Recent zbMATH articles in MSC 57Khttps://www.zbmath.org/atom/cc/57K2021-06-15T18:09:00+00:00WerkzeugOn the arithmetic and the geometry of skew-reciprocal polynomials.https://www.zbmath.org/1460.111232021-06-15T18:09:00+00:00"Liechti, Livio"https://www.zbmath.org/authors/?q=ai:liechti.livioSummary: We reformulate questions due to \textit{D. H. Lehmer} [Ann. Math. (2) 34, 461--479 (1933; Zbl 0007.19904)] and \textit{A. Schinzel} and \textit{H. Zassenhaus} [Mich. Math. J. 12, 81--85 (1965; Zbl 0128.03402)] in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails understanding that the difference between orientation-preserving and orientation-reversing mapping classes is at least as complicated as answering these questions.New computations of the superbridge index.https://www.zbmath.org/1460.570062021-06-15T18:09:00+00:00"Shonkwiler, Clayton"https://www.zbmath.org/authors/?q=ai:shonkwiler.claytonThis paper raises from 29 to 49 the number of prime knots with ten or fewer crossings for which the superbridge index is known, adding also 10 more to the 24 previously known for primes with more than ten crossings. Tables are included that credit numerous authors with prior work along these lines and also indicate the known bounds on unknown indices for the remaining 200 primes through ten crossings. Curiously, there are still no known 12-crossing examples, just three alternating ones with more than ten crossings (all dealt with successfully by \textit{N. H. Kuiper} [Math. Ann. 278, 193--209 (1987; Zbl 0632.57006)]), and apparently none with more than one minimal crossing writhe. (Readers should note that the author, unlike Rolfsen, has renumbered the last four 10-crossing knots in his table.) For the 10 new examples with 13 and 14 crossings [Appendix D] diagrams are included which exhibit homomorphisms of the knot group onto the symmetric group on five letters, in order to show that they each have bridge number greater than 3. The corresponding 5-sheeted covering spaces (and their related ``simple'' 4- and 3-fold descendants) are interesting in their own right, in that they often admit visualizable surfaces bounded by the branch curve of index 2, on which one can easily see its linking numbers with other branches.
Update: This reviewer is informed that the author has recently added an 84th prime knot to the list of all those with known superbridge index [Appendix B].
Reviewer: Kenneth A. Perko Jr. (Scarsdale)Theory of spatial graphs and Alexander polynomials.https://www.zbmath.org/1460.050552021-06-15T18:09:00+00:00"Wu, Zhongtao"https://www.zbmath.org/authors/?q=ai:wu.zhongtaoSummary: We give a brief survey of the theory of spatial graph and explain the joint work with Bao that the author discussed at the first Annual Meeting of International Consortium of Chinese Mathematicians (ICCM 2017).
For the entire collection see [Zbl 1454.00056].Large color \(R\)-matrix for knot complements and strange identities.https://www.zbmath.org/1460.570142021-06-15T18:09:00+00:00"Park, Sunghyuk"https://www.zbmath.org/authors/?q=ai:park.sunghyukA geometric invariant of virtual \(n\)-links.https://www.zbmath.org/1460.570112021-06-15T18:09:00+00:00"Winter, Blake K."https://www.zbmath.org/authors/?q=ai:winter.blake-kVirtual \(n\)-links were introduced by the author in [J. Knot Theory Ramifications 24, No. 14, Article ID 1550062, 38 p. (2015; Zbl 1331.57027)] as a generalisation of the topological interpretation of Kauffman's virtual links. Let \(V_n\) be the set of pairs \((M \times [0,1],N)\), where \(M\) is a compact \((n+1)\)-manifold and \(N\) is an \(n\)-manifold embedded in the interior of \(M \times [0,1]\). Define a relation \(\sim\) on \(V_n\) by declaring that two such pairs \((M_1 \times [0,1],N_1)\) and \((M_2 \times [0,1],N_2)\) are \(\sim\) related if there exists an embedding \(f:M_1 \hookrightarrow M_2\) such that \(f \times id_{[0,1]}(N_1)=N_2\). A virtual \(n\)-link is then defined as an equivalence class under the equivalence relation generated by \(\sim\) together with smooth isotopy.
The paper under review is a continuation of this work. For a virtual \(n\)-link \(K\), the author defines a new virtual link \(VD(K)\), which is invariant under virtual equivalence of \(K\). The homotopy type of the Dehn space of \(VD(K)\) is therefore an invariant of \(K\). Some applications to higher-dimensional virtual links are also given. Being geometric, the invariant is easy to generalize to higher dimensions. \par
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)Multivariate Alexander quandles. II: The involutory medial quandle of a link (corrected).https://www.zbmath.org/1460.570082021-06-15T18:09:00+00:00"Traldi, Lorenzo"https://www.zbmath.org/authors/?q=ai:traldi.lorenzoThe disk complex and topologically minimal surfaces in the 3-sphere.https://www.zbmath.org/1460.570182021-06-15T18:09:00+00:00"Campisi, Marion"https://www.zbmath.org/authors/?q=ai:campisi.marion-moore"Torres, Luis"https://www.zbmath.org/authors/?q=ai:torres.luis-caicedo|torres.luis-miguelVirtual mosaic knot theory.https://www.zbmath.org/1460.570092021-06-15T18:09:00+00:00"Ganzell, Sandy"https://www.zbmath.org/authors/?q=ai:ganzell.sandy"Henrich, Allison"https://www.zbmath.org/authors/?q=ai:henrich.allison-kCurved Rickard complexes and link homologies.https://www.zbmath.org/1460.570152021-06-15T18:09:00+00:00"Cautis, Sabin"https://www.zbmath.org/authors/?q=ai:cautis.sabin"Lauda, Aaron D."https://www.zbmath.org/authors/?q=ai:lauda.aaron-d"Sussan, Joshua"https://www.zbmath.org/authors/?q=ai:sussan.joshuaA Rickard complex is a certain complex of bimodules which, in the context of BGG category \(\mathcal{O}\) and, more generally, categorified quantum groups, generates a categorical braid group action. The paper under review studies certain deformations of Rickard complexes, which are called curved Rickard complexes in the paper. This gives rise to deformations of various types of link homologies and allows one to conceptually connect such deformations to the classical (undeformed) versions of these homology theories.
