Recent zbMATH articles in MSC 57Khttps://zbmath.org/atom/cc/57K2022-11-17T18:59:28.764376ZWerkzeugConway's mathematics after Conwayhttps://zbmath.org/1496.000032022-11-17T18:59:28.764376Z"Ryba, Alex"https://zbmath.org/authors/?q=ai:ryba.alexander-j-eFrom the text: Mathematicians always queued to hear John Conway speak and delved into his writing. They expected to be en-
tertained by beautiful mathematics and believed that they would emerge with valuable enlightenment. John welcomed this attention and considered it his duty to make his mathematics elegant. What really made his mathematics valuable was his wealth of insight. Wherever he worked, he opened up avenues for us to follow.On webs in quantum type \(C\)https://zbmath.org/1496.180202022-11-17T18:59:28.764376Z"Rose, David E. V."https://zbmath.org/authors/?q=ai:rose.david-e-v"Tatham, Logan C."https://zbmath.org/authors/?q=ai:tatham.logan-cThe authors give a linear pivotal category \(\mathbf{Web}(\mathfrak{sp}_{6}) \) defined by diagrams and relations, and conjecture its equivalence to the full subcategory \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) of finite-dimensional representations of \( \mathfrak{sp}_{6} \) tensor-generated by fundamental representations. This is a step towards generalizing to type \(C\) case of the main open problem from Kuperberg's Spider for rank 2 Lie algebras [\textit{G. Kuperberg} Commun. Math. Phys. 180, No. 1, 109--151 (1996; Zbl 0870.17005)]. They prove a number of results that support the conjecture. Namely, they construct a functor from \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) that is full and essentially surjective (they prove that the well-known surjection from the BMW algebra to the representation category factors through the web category). Moreover they prove that all \(\Hom\)-spaces in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) are finite-dimensional and that the endomorphism algebra of the tensor unit in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) is one-dimensional. As a consequence, the authors give a new approach to the quantum \( \mathfrak{sp}_{6} \) link invariants in the same lines as the Kauffman bracket description of the Jones polynomial. In this paper it is also given a thickening of \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to a category constructed using ladders that is one of the ingredients is proving the results above.
Reviewer's remark: The conjecture that is the subject of this paper is a consequence of the results in \url{arXiv:2103.14997} for \(sp_{2n}\) by the authors together with \textit{E. Bodish} and \textit{B. Elias}.
Reviewer: Pedro Vaz (Louvain-la-Neuve)Chaos and integrability in \(\operatorname{SL}(2,\mathbb{R})\)-geometryhttps://zbmath.org/1496.370312022-11-17T18:59:28.764376Z"Bolsinov, Aleksei V."https://zbmath.org/authors/?q=ai:bolsinov.alexey-v"Veselov, Aleksandr P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Ye, Yiru"https://zbmath.org/authors/?q=ai:ye.yiruOn the induced geometry on surfaces in 3D contact sub-Riemannian manifoldshttps://zbmath.org/1496.530412022-11-17T18:59:28.764376Z"Barilari, Davide"https://zbmath.org/authors/?q=ai:barilari.davide"Boscain, Ugo"https://zbmath.org/authors/?q=ai:boscain.ugo"Cannarsa, Daniele"https://zbmath.org/authors/?q=ai:cannarsa.danieleLet \(M\) be a 3-dimensional manifold with a contact sub-Riemannian structure \((D,g)\), and let \(d_{sR}\) be the induced sub-Riemannian distance on \(M\). If \(S \subset M\) is a 2-dimensional submanifold (i.e., an embedded surface), there is induced a distance \(d_S\) on \(S\), where \(d_S(x,y)\) is defined as the infimum of the lengths of all horizontal paths that lie in \(S\) and join \(x\) and \(y\). Note that \(d_S\) is not the restriction of \(d_{sR}\) to \(S\), nor is it the sub-Riemannian distance induced by the restriction of \((D,g)\) to \(S\) (which is not bracket-generating). In this paper, the authors study sufficient conditions for \(d_S\) to be finite on \(S\), which is to say that any two points of \(S\) are joined by a finite-length horizontal path lying in \(S\).
