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We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3 -stick unknots with ribbon linking numbers ± 1 and ± 3 , and we prove that the minimum folded ribbonlength for n -gons with obtuse interior angles is achieved when the n -gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number ± n is bounded from above by 2 n. [ABSTRACT FROM AUTHOR]

I point out a conceptually simple way of establishing one of the main lemmas in the paper "Solvability of equations in elementary functions" by A. Kanel-Belov, A. Malistov and R. Zaytsev. [ABSTRACT FROM AUTHOR]

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants c 1 , c 2 > 0 such that the ribbonlength is bounded above by c 1 Cr (K) 2 , and also by c 2 Cr (K) 3 / 2 . We use a different method for each bound. The constant c 1 is quite small in comparison to c 2 , and the first bound is lower than the second for knots and links with Cr (K) ≤ 12,748. [ABSTRACT FROM AUTHOR]

We note a simple algebraic proof of Frolkina's result that ℝ 3 does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots. [ABSTRACT FROM AUTHOR]

The paper constructs an example of a (non-ambient) isotopy of a (topological) knot in ℝ 3 which does not extend to an isotopy of a two-component link whose second component is a meridianal ring of this knot. By constructing the example, the paper answers a question raised in 2005 by Melikhov. Apart from presenting the corresponding example, the paper will discuss the consequences that the discovery of this example has for attacking a problem that was in literature asked in 1974 by Rolfsen: Namely deciding, whether there exist knots (and it is clear that at most everywhere wild knots might have such a property) that even with respect to ordinary (not necessarily ambient) isotopy are non-equivalent to the trivial knot. [ABSTRACT FROM AUTHOR]

In this paper, we answer a question raised in "Peripheral elements in reduced Alexander modules" [J. Knot Theory Ramifications 31 (2022) 2250058]. We also correct a minor error in that paper. [ABSTRACT FROM AUTHOR]

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard diagrams of 3 -manifolds. In this paper, for a knot inside S 3 , we further introduce an algorithm that computes an upper bound of the dimension of instanton knot homology from knot diagrams. We test the algorithm with all knots up to seven crossings as well as a more complicated knot 1 0 1 5 3 . In the second half of the paper, we show that if the instanton knot Floer homology of a knot has a specified form, then the knot must be an instanton L-space knot. [ABSTRACT FROM AUTHOR]

In this paper, we show that every spinal open book decomposition of a closed oriented 3 -manifold M spinal open book embeds into a certain spinal open book decomposition of # k S 2 × ̃ S 3 , the connected sum of k copies of the twisted S 3 -bundle over S 2 , where k depends on the spinal open book decomposition of M. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented 3 -manifolds into the trivial spinal open book of the 5 -sphere S 5 . Finally, we show that given a closed oriented 3 -manifold M , there exists a spinal open book for M such that M spinal open book embeds into the trivial spinal open book of S 5. In particular, this gives another proof of Hirsch's theorem which states that every closed orientable 3 -manifold embeds in S 5. [ABSTRACT FROM AUTHOR]

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, 3 -manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More 1 -cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order 1 , preprint (2018), arXiv:1810.05747]. [ABSTRACT FROM AUTHOR]

This paper discusses some geometric ideas associated with knots in real projective 3-space ℝ P 3 . These ideas are borrowed from classical knot theory. Since knots in ℝ P 3 are classified into three disjoint classes: affine, class- 0 non-affine and class- 1 knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in ℝ P 3 . We then study the notion of companionship of knots in ℝ P 3 and using it we provide geometric criteria for a knot to be affine. We also define a notion of "genus" for knots in ℝ P 3 and study some of its properties. We prove that this genus detects knottedness in ℝ P 3 and gives some criteria for a knot to be affine and of class- 1. We also prove a "non-cancellation" theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class- 0 non-affine knots with genus 1. Thus we highlight that, ℝ P 3 admits a knot theory with a truly different flavor than that of S 3 or ℝ 3 . [ABSTRACT FROM AUTHOR]

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an S -homotopy class of a nanoword into the S -homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups). [ABSTRACT FROM AUTHOR]

In the first of this series of papers, we looked at various generalized knot theories. In this follow up paper, we look at the theory of generalized braids and show how they are related to generalized knots. The twin groups are related to the theory of planar doodles and we prove a Markov-type theorem for them. An example and invariants are given to show that this is best possible. We also give conditions for when an exchange move is dependent on the other moves. [ABSTRACT FROM AUTHOR]

This paper, in homage to Patrick Dehornoy, tells the birth of Garside groups. [ABSTRACT FROM AUTHOR]

In this paper we study isotopy classes of closed connected orientable surfaces in the standard 3 -sphere. Such a surface splits the 3 -sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a 2 -component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial 1 -handles can recover the original surface. Therefore, we can reduce the study of surfaces in the 3 -sphere to that of 2 -component handlebody-links up to stabilizations. Then, by using G -families of quandles, we construct invariants of 2 -component handlebody-links up to attaching trivial 1 -handles, which lead to invariants of surfaces in the 3 -sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the 3 -sphere. [ABSTRACT FROM AUTHOR]