Reviewer: Volodymyr Mazorchuk (Uppsala)Knotting probability of an arc diagram.https://www.zbmath.org/1460.570042021-06-15T18:09:00+00:00"Kawauchi, Akio"https://www.zbmath.org/authors/?q=ai:kawauchi.akioA spatial arc \(L\) is an oriented polygonal arc in the 3-space \(\mathbb{R}^3\), and this paper is concerned with the question: How a linear object such as a spatial arc can be considered as knotted? This question is motivated by examples such as a protein or a linear polymer in science. The author defines the knotting probability \(p(D)\) of an arc diagram \(D\) (a diagram of the spatial arc \(L\)) as the quadruplet of knotting probabilities (all are rational numbers) that can be determined from \(D\) uniquely up to isomorphisms. The definition of the knotting probability \(p(D)\) of an arc diagram uses an argument on a chord diagram derived from a ribbon surface-knot in the 4-space \(\mathbb{R}^4\). For simple arc diagrams \(D\) the knotting probability is easily calculable and several examples are given. It should be noted that this knotting probability is very different from the notion of a knotting probability of a random polygon or random polygonal arc. The knotting probability defined here can behave differently than one would expect. For example, while the knotting probability \(p(D)\) is independent of a choice of the orientation of an arc diagram \(D\), the knotting probabilities of an arc diagram and its mirror image are generally different.
Reviewer: Claus Ernst (Bowling Green)Algebra of quantum \(\mathcal{C} \)-polynomials.https://www.zbmath.org/1460.810932021-06-15T18:09:00+00:00"Mironov, Andrei"https://www.zbmath.org/authors/?q=ai:mironov.andrei-d|mironov.andrei-evgenevich"Morozov, Alexei"https://www.zbmath.org/authors/?q=ai:morozov.alexei-yurievichSummary: Knot polynomials colored with symmetric representations of \(\mathrm{SL}_q(N)\) satisfy difference equations as functions of representation parameter, which look like quantization of classical \(\mathcal{A} \)-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum \(\mathcal{C} \)-polynomials. It turns out that, for each knot, one can actually derive two difference equations of a finite order for these coefficients, those with shifts in spin \(n\) of the representation and in \(A = q^N \). Thus, the \(\mathcal{C} \)-polynomials are much richer and form an entire ring. We demonstrate this with the examples of various defect zero knots, mostly discussing the entire twist family.Studying complex manifolds by using groups \(G_n^k\) and \(\Gamma_n^k\).https://www.zbmath.org/1460.320622021-06-15T18:09:00+00:00"Manturov, Vassily Olegovich"https://www.zbmath.org/authors/?q=ai:manturov.vassily-olegovich"Wan, Zheyan"https://www.zbmath.org/authors/?q=ai:wan.zheyanPoincaré duality in dimension 3. In memory of Charles B. Thomas.https://www.zbmath.org/1460.570012021-06-15T18:09:00+00:00"Hillman, Jonathan A."https://www.zbmath.org/authors/?q=ai:hillman.jonathan-arthurThe two main objects considered in the present book are Poincaré duality complexes and Poincaré duality groups, concentrating mainly on the ``critical dimension'' 3. Poincaré duality complexes first appeared in a paper by C.T.C. Wall from 1967 and model the homotopy types of closed manifolds, satisfying Poincaré duality; the notion of Poincaré duality groups on the other hand was introduced by F.E.A. Johnson and Wall in 1972 as an algebraic analogue of the notion of closed \textit{aspherical} manifold: a group \(\pi\) is an \(n\)-dimensional Poincaré duality group (a \(\mathrm{PD}_n\)-group for short) if and only if its classifying space (or Eilenberg-MacLane space, or \(K(\pi,1)\), i.e. aspherical (contractible universal covering) and with fundamental group \(\pi\)) satisfies Poincaré duality of dimension \(n\); an algebraic definition, using homological algebra, was given shortly thereafter by \textit{R. Bieri} [Homological dimension of discrete groups. London: Queen Mary College, University of London (1976; Zbl 0357.20027)] which, together with the book of [\textit{K. S. Brown}, Cohomology of groups. New York, NY: Springer (1982; Zbl 0584.20036)], is one of the classical texts on the cohomology of groups.
Concerning dimension 2, by results of Eckmann, Müller and Linnell around 1980, every \(\mathrm{PD}_2\)-group is isomorphic to the fundamental group of a closed surface. On the other hand, in dimension 4 and higher dimensions there is a plethora of exotic examples, e.g. for every \(n \ge 4\) there is a finitely generated \(\mathrm{PD}_n\)-group which is not finitely presented and hence not isomorphic to the fundamental group of a closed aspherical \(n\)-manifold, and there are uncountably many \(\mathrm{PD}_4\)-groups (Wall conjectured that every \textit{finitly presented} \(\mathrm{PD}_n\)-groups is the fundamental group of a closed, aspherical \(n\)-manifold). So this directs attention to the critical dimension 3 where it is not known whether every \(\mathrm{PD}_3\)-group is isomorphic to the fundamental group of a closed aspherical 3-manifold. The fundamental group plays a crucial role for aspherical 3-manifolds: after a long history and in particular the geometrization of 3-manifolds, closed aspherical 3-manifolds are well understood and determined by their fundamental groups; however, no purely group-theoretical characterization of their fundamental groups is known, so the groups themselves remain somewhat mysterious. On the other hand, \(\mathrm{PD}_3\)-groups is a rather sophisticated notion, so one would like to find also some more basic and concrete group-theoretical properties which determine 3-manifold groups; such properties and their possible generalizations to \(\mathrm{PD}_3\)-groups are also an important point of the present book. An account on algebraic properties of 3-manifold groups, after the solution of almost all of Thurston's famous problems on 3-manifolds and Kleinian groups, can be found in a book by \textit{M. Aschenbrenner} et al. [3-manifold groups. Zürich: European Mathematical Society (EMS) (2015; Zbl 1326.57001)] (an interesting property of 3-manifold groups is e.g. coherence, i.e. every finitely generated subgroup is finitely presented, and coherent groups are discussed in detail in a survey by [\textit{D. T. Wise}, Ann. Math. Stud. 205, 326--414 (2020; Zbl 1452.57019)], as a class of groups close in nature to the fundamental groups of 3-manifolds).
