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Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$., Comment: 10 pages

Plans’ theorem states that, for odd n, the first homology group of the n-fold cyclic covering of the three-dimensional sphere branched over a knot is the direct product of two copies of an Abelian group. A similar statement holds for even n. In this case, one has to factorize the homology group of n-fold covering by the homology group of two-fold covering of the knot. The aim of this paper is to establish similar results for Jacobians (critical group) of a circulant graph. Moreover, it is shown that the Jacobian group of a circulant graph on n vertices reduced modulo a given finite Abelian group is a periodic function of n.

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category., 18 pages, 6 figures, to appear in Trans. Amer. Math. Soc

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.

Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.

Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the \nu^-(K). Moreover, the difference l(K)-\nu^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance., Comment: 18 pages; Corrected Typos

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms., Comment: 28 pages, 16 figures, 1 table

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level number. We then find braid descriptions for $(1,1)$-positions of all $2$-bridge knots providing upper bounds for their level numbers, and also show that the $(-2, 3, 7)$-pretzel knot has level number two., Comment: 24 pages, 17 figures

Knotting is common in long polymers and significantly affects the polymer behavior. It is generally believed that knotting should increase polymer bending energy. However, many recent studies found...

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the trivial knot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the trivial knot of treewidth 2 can always be reduced to the trivial diagram with at most n untwist and unpoke Reidemeister moves.

We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show that $\operatorname{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants and bounded above by the topological $4$-genus $g_4^{\operatorname{top}}(K)$. We provide an infinite family of examples with $\operatorname{sn}(K), Comment: 40 pages, 11 figures, implements suggestions from a referee report and corrects a mistake in the appendix

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

We survey Ozsv\'ath-Szab\'o's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts ($\mathcal{A}_\infty$-modules, type $D$-structures, box tensor, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsv\'ath and Szab\'o's analytic construction of the type $D$-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the $DA$-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in $S^3$. The boundary DGAs $\mathcal{B}(n,k)$ and $\mathcal{A}(n,k)$ of [7] are replaced here by an associative algebra $\mathcal{C}(n)$. These are the notes of two lecture series delivered by Peter Ozsv\'ath and Zolt\'an Szab\'o at Princeton University during the summer of 2018., Comment: 24 pages, 12 figures, Minor errors have been corrected and the exposition has been improved. To appear in Periodica Mathematica Hungarica

In a previous paper, V\'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\hat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}} (T_1), \dotsc, \widetilde{\mathrm{CT}} (T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology., Comment: 72 pages, 48 figures, 5 tables. Minor revisions

We present a new method to estimate individual knot volumes based on a knot geometry model coupled with observations on branch characteristics. X-ray computer tomography and image analysis were used to measure the volume and geometry of 424 knots of Sitka spruce (Picea sitchensis (Bong.) Carrière). Knot geometry can be described mathematically by deriving functions for relative vertical position, diameter, and slope dependent on radial position in the stem. These functions were parameterized using “seemingly unrelated regression” and mixed-modelling techniques. This provided a base model for typical knots. To estimate individual knot volume, we used available data for branch diameter and insertion angle to obtain conditional predictions. We imputed the most likely knot trajectory, as relative vertical position cannot be measured on branches. The model explained up to 96% of the variability in knot volume by incorporating the branch measurements, in contrast to the 43% explained using the typical knot model. Knot volume assessment based only on conditional predictions of diameter and marginal predictions of vertical position also accounted for 96% of the variability. Therefore, measurements of branch diameter alone would be enough to obtain highly precise predictions of individual knot volume. This estimator is a first step towards a knot model to be used for the management of Sitka spruce in Great Britain.

In this paper, we evaluate Button’s claim that knot is a nasty connective. Knot’s nastiness is due to the fact that, when one extends the set $$\{ \lnot , \vee , \wedge , \rightarrow \}$$ with knot, the connective provides counterexamples to a number of classically valid operational rules in a sequent calculus proof system. We show that just as going non-transitive diminishes tonk’s nastiness, knot’s nastiness can also be reduced by dropping Reflexivity, a different structural rule. Since doing so restores all other rules in the system as validity-preserving, we are inclined to conclude that there, knot is not that nasty. However, since motivating non-reflexivity is harder than motivating non-transitivity, we also acknowledge that disagreement with our conclusion is possible.