The first part of the paper is concerned with local properties of the characteristic foliation induced by the distribution \(D \cap TS\), in a neighborhood of a characteristic point \(p\) (i.e., a point where \(S\) is tangent to \(D\)). The authors introduce a scalar quantity \(\widehat{K}_p\), defined in terms of the Gaussian curvature of \(S\) with respect to Riemannian approximations to the sub-Riemannian geometry \((D,g)\), but which is shown to be independent of the choice of Riemannian approximation (Theorem 1.1.). The value of \(\widehat{K}_p\) turns out to relate to the eigenvalues of \(DX(p)\) (Proposition 1.2), where \(X\) is the characteristic vector field, and thus is useful in classifying the qualitative behavior of the foliation near \(p\) as a saddle, saddle-node, node, or focus (Corollaries 4.5 and 4.7). This is then used to address the question of whether or not a horizontal path in \(S\) that approaches \(p\) will have finite length (Proposition 1.3).
In the second part of the paper (Theorem 1.5), the authors prove that if a compact connected \(C^2\) surface \(S \subset M\) satisfies the following set of conditions, then the induced distance \(d_S\) is finite on \(S\):
\begin{itemize}
\item The contact sub-Riemannian structure \((M,D,g)\) is coorientable (i.e., the distribution \(D\) is the kernel of a globally defined contact one-form \(\omega\)) and is tight (admits no overtwisted disk);
\item The surface \(S\) is homeomorphic to the sphere \(S^2\);
\item The characteristic points of \(S\) are isolated.
\end{itemize}
Reviewer: Nathaniel Eldredge (Storrs)Knot theory and statistical mechanicshttps://zbmath.org/1496.570012022-11-17T18:59:28.764376Z"Kauffman, Louis H."https://zbmath.org/authors/?q=ai:kauffman.louis-hirschSummary: This chapter discusses connections between knot theory and statistical mechanics and quantum amplitudes.
For the entire collection see [Zbl 1491.46002].Conway's knotty pasthttps://zbmath.org/1496.570022022-11-17T18:59:28.764376Z"Adams, Colin"https://zbmath.org/authors/?q=ai:adams.colin-cThe author gives an overview of John Conways contributions to knot theory. ``Conway's publications in knot theory
consist of just three papers, one with him as the sole author
and two coauthored with Cameron Gordon. And yet his
influence on the field has been and continues to be
dramatic.''Rokhlin's theorem, a problem and a conjecturehttps://zbmath.org/1496.570032022-11-17T18:59:28.764376Z"Sullivan, Dennis"https://zbmath.org/authors/?q=ai:sullivan.dennis-pThis is a short sketch of mathematical questions and developments that arose following Rokhlin's theorem on divisibility of the signature of a closed, smooth spin 4-manifold.
For the entire collection see [Zbl 1492.57002].A Möbius invariant discretization of O'Hara's Möbius energyhttps://zbmath.org/1496.570042022-11-17T18:59:28.764376Z"Blatt, Simon"https://zbmath.org/authors/?q=ai:blatt.simon|blatt.simon.1"Ishizeki, Aya"https://zbmath.org/authors/?q=ai:ishizeki.aya"Nagasawa, Takeyuki"https://zbmath.org/authors/?q=ai:nagasawa.takeyukiThe authors introduce a new discretization of O'Hara's Möbius energy. In contrast to the known discretizations of \textit{J. K. Simon} [J. Knot Theory Ramifications 3, No. 3, 299--320 (1994; Zbl 0841.57017)] and of \textit{D. Kim} and \textit{R. Kusner} [Exp. Math. 2, No. 1, 1--9 (1993; Zbl 0818.57007)] the new discretization is invariant under Möbius transformations of the surrounding space. Moreover, this energy is minimized by polygons with vertices on a circle. The starting point for this new discretization is the so-called cosine formula of Doyle and Schramm. In addition, the authors then show \(\Gamma\)-convergence of the discretized energy to the Möbius energy provided that the fineness of the polygons is going to 0. (Here a map \(p : \mathbb{R/Z}\to \mathbb{R}^n\) a closed polygon with the \(m\) vertices \(p(\theta_i)\in\mathbb{R}^n\), \(i=1,\ldots,m\) if there are points \(\theta_i \in[0,1)\), \(\theta_1 <\theta_2 <\dots<\theta_m\) such that \(p\) is linear between two neighboring points \(\theta_i\) and \(\theta_{i+1}\) with \(\theta_{m+1}=\theta_1\). The fineness is then defined as \(\max|\theta_{i+1}-\theta_i|\).)