Concerning 3-dimensional Poincaré duality complexes instead, every orientable \(\mathrm{PD}_3\)-complex is a connected sum of indecomposables (with respect to internal connected sums and boundary-connected sums, algebraically with respect to free products), and the indecomposables are either aspherical or have virtually free fundamental group, isomorphic to the fundamental groups of certain finite graphs of finite groups (trees with nontrivial cyclic edge groups and finite vertex groups of cohomological period dividing 4, e.g. dihedral groups which occur as fundamental groups of finite \(\mathrm{PD}_3\)-complexes but not of 3-manifolds). Explicit examples discussed in the book have as fundamental group the dihedral or symmetric group \(S_3\) of order 6, and the free products with amalgamation \(S_3 *_{\mathbb Z/2\mathbb Z}S_3\) and \(S_3 *_{\mathbb Z/2\mathbb Z}\mathbb Z/4\mathbb Z\) which are fundamental groups of \(\mathrm{PD}_3\)-complexes but not of 3-manifolds. ``This book shall give an account of the reduction to indecomposables for \(\mathrm{PD}_3\)-complexes, and what is presently known about them'' (following work of Hendriks, Swarup, C.B. Thomas, Turaev, Crisp and the present author). This is the content of the first seven chapters of the book (containing also a discussion of the finite groups with periodic cohomology, related to the spherical space form problem). However, ``the primary interest is in the aspherical case'', and in particular in a discussion of possible approaches to prove that every \(\mathrm{PD}_3\)-group is a 3-manifold group.
The book has about 160 pages, a final appendix with 64 open questions and seven pages of references; the titles of the 12 chapters as follows:
1. Generalities; 2. Classification, Realization and Splitting; 3. The relative case; 4. The Centralizer Condition; 5. Orientable \(\mathrm{PD}_3\)-complexes with \(\pi\) virtually cyclic; 6. Indecomposable orientable \(\mathrm{PD}_3\)-complexes; 7. Nonorientable \(\mathrm{PD}_3\)-complexes; 8. Asphericity and 3-manifolds; 9. Centralizers, normalizers and ascendant subgroups; 10. Splitting along \(\mathrm{PD}_2\)-subgroups; 11. The Tits Alternative; 12. Homomorphisms of nonzero degree.
``Our general approach is to prove most assertions which are specific about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results.'' The book is located in the rich and fascinating field of intersections of low-dimensional topology, 3-manifold topology, group theory, homological algebra and cohomology of groups. Some of the main topics discussed are various group-theoretical properties of 3-manifold groups and their possible generalizations to \(\mathrm{PD}_3\)-groups. The book is densely written (a reader needs a solid background in cohomology of groups), contains a lot of information and is stimulating to read (maybe starting spontaneously with one of the many subsections of some chapter); it is the first book treating \(\mathrm{PD}_3\)-complexes and \(\mathrm{PD}_3\)-groups in a systematic way.
Reviewer: Bruno Zimmermann (Trieste)The Kodaira dimension of contact 3-manifolds and geography of symplectic fillings.https://www.zbmath.org/1460.530752021-06-15T18:09:00+00:00"Li, Tian-Jun"https://www.zbmath.org/authors/?q=ai:li.tian-jun|li.tianjun"Mak, Cheuk Yu"https://www.zbmath.org/authors/?q=ai:mak.cheuk-yuIf \((Y,\xi)\) is a closed, connected, and co-oriented contact manifold, then a symplectic manifold \((N,\omega_N)\) with boundary \(\partial N\) and a locally defined outward Liouville vector field \(V\) along \(\partial N\) such that the induced contact manifold \((\partial N,\xi_N)\) is contactomorphic to \((Y,\xi)\) is called a symplectic filling of \((Y,\xi)\).
A symplectic capping, or cap, \((P,\omega_P)\) of \((Y,\xi)\) is defined in exactly the same way as a symplectic filling with the requirement that \(V\) points inward instead onward. The induced contact 1-form is denoted as \(\alpha_P\) and \((P,\omega_P,\alpha_P)\) forms a concave symplectic pair.
A Stein filling of \((Y, \xi)\) is a \(4\)-manifold \((N,\omega_N)\) that is a Stein domain such that \((Y, \xi)\) is contactomorhpic to \(\partial N\) with the contact structure induced by the complex tangencies. The characteristic numbers for \(N\) are the Betti number \(b_1(N)\), signature \(\sigma(N)\), and the Euler characteristic \(\chi(N)\).
If \((X,\omega)\) is a closed symplectic 4-manifold and \(D\) is a smooth symplectic surface in \(X\), then \(D\) is called maximal if any symplectic exceptional class in \((X,\omega)\) pairs positively with \([D]\).
If \((P,\omega_P)\) is a concave symplectic manifold and \(\alpha_P\) is a contact 1-form on \(\partial P\) induced by an inward pointing Liouville vector field, then \((P,\omega_P,\alpha_P)\) is called a concave symplectic pair with rational period if \(\frac1{2\pi}[(\omega_P,\alpha_P)]\in H^2(P,\partial P;\mathbb{Q})\).
