, improving on existing methods.
problem Detecting causality in (2+1)-dimensional spacetimes using existing methods.
method Combining Alexander-Conway polynomial with symplectic quandles.
result Symplectic quandles can distinguish between different types of links, suggesting their ability to detect causality.
Alexander quandles fail to distinguish certain links, thus not detecting causality.
problem Detecting causality in spacetimes using link polynomials.
method Examined Alexander quandles' ability to distinguish specific links.
result Alexander quandles cannot distinguish the connected sum of two Hopf links and Allen-Swenberg Links.
We prove that, for a link L in a rational homology 3--sphere, the link Floer homology detects the Thurston norm of its complement. This generalizes the previous results due to Ozsváth, Szabó and the author.
Quandle coloring detects causality in spacetime links.
problem Detecting causality in spacetime links using quandle colorings.
method Conjugation quandle of dihedral group D5 combined with Alexander-Conway polynomial.
result The conjugation quandle over D5 distinguishes causally related and unrelated events.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
Study detects a specific type of link using annular Khovanov homology.
problem Detecting a specific type of three-strand weaving link.
method Combines braid detection with rigidity theorem to determine (σ1σ2−1)N up to conjugacy. result Annular Khovanov homology detects the underlying unoriented annular link KN. Study shows surgeries on specific links result in L-spaces.
problem Understanding surgeries on specific links and their outcomes.
method Proving surgeries on chainmail links yield L-spaces using alternating surgeries.
result Alternating surgeries on chainmail links result in total L-spaces.
Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …
Modularity-aware GAE and VGAE improve community detection and link prediction.
problem Improving community detection with GAE and VGAE in the absence of node features.
method Introducing a modularity-aware message passing scheme and regularizer to GAE and VGAE encoders.
result Jointly addressing community detection and link prediction with high accuracy is possible.
Method detects intersections between ellipses for Borromean linking.
problem Detecting Borromean linking between ellipses.
method Transforming one ellipse to a unit circle, examining intersections.
result Efficiently determines Borromean linking between ellipses.
Method detects and locates eavesdropping in optical links.
problem Detect and locate eavesdropping in optical links with small power losses.
method Cluster-based approach using OPM data at receiver and in-line OPM data for localization.
result Subtle eavesdropping losses can be detected and localized using OPM data.
Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.
problem Detecting checkerboard colorability of virtual links.
method Using odd writhe and arrow polynomial.
result Proves 6 virtual knots are not checkerboard colorable.
A cobordism between links in thickened surfaces consists of a surface S and a 3-manifold M, with S properly embedded in M×I. We show that there exist links in thickened surfaces such that if (S,M) is a cobordism between them in which S is simple, then M must be complex. That is, there…
Outliers arise in networks due to different reasons such as fraudulent behavior of malicious users or default in measurement instruments and can significantly impair network analyses. In addition, real-life networks are likely to be incompletely observed, with missing links due to individual non-response or machine fai…
The paper classifies links with low rank knot Floer and Khovanov homologies.
problem Detecting and classifying links with low rank knot Floer and Khovanov homologies.
method Generalized link Floer homology, used to obtain rank bounds and classify links.
result Knot Floer homology detects T(2,8) and T(2,10). Medial quandles fail to distinguish certain links, highlighting their limitations.
problem Detecting causality in spacetime using quandles.
method Investigated medial quandles' ability to distinguish specific links and knots.
result Medial quandles fail to distinguish certain links, including the connected sum of two Hopf links from an infinite series of relevant three-component links.
Study instanton Floer homology for links in RP^3 and use it to detect knots.
problem Detecting knots in RP3 using instanton Floer homology. method Compute instanton Floer homology for links in RP3 and use spectral sequences. result Khovanov homology detects the unknot and projective unknot in RP3. New method detects novel node categories in graphs with distribution shifts.
problem Detecting novel node categories in graphs with distribution shifts.
method Recall-Constrained Optimization with Selective Link Prediction (RECO-SLIP).
result RECO-SLIP outperforms existing methods in detecting novel node categories.
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer h…
The paper extends techniques to create non-left-orderable manifolds from links with multiple boundary components.
problem Creating infinite families of non-left-orderable manifolds from Dehn fillings.
method Order-detection of slopes to generalize techniques for multiple boundary components.
result Produces an infinite family of non-left-orderable Dehn fillings for hyperbolic links, including the Whitehead link.
This paper improves hierarchical community detection efficiency using local structural properties.
problem Efficiency of hierarchical community detection methods in large networks.
method Use of local structural network properties as proxies to improve efficiency.
result Achieves competitive results in modularity with improved efficiency.
In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used …
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…
New method detects essential tori in mixed singularity links.
problem Detecting essential tori in mixed singularity link complements.
method Analyzing properties of defining mixed polynomials.
result Explicit criteria for essential tori existence.
An explicit polynomial in the linking numbers lij and Milnor's triple linking numbers μ(rst) on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…
We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several speci…
We prove the existence of a degree 7 Vassiliev invariant of long (or string) two-component links which is not preserved under the simultaneous change of orientation of both components. The non-invertibility of this invariant can be detected by the standard weight system with values in the tensor square of the universal…
We prove that any link in S3 whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either Z or Z/2Z.
If L_1 and L_2 are two Brunnian links with all pairwise linking numbers 0, then we show that L_1 and L_2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L_1 is a Brunnian link with all pairwise linking numbers 0, and t…
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. W…
In this paper, we consider signal detection algorithms in a multiple-input multiple-output (MIMO) decode-forward (DF) relay channel with one source, one relay, and one destination. The existing suboptimal near maximum likelihood (NML) detector and the NML with two-level pair-wise error probability (NMLw2PEP) detector a…
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
A new method detects communities in multi-relational networks.
problem Detect communities in heterogeneous multi-relational networks.
method Message passing based algorithm on a hierarchical structure of homogeneous networks.
result Effectiveness of the proposed method confirmed on synthetic and real-world networks.
Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.
problem Detecting and obstructing sliceness of knots using surface invariants.
method Defining an invariant from Khovanov homology of boundary links.
result The invariant can obstruct sliceness of knots and detect slices for specific knots.
The Links--Gould polynomial detects causality in spacetimes.
problem Detecting causality in spacetimes using link invariants.
method Using the Links--Gould polynomial, which specializes to the Alexander--Conway polynomial.
result The Links--Gould polynomial distinguishes Allen-Swenberg links and detects causality.