Consider two parallel lines ℓ 1 and ℓ 2 in ℝ 3 . A rail arc is an embedding of an arc in ℝ 3 such that one endpoint is on ℓ 1 , the other is on ℓ 2 , and its interior is disjoint from ℓ 1 ∪ ℓ 2 . Rail arcs are considered up to rail isotopies, ambient isotopies of ℝ 3 with each self-homeomorphism mapping ℓ 1 and ℓ 2 onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs. [ABSTRACT FROM AUTHOR]

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient ℤ / n ℤ (n ≥ 2) associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient ℤ / 2 ℤ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient ℤ / n ℤ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient ℤ / n ℤ) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542]. [ABSTRACT FROM AUTHOR]

In this paper, mosaic tiles have been used to build knots and define invariants since they were introduced by Lomonaco and Kauffman to describe quantum knots. In this paper, we propose a modified set of tiles to describe the front projections of Legendrian knots. We explore the effect of stabilization on the mosaic number of a Legendrian knot by defining and classifying "two-cell" stabilizations. Inspired by a construction of Ludwig et al. in 2013, we provide an infinite family of Legendrian unknots whose mosaic numbers are realized only in non-reduced projections. Finally, we provide a census of known mosaic numbers for Legendrian unknots and trefoils. [ABSTRACT FROM AUTHOR]

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal n -chart with two crossings I: Complementary domains of Γ 1 ∪ Γ n − 1 , J. Knot Theory Ramifactions27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal n -chart with two crossings II: Neighbourhoods of Γ 1 ∪ Γ n − 1 , Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math.113 (2019) 1693–1738, arXiv:1709.08827v2 ] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links. [ABSTRACT FROM AUTHOR]

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices. [ABSTRACT FROM AUTHOR]

Let n be the set of un-oriented and rational links with crossing number n , a precise formula for | n | was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let Λ n be the set of oriented rational links with crossing number n and let Λ n (d) be the set of oriented rational links with crossing number n (n ≥ 2) and deficiency d. In this paper, we derive precise formulas for | Λ n | and | Λ n (d) | for any given n and d and show that Λ n (d) = F n − d − 1 (d) + 1 + (− 1) n d 2 F ⌊ n 2 ⌋ − ⌊ d + 1 2 ⌋ (⌊ d 2 ⌋) , where F n (d) is the d th convolution of the convolved Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case). [ABSTRACT FROM AUTHOR]

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya's combinatorial description of the Heegaard Floer contact invariant. [ABSTRACT FROM AUTHOR]

The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree. [ABSTRACT FROM AUTHOR]

Gukov–Pei–Putrov–Vafa conjectured the existence of q-series whose radial limits are Witten–Reshetikhin–Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov–Pei–Putrov–Vafa and Gukov–Manolescu constructed homological blocks. In this paper, we construct indefinite false theta functions which are candidates of homological blocks for some plumbed 3-manifolds which are not weakly negative definite. Moreover we prove that, for the Poincaré homology sphere, our indefinite false theta function coincides with the original homological block. [ABSTRACT FROM AUTHOR]

As a generalization of the classical knots, knotoids deal with the open ended knot diagram in a surface. In recent years, many polynomial invariants for knotoids appeared, such as the bracket polynomial, the index polynomial and the nth polynomial. In this paper, we introduce a new polynomial invariant F-polynomial for knotoids and discuss some properties of the F-polynomial. Finally, we construct a family of knotoid diagrams which can be distinguished from each other by the F-polynomial but cannot be distinguished by the index polynomial and the nth polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module. [ABSTRACT FROM AUTHOR]

The 0 -trace of a knot is the 4 -manifold represented by the 0 -framing of the knot. In this paper, we survey methods constructing a pair of knots with diffeomorphic 0 -traces. In particular, we focus on Gompf–Miyazaki's dualizable pattern, Abe–Jong–Omae–Takeuchi's band presentation, and RGB-diagram given by Piccirillo and named by the author, and we draw the relations among these methods directly. As an application, we give a sufficient condition that two knots obtained by Abe–Jong–Omae–Takeuchi's method coincide. [ABSTRACT FROM AUTHOR]

In this paper, we describe a method of making an invariant of virtual links, generalizing Jones–Kauffman polynomial of virtual links. [ABSTRACT FROM AUTHOR]

In this paper, we present those properties of planar doodles, especially when regarded as four-valent graphs, that enable us to classify them into prime and super-prime doodles by analogy to a knot sum. We describe a method for partially characterizing a doodle diagram by a doodle code that describes the complementary regions of the diagram and use that code to enumerate all possible prime and super-prime doodle diagrams via their dual graph. In addition, we explore the relationship between planar doodles and twin groups, and note that a theorem of Tutte means that super-prime doodles have a Hamiltonian circuit and we give examples of prime doodles without a Hamiltonian circuit. [ABSTRACT FROM AUTHOR]