If \((P,\omega_P,\alpha_P)\) is a concave symplectic pair with rational period, then a closed symplectic hypersurface \(D\) is called a Donaldson hypersurface of \((P,\omega_P,\alpha_P)\) if it is Lefschetz dual to an integral multiple of \(\frac1{2\pi}[(\omega_P,\alpha_P)]\). A symplectic cap is called a Donaldson cap if it admits a Donaldson hypersurface, and a Donaldson cap with a chosen Donaldson hypersurface is called a polarized cap.
If \(X\) is a closed, oriented smooth 4-manifold and \(\mathcal{E}_X\) is the set of cohomology classes whose Poincaré duals are represented by smoothly embedded spheres of self-intersection \(-1\), then \(X\) is said to be minimal if \(\mathcal{E}_X\) is the empty set.
A compact-oriented manifold \(W\) such that \(\partial W=Y_+\cup (-Y_-)\) is called an oriented cobordism from a closed oriented manifold \(Y_-\) to another \(Y_+\). An oriented cobordism \(W\) with a symplectic structure \(\omega\) compatible with the orientation is called a symplectic cobordism if \((W,\omega)\) is symplectic concave at \(Y_-\) and convex at \(Y_+\). A symplectic cobordism \((W,\omega)\) is called exact if the locally defined Liouville vector fields near \(Y_+\) and \(Y_-\) extend to a global Liouville vector field.
In [Mich. Math. J. 51, No. 2, 327--337 (2003; Zbl 1043.53066)], \textit{A. Stipsicz} showed that the set \(\{2\chi(N)+3\sigma(N)\}\subset\mathbb{Z}\) is bounded from below for any Stein filling \((N,\omega_N)\) of \((Y,\xi)\). As a nice corollary, they showed that if \(b_2^+=0\) for any Stein filling \((N,\omega_N)\) of \((Y,\xi)\), then the set \(\{(b_1(N),\chi(N),\sigma(N))\}\) is finite.
In [Proc. Lond. Math. Soc. (3) 114, No. 1, 159--187 (2017; Zbl 1373.57046)], the present authors and \textit{K. Yasui} introduced three types of caps, namely, Calabi-Yau caps, uniruled caps, and adjunction caps for contact manifolds.
A Calabi-Yau cap of a contact \(3\)-manifold \((Y,\xi)\) is a compact symplectic manifold \((P,\omega_P)\) which is a strong concave filling of \((Y,\xi)\) such that \(c_1(P)\) is torsion.
An uniruled cap of a contact \(3\)-manifold \((Y,\xi)\) is a symplectic concave filling \((P,\omega_P)\) of \((Y,\xi)\) such that \(c_1(P)\cdot[(\omega_P,\alpha_P)]>0\) for some Liouville one-form \(\alpha_P\), where \([(\omega_P,\alpha_P)]\) is a relative cohomology class in \(H^2(P,\partial P,\mathbb R)\).
Another type of caps called adjunction caps is based on an observation that existence of a smoothly embedded surface in a closed symplectic manifold with sufficiently large self-intersection number relative to the genus implies that the symplectic manifold is uniruled.
These authors also proved that if a contact \(3\)-manifold \((Y,\xi)\) admits a Calabi-Yau cap, or a uniruled cap, or an adjunction cap \((P,\omega_P)\), then the set \(\{(b_1(P),\chi(P),\sigma(P))\}\) is finite.
If, moreover, a Calabi-Yau cap \((P,\omega_P)\) cannot be embedded in an uniruled manifold, then all exact fillings of \((Y,\xi)\) have torsion first Chern class.
In this paper, the authors introduce the Kodaira dimension of contact 3-manifolds, establish the geography of various kinds of symplectic fillings, find a lower bound of \(2\chi+3\sigma\) for all of its minimal symplectic fillings, and discuss various aspects of exact self-cobordisms of fillable contact 3-manifolds.
The Kodaira dimension of a contact 3-manifold \((Y,\xi)\) is defined as follows
\[\mathrm{Kod}(Y,\xi)=\left\{
\begin{array}{rl}
-\infty & \text{if it admits a uniruled cap} \\
0 & \text{if it admits a Calabi-Yau cap but admits no uniruled caps} \\
1 & \text{if it admits neither a Calabi-Yau cap nor a uniruled cap}
\end{array}
\right.\]
The authors show that the symplectic filling version of Stipsicz's result holds for any contact 3-manifold with \(\mathrm{Kod}=-\infty\), and the exact filling version of Stipsicz's result holds for any contact 3-manifold with \(\mathrm{Kod}=0\). Moreover, explicit homological bounds can be obtained in the case \(\mathrm{Kod}=-\infty\) given a uniruled cap and in the case \(\mathrm{Kod}=0\) given a Calabi-Yau cap. There are many contact 3-manifolds with \(\mathrm{Kod}=1\).
Making use of the notions of a maximal surface and a Donaldson cap, the authors show in the main result of the paper that for any contact 3-manifold \((Y,\xi)\), the set \(\{2\chi(N)+3\sigma(N)\}\subset\mathbb{Z}\) is bounded from below when \((N,\omega_N)\) is a minimal symplectic filling of \((Y,\xi)\). Moreover, they explicitly calculate the lower bound for a polarized symplectic cap.
This result, together with the above results for \(\mathrm{Kod}=-\infty\) and \(\mathrm{Kod}=0\) contact 3-manifolds, provides a comprehensive geography picture for various fillings of contact 3-manifolds with a fixed Kodaira dimension.
The authors also prove that for any contact 3-manifold \((Y,\xi)\), there exists a symplectic cap \((P,\omega_P)\) of \((Y,\xi)\) such that for any minimal symplectic filling \((N,\omega_N)\) of \((Y,\xi)\), the glued symplectic manifold \((N\cup P,\omega)\) is minimal. In particular, any minimal convex 4-manifold embeds into a minimal closed symplectic 4-manifold.