The knot is one of the most remarkable topological features identified in an increasing number of proteins with important functions. However, little is known about how the knot is formed during protein folding, and untied or maintained in protein unfolding. By means of all-atom molecular dynamics simulation, here we employ methyltransferase YbeA as the knotted protein model to analyze changes of the knotted conformation coupled with protein unfolding under thermal and mechanical denaturing conditions. Our results show that the trefoil knot in YbeA is occasionally untied via knot loosening rather than sliding under enhanced thermal fluctuations. Through correlating protein unfolding with changes in the knot position and size, several aspects of barriers that jointly suppress knot untying are revealed. In particular, protein unfolding is always prior to knot untying and starts preferentially from separation of two α-helices (α1 and α5), which protect the hydrophobic core consisting of β-sheets (β1–β4) from exposure to water. These β-sheets form a loop through which α5 is threaded to form the knot. Hydrophobic and hydrogen bonding interactions inside the core stabilize the loop against loosening. In addition, residues at N-terminal of α5 define a rigid turning to impede α5 from sliding out of the loop. Site mutations are designed to specifically eliminate these barriers, and easier knot untying is achieved under the same denaturing conditions. These results provide new molecular level insights into the folding/unfolding of knotted proteins.

This paper introduces techniques for computing a variety of numerical invariants associated to a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of a regular neighborhood of the binding. From this front projection, we compute the rotation number for any null-homologous Legendrian knot as a count of oriented cusps and linking or intersection numbers; in the case that the manifold has non-trivial second homology, we can recover the rotation number with respect to a Seifert surface in any homology class. We also provide explicit formulas for computing the necessary intersection numbers from the front projection, and we compute the Euler class of the contact structure supported by the open book. Finally, we introduce a notion of Lagrangian projection and compute the classical invariants of a null-homologous Legendrian knot from its projection to a fixed page., 18 pages, 8 figures; V2: Minor corrections, to appear in Michigan Math. J

We show that the equivariant and non-equivariant non-orientable 4-genus of [Formula: see text]-periodic knots may differ, for any choice of [Formula: see text]. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic knot. These stand in contrast to Edmonds’ acclaimed result by which the equivariant and non-equivariant Seifert genus of a periodic knot agree.

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these results and prove that free products of residually finite quandles are residually finite provided their associated groups are residually finite. As associated groups of link quandles are link groups, which are known to be residually finite, it follows that link quandles are residually finite., Comment: 11 pages, final version, to appear in Monatshefte f\"ur Mathematik

Combinatorial two-player games have recently been applied to knot theory. Examples of this include the Knotting-Unknotting Game and the Region Unknotting Game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the game. In this paper, we introduce the Linking-Unlinking Game which is played on two-component link shadows. We then present winning strategies for the Linking-Unlinking Game played on all shadows of two-component rational tangle closures and played on a large family of general two-component link shadows., Comment: 34 pages, 30 figures

The topological classification of nodal links and knot has enamored physicists and mathematicians alike, both for its mathematical elegance and implications on optical and transport phenomena. Central to this pursuit is the Seifert surface bounding the link/knot, which has for long remained a mathematical abstraction. Here we propose an experimentally realistic setup where Seifert surfaces emerge as boundary states of 4D topological systems constructed by stacking 3D nodal line systems along a 4th quasimomentum. We provide an explicit realization with 4D circuit lattices, which are freed from symmetry constraints and are readily tunable due to the dimension and distance agnostic nature of circuit connections. Importantly, their Seifert surfaces can be imaged in 3D via their pronounced impedance peaks, and are directly related to knot invariants like the Alexander polynomial and knot Signature. This work thus unleashes the great potential of Seifert surfaces as sophisticated yet accessible tools in exotic bandstructure studies. Although characterizing the Seifert surface is central to the realization of novel phases in topological materials, this has been a purely mathematical construction so far. This paper proposes an experimentally realistic scheme for the realization and imaging of 4D-embedded Seifert surfaces, opening up a way for experimental topological characterization

Let K be a nontrivial knot in S 3 with the exterior E ( K ) , and denote π 1 ( E ( K ) ) by G ( K ) . We prove that for any hyperbolic knot K and any nontrivial element g ∈ G ( K ) , there are only finitely many Dehn fillings of E ( K ) which trivialize g. We also demonstrate that there are infinitely many nontrivial elements in G ( K ) which cannot be trivialized by nontrivial Dehn fillings.

An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in R 4 , respectively. The triple point number t ( F ) of an embedded/immersed surface-knot F is the minimum number of triple points required for a diagram of F. Satoh proved that (i) there does not exist an embedded surface-knot F such that t ( F ) = 1 , and (ii) there does not exist an embedded 2-knot F such that t ( F ) = 2 or 3 . In this paper, we prove similar results for immersed surface-knots with some conditions.