In this paper, we discuss applications of neural networks to recognizing knots and, in particular, to the unknotting problem. One of the motivations for this study is to understand how neural networks work on the example of a problem for which rigorous mathematical algorithms for its solution are known. We represent knots by rectangular Dynnikov diagrams and apply neural networks to distinguish a given diagram's class from a finite family of topological types. The data presented to the program is generated by applying Dynnikov moves to initial samples. The significance of using these diagrams and moves is that in this context the problem of determining whether a diagram is unknotted is a finite search of a bounded combinatorial space. In this way, this paper provides a foundation for further work where the neural network itself will learn to use the Dynnikov moves for knot recognition. Source code of the programs is available at https://github.com/nerusskikh/deepknots. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously known criteria, among which is the Khovanov homology criterion of Borodzik and the second author. [ABSTRACT FROM AUTHOR]

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set E A T k of all essential alternating rational 3-tangles for positive integer k , the authors showed that all knot diagrams in the numerator closure set C N (E A T 2 l + 1) and the denominator closure set C D (E A T 2 l + 2) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set A A T 4 n of additions of alternating rational tangles in E A T 4 . The set A A T 4 n generalizes E A T k and contains it as a subset for some k. We show that any closure set C (A A T 4 n) on A A T 4 n so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in A A T 4 n + 1 is constructed inductively from a tangle diagram in A A T 4 n by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary and sufficient combinatorial conditions for a link to be projective, that is, a link in ℝ ℙ 3 . This characterization is closely related to the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the notion of symmetric cycle, an interesting issue arising in projective links leading us to an easy necessary condition for alternating anti-inversely symmetric links. [ABSTRACT FROM AUTHOR]

In this paper, we find that for each given spatial graph, the number of its maximal trees is finite, and the quotient space corresponding to some maximal tree is also a spatial graph and it is homeomorphic to a disjoint union of wedge sum of circles. That means this quotient space is orientable and colorable. Thus, we define two equivalence invariants of spatial graphs, the coloring number bracket and the Alexander invariant set, which are coming from the coloring invariant and Alexander matrix. [ABSTRACT FROM AUTHOR]

An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all n ≥ 2 the H (n) -move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation H (n) -move to virtual knots and links. Virtualization and forbidden move are well-known unknotting operations for virtual knots and links. We also show that virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and H (n) -moves. [ABSTRACT FROM AUTHOR]

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic PSL (2 , ℂ) -representation to boundary-unipotent SL (2 , ℂ) -representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group. [ABSTRACT FROM AUTHOR]

In this paper, we construct a polynomial invariant of Kauffman type for kd(1)-linkoids and compute the polynomials for knotoids with up to three crossings. As a consequence, it is shown that the polynomial is different from the previous one in [Y. Miyazawa, A polynomial invariant of Kauffman type for knotoids, preprint]. [ABSTRACT FROM AUTHOR]

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner. [ABSTRACT FROM AUTHOR]

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology. [ABSTRACT FROM AUTHOR]

It is known that the category H of nilpotent weight modules over the quantum group associated with the super Lie algebra (2 | 1) is a relative pre-modular G -category. Its modified trace enables one to define an invariant of 3 -manifolds. In this paper, we show that the category H is a relative modular G -category which allows one to constructing a family of non-semi-simple extended topological quantum field theories, and the family of TQFTs is anomaly free. The quantum group associated with (2 | 1) is considered at odd roots of unity. [ABSTRACT FROM AUTHOR]

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type (7) if there exists a label m such that w (Γ) = 7 , w (Γ m ∩ Γ m + 1) = 7 where w (G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (7). [ABSTRACT FROM AUTHOR]

In this paper, we generalize the well-known constructions of invariants of framed links and 3-manifolds based on Hopf algebras to multiplier Hopf algebras. Results include a structural characterization of ribbon multiplier Hopf algebras from which invariants of framed links arise from the usual functorial construction to the ribbon category of modules, and the extension of Hennings construction of invariants of framed links and 3-manifolds obtained from Hopf algebras to algebraic quantum groups equipped with an appropriate trace. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two monoidal supercategories: the odd dotted Temperley–Lieb category ℒ o , • (δ) , which is a generalization of the odd Temperley–Lieb category studied by Brundan and Ellis in [Monoidal Supercategories, Commun. Math. Phys. 351 (2017) 1045–1089], and the odd annular Bar-Natan category ℬ o () , which generalizes the odd Bar-Natan category studied by Putyra in [A 2 -category of chronological cobordisms and odd Khovanov homology, Banach Center Publ. 103 (2014) 291–355]. We then show there is an equivalence of categories between them if δ = 0. We use this equivalence to better understand the action of the Lie superalgebra (1 | 1) on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and the second author in [An action of (1 | 1) on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711–742]. [ABSTRACT FROM AUTHOR]

A genus two handlebody-knot is a genus two handlebody embedded in the three-sphere. In this paper, we describe some necessary conditions for a two-component link to be a constituent link of a genus two handlebody-knot. We give a lower bound of the crossing number of constituent links of a genus two handlebody-knot such that two-variable Alexander polynomial of the link is nonzero. [ABSTRACT FROM AUTHOR]