By combining this result with the previous one, the authors show that for any fillable contact 3-manifold \((Y,\xi)\), the set \(
\{2\chi(W)+3\sigma(W)\}\subset\mathbb{Z}\) is bounded below by \(0\) when \((W,\omega_W)\) is an exact cobordism from \((Y,\xi)\) to itself. In particular, if it is also bounded above, then the set is \(\{0\}\).
Reviewer: Andrew Bucki (Edmond)Erratum to: ``Multivariate Alexander quandles. II. The involutory medial quandle of a link''.https://www.zbmath.org/1460.570072021-06-15T18:09:00+00:00"Traldi, Lorenzo"https://www.zbmath.org/authors/?q=ai:traldi.lorenzoSubquandles of affine quandles.https://www.zbmath.org/1460.200142021-06-15T18:09:00+00:00"Jedlička, Přemysl"https://www.zbmath.org/authors/?q=ai:jedlicka.premysl"Pilitowska, Agata"https://www.zbmath.org/authors/?q=ai:pilitowska.agata"Stanovský, David"https://www.zbmath.org/authors/?q=ai:stanovsky.david"Zamojska-Dzienio, Anna"https://www.zbmath.org/authors/?q=ai:zamojska-dzienio.annaA quandle is a groupoid \((Q, \cdot )\) which is: (a) idempotent (i.e. it obeys the identity \(x \cdot x = x\) for all \(x\in Q\)); (b) left unique solvable (i.e. for all \(a, b\in Q\), there exists a unique \(x\in Q\) such that \(a \cdot x = b\), which is usually denoted by \(x = a\backslash b\)) and (c) left distributive (i.e. it satisfies the identity \(x \cdot (y \cdot z) = (x \cdot y) \cdot (x \cdot z)\) for all \(x,y,z\in Q\)). A quandle \((Q, \cdot )\) is called medial (or entropic) if it satisfies the identity \((x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)\) for all \(x,y,u,v\in Q\). Let \((G, +)\) be an abelian group, \(\alpha\) an automorphism of \((G, +)\), and define a binary operation (\(\star\)) on the \(G\) by \(a \star b = (1 - \alpha)(a) + \alpha(b)\) for all \(a,b\in G\). Then, \((G, \star)\) is a medial quandle and is usually denoted by \(\mathrm{Aff}(A, f)\) and said to be affine over the group \((G, +)\). A quandle that is embeddable into an affine quandle is referred to as quasi-affine. Free medial quandles are quasi-affine, but not affine. Affine quandles (also known as Alexander quandles) play an important role in quandle theory from algebraic perspective and in applications in knot theory.
In this article under review, the authors consider the structure and properties of quandles that are embeddable into affine quandles (i.e. that are isomorphic to a subquandle of an affine quandles). Their motivation is mainly algebraic with attention on computational aspects. They consider a structural theorem which is achieved using a special kind of central extension. They also establish a computationally feasible characterization of quasi-affine quandles. The key property behind their results is abelianness and semiregularity of the displacement group which are reflections of their main results (Theorem 2.2 and Theorem 2.3). These results serve as confirmation of a conjecture for the class of quandles; an open problem of whether every idempotent algebraic structure satisfying certain syntactic condition called abelianness is quasi-affine. They present polynomial-time Algorithm 7.1 and Algorithm 7.4 for recognition of affine and quasi-affine quandles (in multiplication table form). These two algorithms are based on the properties of the displacement group as described in Theorem 2.2 and Theorem 2.3.
Reviewer: Temitope Gbolahan Jaiyeola (Ile-Ife)Higher depth quantum modular forms and plumbed 3-manifolds.https://www.zbmath.org/1460.110512021-06-15T18:09:00+00:00"Bringmann, Kathrin"https://www.zbmath.org/authors/?q=ai:bringmann.kathrin"Mahlburg, Karl"https://www.zbmath.org/authors/?q=ai:mahlburg.karl"Milas, Antun"https://www.zbmath.org/authors/?q=ai:milas.antunSummary: In this paper, we study new invariants \(\widehat{Z}_{a}(q)\) attached to plumbed 3-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa [\textit{S. Gukov} et al., J. Knot Theory Ramifications 29, No. 2, Article ID 2040003, 85 p. (2020; Zbl 1448.57020)]. These remarkable \(q\)-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed 3-manifold. Here, we investigate the series \(\widehat{Z}_0(q)\) for unimodular plumbing H-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, \( \widehat{Z}_0(q)\) is a depth two quantum modular form on \({\mathbb{Q}} \).Quasiconvexity and Dehn filling.https://www.zbmath.org/1460.570202021-06-15T18:09:00+00:00"Groves, Daniel"https://www.zbmath.org/authors/?q=ai:groves.daniel-p"Manning, Jason Fox"https://www.zbmath.org/authors/?q=ai:manning.jason-foxThe authors introduce a new condition on relatively hyperbolic Dehn filling, which they call \(H\)-wide and which they use in the study of quasiconvex subgroups \(H\) in a wider generality than the known techniques. This allows them to control the behavior of relatively quasiconvex subgroups which need not be full. As an application, combining their results with work of \textit{D. Cooper} and \textit{D. Futer} [Geom. Topol. 23, No. 1, 241--298 (2019; Zbl 1444.57013)], they provide a new proof of an unpublished result due of Wise stating that the fundamental group of a non-compact finite-volume hyperbolic 3-manifold is always virtually compact special. As a consequence of this result and of a fibering criterion due to \textit{I. Agol} [J. Topol. 1, No. 2, 269--284 (2008; Zbl 1148.57023)], they prove that any non-compact finite-volume hyperbolic 3-manifold has a finite-sheeted cover which fibers over the circle. As another consequence of their main result, they obtain generalizations of results by \textit{I. Agol} [Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019), Appendix A] with which they control the ```relative height'' of relatively quasiconvex subgroups under Dehn filling, and they use this to prove a result used by \textit{H. Wilton} and \textit{P. Zalesskii} [Geom. Topol. 21, No. 1, 345--384 (2017; Zbl 1361.57023)] in their work on profinite rigidity of 3-manifold groups.