This paper explains a research on the character varieties and the Kauffman bracket skein module (KBSM) of knot exteriors, which has been done by the author in his stay at the University of California, Riverside, 2004-2006. As a main consequence, for any 2-bridge knot, we gave a representation theoretical proof to the conjecture that degree 0 abelian knot contact homology H C 0 a b ( K ) of a knot K in the 3-space R 3 is isomorphic to the S L 2 ( C ) -character ring of the fundamental group of the 2-fold branched cover of the 3-sphere S 3 branched along K. (This result was first shown by L. Ng [23] by using cord formalism.) Based on this result, deep studies of the conjecture has been done in [18] , [19] , [20] .

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity., 14 pages, 13 figures

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.

We introduce the Alexander–Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector. © 2019. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: http://dx.doi.org/10.4064/fm519-6-2018

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study we ask if non-native interactions are strictly necessary to tangle a protein, or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous, interactions. In order to address this problem we conducted extensive Monte Carlo simulations of lattice models of proteinlike sequences designed to fold into a pre-selected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting, and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack, because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is enroute to the native structure. Our results thus show that - at least in the context of the adopted model - non-native interactions are not necessary to knot a protein. However, when they are taken into account into protein energetics it is possible to find specific, non-local non-native interactions that operate as a scaffold that assists the knotting step.

We use Langevin dynamics simulations to model, at atomistic resolution, how various natively–knotted proteins are unfolded in repeated allosteric translocating cycles of the ClpY ATPase. We consider proteins representative of different topologies, from the simplest knot (trefoil 31), to the three–twist 52 knot, to the most complex stevedore, 61, knot. We harness the atomistic detail of the simulations to address aspects that have so far remained largely unexplored, such as sequence–dependent effects on the ruggedness of the landscape traversed during knot sliding. Our simulations reveal the combined effect on translocation of the knotted protein structure, i.e. backbone topology and geometry, and primary sequence, i.e. side chain size and interactions, and show that the latter can even dominate translocation hindrance. In addition, we observe that, due to the interplay between the knotted topology and intramolecular contacts, the transmission of tension along the peptide chain occurs very differently from homopolymers. Finally, by considering native and non–native interactions, we examine how the disruption or formation of such contacts can affect the translocation processivity and concomitantly create multiple unfolding pathways with very different activation barriers.

The knotting of biomolecules impacts their function, and enables them to carry out new tasks. Likewise, complex topologies underpin the operation of many synthetic molecular machines. The ability to generate and control more complex knotted architectures is essential to endow these machines with more advanced functions. Here we report the synthesis of a molecular knot with eight crossing points, consisting of a single organic loop woven about six templating metal centres, via one-pot self-assembly from a simple pair of dialdehyde and diamine subcomponents and a single metal salt. The structure and topology of the knot were established by NMR spectroscopy, mass spectrometry and X-ray crystallography. Upon demetallation, the purely organic strand relaxes into a symmetric conformation, whilst retaining the topology of the original knot. This knot is topologically chiral, and may be synthesised diastereoselectively through the use of an enantiopure diamine building block.

In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in $S^3$ carries a natural action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on 3 parameters $q, t_1, t_2$. As a consequence, for a knot $K$ satisfying this conjecture, we defined a three-variable polynomial invariant $J^K_n(q,t_1,t_2)$ generalizing the classical colored Jones polynomials $J^K_n(q)$. In this paper, we give explicit formulas and provide a quantum group interpretation for the generalized Jones polynomials $J^K_n(q,t_1,t_2)$. Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by K.\ Habiro: as in the classical case, they imply the integrality of $J^K_n(q,t_1,t_2)$ and, in fact, make sense for an arbitrary knot $K$ independent of whether or not it satisfies our earlier conjecture. When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of $J^K_n(q,t_1)$ are determined by Macdonald orthogonal polynomials of type $A_1$., Comment: 23 pages, minor corrections in v2

Arthroscopic knot tying is a crucial component for a successful arthroscopic shoulder surgery. Knot tying should not be difficult to master or time consuming to perform. This study describes a new sliding locking knot for arthroscopic shoulder surgery and we named it Banarji knot, in the name of the author. It is a low profile, non-bulky, and double locking knot, which makes it a more secure knot.