Reviewer: Athanase Papadopoulos (Strasbourg)Limits of canonical forms on towers of Riemann surfaces.https://www.zbmath.org/1460.140662021-06-15T18:09:00+00:00"Baik, Hyungryul"https://www.zbmath.org/authors/?q=ai:baik.hyungryul"Shokrieh, Farbod"https://www.zbmath.org/authors/?q=ai:shokrieh.farbod"Wu, Chenxi"https://www.zbmath.org/authors/?q=ai:wu.chenxiLet \(S\) be a Riemann surface of genus \(g \geq 2\), and consider an ascending sequence \(\{S_n \rightarrow S\}\) of finite Galois covers converging to the universal cover. Then, \textit{D. A. Kazhdan} proved in [in: Lie Groups Represent., Proc. Summer Sch. Bolyai Janos math. Soc., Budapest 1971, 151--217 (1975; Zbl 0308.14007)] that the \((1,1)\)-forms on \(S\) inherited from canonical forms via \(\{S_n \rightarrow S\}\), converge uniformly to a multiple of the hyperbolic \((1,1)\)-form of the universal cover. In the present paper the authors generalize the result of Kazhdan, by replacing the universal cover with any infinite Galois cover. Their main result is Theorem A (Theorem 5.4), according to which if \(S'\) is any infinite Galois cover of \(S\), and \(\{S_n \rightarrow S\}\) is a sequence of finite Galois covers converging to \(S'\), then the sequence of \((1,1)\)-forms on \(S\) induced from the canonical forms on \(S_n\) converges uniformly to the \((1,1)\)-form induced from the canonical form on \(S'\). In order to prove it, Theorem 5.3 gives a weaker result, on the strong convergence of the associated measures attached to the \((1,1)\)-forms. Then, the uniform convergence of the forms follows from an analytic argument. For proving that Theorem 5.3, a Gauss-Bonnet type result (Theorem B) is relevant.
Results on metric graphs, very close to Theorems A and B, were obtained by the second and third listed authors in [Invent. Math. 215, No. 3, 819--862 (2019; Zbl. 1440.14284)].
Reviewer: José Javier Etayo (Madrid)A census of exceptional Dehn fillings.https://www.zbmath.org/1460.570212021-06-15T18:09:00+00:00"Dunfield, Nathan M."https://www.zbmath.org/authors/?q=ai:dunfield.nathan-mA \textit{1-cusped hyperbolic 3-manifold} is a compact orientable 3-manifold \(M\) such that \(\partial M\) is a torus and the interior of \(M\) admits a hyperbolic metric of finite volume. By one of the most beautiful results of Thurston, except for finitely many slopes \(\alpha\) on \(\partial M\) the closed 3-manifold \(M(\alpha)\) obtained by Dehn filling of slope \(\alpha\) on \(M\) is again hyperbolic; the finitely many remaining cases are called \textit{exceptional Dehn fillings}, and in these cases \(M(\alpha)\) contains an essential 2-sphere (and is a connected sum or \(S^2 \times S^1\)), or an essential torus (and has a nontrivial JSJ-decomposition or belongs to the Sol-geometry), or is Seifert fibered (including 3-manifolds with finite fundamental group by the geometrization of 3-manifolds). There is an extensive literature on the classification of the exceptional Dehn fillings for various classes of hyperbolic 3-manifolds.
The main result of the present paper states that there are precisely 205,822 exceptional Dehn fillings on the 59,107 1-cusped hyperbolic 3-manifolds admitting an ideal triangulation with at most nine ideal tetrahedra. The author presents various figures and tables with statistical data about these manifolds, derives evidence for some standard conjectures and adds new observations. The paper is also a good survey on the literature on exceptional Dehn fillings and on the various computer software involved in the enumeration and classification of hyperbolic and non-hyperbolic 3-manifolds.
For the entire collection see [Zbl 1458.57001].
Reviewer: Bruno Zimmermann (Trieste)The Dehn twist on a sum of two \(K3\) surfaces.https://www.zbmath.org/1460.570222021-06-15T18:09:00+00:00"Kronheimer, P. B."https://www.zbmath.org/authors/?q=ai:kronheimer.peter-benedict"Mrowka, T. S."https://www.zbmath.org/authors/?q=ai:mrowka.tomasz-sSummary: \textit{D. Ruberman} [Math. Res. Lett. 5, No. 6, 743--758 (1998; Zbl 0946.57025)] gave the first examples of self-diffeomorphisms of four-manifolds that are isotopic to the identity in the topological category but not smoothly so. We give another example of this phenomenon, using the Dehn twist along a \(3\)-sphere in the connected sum of two \(K3\) surfaces.Farey recursion and the character varieties for 2-bridge knots.https://www.zbmath.org/1460.570192021-06-15T18:09:00+00:00"Chesebro, Eric"https://www.zbmath.org/authors/?q=ai:chesebro.ericThe author uses Farey recursion to define a set of polynomials that describe the (P)SL\(_{2}\mathbb{C}\) character varieties of all 2-bridge knots and the diagonal character varieties of all 2-bridge links. This makes the computation fairly easy and efficient using a computer.
For the entire collection see [Zbl 1458.57001].
Reviewer: Khaled Qazaqzeh (Irbed)A note on the weak splitting number.https://www.zbmath.org/1460.570022021-06-15T18:09:00+00:00"Cavallo, Alberto"https://www.zbmath.org/authors/?q=ai:cavallo.alberto|cavallo.alberto.1"Collari, Carlo"https://www.zbmath.org/authors/?q=ai:collari.carlo"Conway, Anthony"https://www.zbmath.org/authors/?q=ai:conway.anthonyThe weak splitting number, \(wsp(L)\), of a link, \(L\), is the minimal number of crossing changes needed to untangle the components of \(L\) from one another. As the changes need not involve two different components, the isotopy type of the components may change, and this seems to make more difficult the establishment of a lower bound for the \(wsp\) of a link than for example giving a lower bound for \(sp(L)\), the fewest number of crossing changes involving different components needed to untangle \(L\).
This paper aims to utilize old, and establish new, lower bounds on the \(wsp\) for links in a table (attributed to Thistlethewaite) of links having \(9\) or fewer crossings. Using those bounds the authors mean to determine the \(wsp\) of the links. There are \(130\) such links and the paper fails in its object in only 2 cases.
A variety of theorems and methods are applied to achieve the authors' end. A recurring term in a number of these theorems is the difference between a link invariant evaluated on a link \(L\) and the same invariant summed over its evaluation on the components of \(L\). For example: ``If \(L = K_{1} \cup \cdots \cup K_\ell\) is an oriented link and \(\omega = (\omega_{1},\dots ,\omega_{\ell}) \in (S^1)^\ell\), then the following inequality holds, \[|\sigma_{L}(\omega) - \sum_{i=1}^{\ell}\sigma_{K_i}(\omega_i)| + |\eta_{L}(\omega) - \sum_{i=1}^{\ell}\eta_{K_i}(\omega_i) - \ell + 1| +3\sum_{i < j}|\ell k(K_{i},K_{j})| \leq 4 (wsp(L))."\] (This result is from Conway's thesis and an alternative proof is given in the paper under review.) The invariants \(\sigma\) and \(\eta\) in the inequality are the signature and nullity from the paper by \textit{D. Cimasoni} and \textit{V. Florens} [Trans. Am. Math. Soc. 360, No. 3, 1223--1264 (2008; Zbl 1132.57004)].
Among the topics appealed to in establishing the \(wsp\)'s of the links are: the \(J\)-function from link Floer homology, a concordance invariant of \textit{J. Hom} and \textit{Z. Wu} [J. Symplectic Geom. 14, No. 1, 305--323 (2016; Zbl 1348.57023)], homotopical considerations, and covering links.
Different bounds for the \(wsp\) obtained by considering different invariants are also compared.
The paper concludes with a table of the \(wsp\) of the links under study and an indication as to how those \(wsp\)s were finally obtained.
Reviewer: Lee P. Neuwirth (Princeton)Contractible, hyperbolic but non-CAT(0) complexes.https://www.zbmath.org/1460.570172021-06-15T18:09:00+00:00"Webb, Richard C. H."https://www.zbmath.org/authors/?q=ai:webb.richard-c-hLet \(S_{g,p}\) denote an orientable surface with genus \(g\) and \(p\) punctures/marked points. In this paper, the author investigates whether several complexes satisfy a linear combinatorial isoperimetric inequality or not. Moreover, he points out that A. Putman asked the question whether the curve complex satisfies a linear combinatorial isoperimetric inequality, and mentioned that this does not immediately follow from hyperbolicity.
Let \(K\) be one of the following complexes:
\begin{itemize}
\item[(1)] The arc complex \(\mathcal{A}(S_{g,p})\), where \(g \geq 2\) and \(p \geq 2\), or, \(g = 1\) and \(p \geq 4\), or \(g = 0\) and \(p \geq 6\).
\item[(2)] The disc complex \(\mathcal{D}_n\) of a handlebody of genus \(n \geq 5\).
\item[(3)] The free splitting complex \(\mathcal{FS}_n\) of a free group of rank \(n \geq 5\).
\end{itemize}
The author shows that \(K\) does not admit a \(\text{CAT(0)}\) metric with finitely many shapes. Furthermore, it does not admit one with bounded, thick shapes. None of the above complexes satisfies any combinatorial isoperimetric inequality; although they are contractible [\textit{J. L. Harer}, Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005); \textit{A. E. Hatcher}, Topology Appl. 40, No. 2, 189--194 (1991; Zbl 0727.57012); \textit{D. McCullough}, J. Differ. Geom. 33, No. 1, 1--65 (1991; Zbl 0721.57008); \textit{A. E. Hatcher}, Comment. Math. Helv. 70, No. 1, 39--62 (1995; Zbl 0836.57003)], hyperbolic [\textit{H. Masur} and \textit{S. Schleimer}, J. Am. Math. Soc. 26, No. 1, 1--62 (2013; Zbl 1272.57015); \textit{M. Handel} and \textit{L. Mosher}, Geom. Topol. 17, No. 3, 1581--1672 (2013; Zbl 1278.20053)], and such that any finite group action must fix some point [\textit{S. P. Kerckhoff}, Ann. Math. (2) 117, 235--265 (1983; Zbl 0528.57008); \textit{J. L. Harer}, Invent. Math. 84, 157--176 (1986; Zbl 0592.57009); \textit{S. Hensel} et al., Geom. Topol. 18, No. 4, 2079--2126 (2014; Zbl 1320.57022)], and so traditional obstructions to being \(\text{CAT(0)}\) do not apply. In this paper, the author's obstruction to being \(\text{CAT(0)}\) with finitely many shapes is new.
To show the above result, the author uses, among other results, the following theorem also proved in the paper: whenever \(K\) is a (not necessarily locally compact) flag simplicial complex equipped with a \(\text{CAT(0)}\) metric with bounded, thick shapes then \(K\) satisfies a quadratic combinatorial isoperimetric inequality.
On the other hand, if \(K\) is the curve complex \(\mathcal{C}(S_{g,p})\) for \(3g+p-3 \geq 3\) or the arc-and-curve complex \(\mathcal{AC}(S_{g,p})\), then the author proves that \(K\) satisfies a linear combinatorial isoperimetric inequality. To prove this he uses the tightening procedure introduced by Masur and Minsky for geodesics, but the author's procedure for loops is more involved.