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition function at or around roots of unity $q=e^{2\pi i \frac{1}{K}}$ with rational level $K=\frac{r}{s}$ where $r$ and $s$ are coprime integers. From the exact expression for the $G=SU(2)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds at other roots of unity obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at other roots of unity. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the standard limit in the volume conjecture., Comment: 22 pages, v2 minor revisions, version to appear in JHEP

This chapter discusses the Casson invariant of a closed three-manifold for a knot in a closed three-manifold. This is a chord diagram version of the method noted in the end of by Lickorish. It also discusses the extension of the construction of the Casson invariant from the universal Vassiliev-Kontsevich invariant by adding the 3T relation and some natural relations. The semi-simple quotient of this representation is isomorphic to a disjoint union of two copies of the natural representation of SL(2, Z). The contribution from the extension of SL(2, Z) appears in the Levi part of this representation, and this part seems to contain information for the Casson invariant. The Kirby moves correspond to Kirby’s handle slide move, where any component can be changed only along a thin line component since the thick line component is a knot and is not applied the surgery.

KNOT is one of the 32 candidates in the second round of NIST’s lightweight cryptography standardization process. The KNOT family consists of bit-slice lightweight Authenticated Encryption with Associated Data (AEAD) and hashing algorithms. In this paper, we evaluate the security for the initialization phase of two members of the KNOT-AEAD family by differential-linear cryptanalysis.

We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums., 12 pages. To appear in J. Inst. Math. Jussieu

The trace of the -framed surgery on a knot in is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable -dimensional knot invariants. For each, this provides conditions that imply a knot is topologically -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice., Compositio Mathematica, 157 (10), ISSN:0010-437X, ISSN:1570-5846

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots., 27 pages, 13 figures. v2 minor corrections

Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology $3_1$, $4_1$, or $5_1$, as well as unknotted polymers that are capable of forming S-loops. We measure the variation of the free energy $F$ with the end-to-end polymer extension length $X$ and examine the effect of varying the polymer topology, persistence length $P$ and cylinder diameter $D$ on the free energy functions. Similarly, we characterize the behavior of the knot span along the channel. We find that increasing the knot complexity increases the typical size of the knot. In the regime of low $X$, where the knot/S-loop size is large, the conformational behavior is independent of polymer topology. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of $F$ with position of a knot in conical channels for various values of the cone angle $\alpha$. The free energy decreases as the knot moves in a direction where the cone widens, and it also decreases with increasing $\alpha$ and with increasing knot complexity. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in $F$ is the change in the size of the hairpins as the knot moves to the wider region of the channel., Comment: 15 pages, 11 figures, supplemental information

We prove the analyticity of smooth critical points for O’Hara’s knot energies [Formula: see text], with [Formula: see text] and [Formula: see text], subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of [Formula: see text] subject to a fixed length constraint are not only [Formula: see text] but also analytic. Our approach is based on Cauchy’s method of majorants and a decomposition of the gradient that was adapted from the Möbius energy case [Formula: see text].

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati\'c's contact class is a well-defined invariant of contact manifolds., Comment: 63 pages, 13 figures; v2: corrected Lemma 3.3 and subsequent material, many other small changes; v3: accepted version, substantially revised to correct the proof of the main theorem

A classic knot is an embedding of an one-dimensional sphere S1 in a real three-dimensional environment, usually R3. Under these conditions it is possible to consider the knot diagram, that is, the planar projection of the embedding. This resembles a curve in which the intersections are exchanged for interruptions in its trace, thus indicating that one arc passes over the other. In Knot Theory we study algebraic invariants extracted from the complement of the embedding, and this complement is visible in the case of embedding S1 --> R3. One of the invariants extracted from the knot diagram is the Alexander Polynomial. In this article we show how the process of determining the Alexander Polynomial of a knot can be transported to an algorithm implemented in Python Language and obtained from a drawn diagram with the aid of a computer mouse.

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define $\hat{Z}^G$ for negative definite plumbed 3-manifolds and $F_K^G$ for torus knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu, there is a surgery formula relating $F_K^G$ to $\hat{Z}^G$ of a Dehn surgery on the knot $K$. Furthermore, specializing to symmetric representations, $F_K^G$ satisfies a recurrence relation given by the quantum $A$-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.

Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory., 26 pages, 4 figures

We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and sampling them with the counting measure on from sets of a fixed number of vertices n. We prove that random rooted knot diagrams are highly composite and hence almost surely knotted (this is the analogue of the Frisch-Wasserman-Delbruck conjecture) and extend this to unrooted knot diagrams by showing that almost all knot diagrams are asymmetric. The model is similar to one of Dunfield, et al.

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3)., 13 pages, 25 figures

A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product of the knot quandle or the knot symmetric quandle of a surface-link in 4-space can be used to classify or construct invariants of 1-handles attaching to the surface-link. We also compute the tensor products for dihedral quandles and their symmetric doubles.

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $$\mathfrak {sl}_N$$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.