Reviewer: Ferihe Atalan (Ankara)Low-dimensional topology. Abstracts from the workshop held February 16--22, 2020.https://www.zbmath.org/1460.000382021-06-15T18:09:00+00:00"Friedl, Stefan (ed.)"https://www.zbmath.org/authors/?q=ai:friedl.stefan"Moriah, Yoav (ed.)"https://www.zbmath.org/authors/?q=ai:moriah.yoav"Purcell, Jessica S. (ed.)"https://www.zbmath.org/authors/?q=ai:purcell.jessica-s"Schleimer, Saul (ed.)"https://www.zbmath.org/authors/?q=ai:schleimer.saulSummary: The workshop brought together experts from across all areas of low-dimensional topology, including knot theory, mapping class groups, three-manifolds and four-manifolds. In addition to the standard research talks we had five survey talks by Burton, Minsky, Powell, Reid, and Roberts leading to discussions of open problems. Furthermore we had three sessions of five-minute talks by a total of thirty-five participants.Ozsváth-Szabó bordered algebras and subquotients of category \(\mathcal{O} \).https://www.zbmath.org/1460.570162021-06-15T18:09:00+00:00"Lauda, Aaron D."https://www.zbmath.org/authors/?q=ai:lauda.aaron-d"Manion, Andrew"https://www.zbmath.org/authors/?q=ai:manion.andrewIn the paper under review it is proved that for \(0\leq k\leq n\), Ozsváth-Szabó's algebra \(B_l(n,k)\) is a graded flat deformation of Sartori's algebra \(A_{n,k}\).
The projection functors between categories of finitely generated projective graded modules over these algebras induce isomorphisms on Grothendieck groups \(K_0\) intertwining the identification of indecomposable projectives with canonical basis elements on each side.
The constructed isomorphism allows to transport a number of other constructions between these two algebras.
Reviewer: Dmitry Artamonov (Moskva)Alexander and Markov theorems for virtual doodles.https://www.zbmath.org/1460.570102021-06-15T18:09:00+00:00"Nanda, Neha"https://www.zbmath.org/authors/?q=ai:nanda.neha"Singh, Mahender"https://www.zbmath.org/authors/?q=ai:singh.mahender|singh.mahender-pSummary: Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to classical knot theory. Alexander and Markov theorems for the genus zero case are known where the role of groups is played by twin groups, a class of right angled Coxeter groups with only far commutativity relations. The purpose of this paper is to prove Alexander and Markov theorems for higher genus case where the role of groups is played by a new class of groups called virtual twin groups which extends twin groups in a natural way.Higher rank \(\hat{Z}\) and \(F_K\).https://www.zbmath.org/1460.570132021-06-15T18:09:00+00:00"Park, Sunghyuk"https://www.zbmath.org/authors/?q=ai:park.sunghyukThis paper studies higher rank analogues, with a Lie group \(G\), of \(\hat{Z}\) and \(F_{K}\), two invariants which have appeared in previous works [\textit{S. Gukov} and \textit{C. Manolescu}, ``A two-variable series for knot complements'', Preprint, \url{arXiv:1904.06057}] and [\textit{S. Gukov} et al., J. Knot Theory Ramifications 29, No. 2, Article ID 2040003, 85 p. (2020; Zbl 1448.57020)]. A definition of \(\hat{Z}\) for negative definite plumbed \(3\)-manifolds is given. It is proved that \(\hat{Z}\) is indeed an invariant. Examples of computations such as for \(0\)-surgery on twist knots are given. In the second part, a definition of a higher rank analogue of \(F_{K}\) for torus knot complements is given. For symmetric representations a relationship of \(F_{K}\) to the \(A\)-polynomial is indicated. The paper contains many more computations, conjectures, and discussions.
Reviewer: Huỳnh Quang Vū (Ho Chi Minh City)RII number of knot projections.https://www.zbmath.org/1460.570032021-06-15T18:09:00+00:00"Ito, Noboru"https://www.zbmath.org/authors/?q=ai:ito.noboru"Takimura, Yusuke"https://www.zbmath.org/authors/?q=ai:takimura.yusukeSummary: Every knot projection is simplified to the trivial spherical curve not increasing double points by using deformations of types 1, 2, and 3 which are analogies of Reidemeister moves of types 1, 2, and 3 on knot diagrams. We introduce RII number of a knot projection that is the minimum number of deformations of negative type 2 among such sequences. By definition, it is invariant under deformations of types 1 and 3. This is motivated by Östlund's conjecture [\textit{O.-P. Östlund}, Invariants of knot diagrams and diagrammatic knot invariants, PhD-Thesis, University of Uppsala (2001)]: Deformations of type 1 and 3 are sufficient to describe a homotopy from any generic immersion of a circle in a two dimensional plane to an embedding of the circle, which implies RII number always would be zero. However, \textit{T. Hagge} and \textit{J. Yazinski} [Banach Cent. Publ. 103, 101--110 (2014; Zbl 1314.14055)] disproved the conjecture by showing the first counterexample with 16 double points, which implies that RII number is nontrivial. This paper shows that RII number can be any nonnegative number.Simple-ribbon concordance of knots.https://www.zbmath.org/1460.570052021-06-15T18:09:00+00:00"Kishimoto, Kengo"https://www.zbmath.org/authors/?q=ai:kishimoto.kengo"Shibuya, Tetsuo"https://www.zbmath.org/authors/?q=ai:shibuya.tetsuo"Tsukamoto, Tatsuya"https://www.zbmath.org/authors/?q=ai:tsukamoto.tatsuyaSummary: In \textit{K. Kishimoto} et al. [J. Math. Soc. Japan 68, No. 3, 1033--1045 (2016; Zbl 1357.57015)], we introduced a special kind of fusions so called simple-ribbon fusions. In this paper we consider concordance of knots induced by simple-ribbon fusions, and show that there are infinitely many ribbon knots which are not obtained from the trivial knot by a finite sequence of simple-ribbon fusions.Virtual tangles and fiber functors.https://www.zbmath.org/1460.570122021-06-15T18:09:00+00:00"Brochier, Adrien"https://www.zbmath.org/authors/?q=ai:brochier.adrien