Reviewer: Claus Ernst (Bowling Green)The tunnel numbers of all 11- and 12-crossing alternating knotshttps://zbmath.org/1496.570052022-11-17T18:59:28.764376Z"Castellano-Macías, Felipe"https://zbmath.org/authors/?q=ai:castellano-macias.felipe"Owad, Nicholas"https://zbmath.org/authors/?q=ai:owad.nicholas-jSummary: Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel numbers of all 1655 alternating 11- and 12-crossing knots and of 881 nonalternating 11- and 12-crossing knots. We also find all 5525~Montesinos knots with 14 or fewer crossings.A meeting with Patrick Dehornoyhttps://zbmath.org/1496.570062022-11-17T18:59:28.764376Z"Kauffman, Louis H."https://zbmath.org/authors/?q=ai:kauffman.louis-hirschFinite involutory quandles of two-bridge links with an axishttps://zbmath.org/1496.570072022-11-17T18:59:28.764376Z"Mellor, Blake"https://zbmath.org/authors/?q=ai:mellor.blakeThe concept of quandles was explicitly presented in the independent works of \textit{D. Joyce} [J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] and \textit{S. V. Matveev} [Math. USSR, Sb. 47, 73--83 (1984; Zbl 0523.57006); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78--88 (1982)]. To each oriented diagram \(D_K\) of an oriented knot \(K\) in \(\mathbb{R}^3\) they associate the quandle \(Q(K)\) which does not change if we apply the Reidemeister moves to the diagram \(D_K\). Joyce and Matveev proved that two knot quandles \(Q(K_1)\) and \(Q(K_2)\) are isomorphic if and only if there is a homeomorphism (possibly reversing orientation) of the ambient space \(\mathbb{R}^3\) which maps \(K_1\) to \(K_2\). The knot quandle is a very strong invariant for knots in \(\mathbb{R}^3\), however, usually it is very difficult to determine if two knot quandles are isomorphic. In [\textit{A. D. Brooke-Taylor} and \textit{S. K. Miller}, J. Aust. Math. Soc. 108, No. 2, 262--277 (2020; Zbl 1482.20039)] it is shown that the isomorphism problem for quandles is, from the perspective of Borel reducibility, fundamentally difficult (Borel complete).
Sometimes homomorphisms from knot quandles to simpler quandles provide useful information that helps determine whether two knot quandles are isomorphic. One of the homomorphisms that provides useful information about the quandle \(Q(K)\) is the homomorphism \(Q(K)\to Q_n(K)\) which sends the quandle \(Q(K)\) to its quotient \(Q_n(K)\) by the relations \((\dots((x*y)*y)*\dots)*y\) for all \(x,y\in Q(K)\), where \(*\) is a quandle operation, and \(y\) appears \(n\) times in the formula. The authors of the paper under review study the quandle \(Q_2(L)\) in the case when \(L\) is a two-bridge link with an axis. In this case the authors calculate the order of \(Q_2(L)\) and give an explicit description of the Cayley graphs for these quandles.
Reviewer: Timur Nasybullov (Novosibirsk)Notes on constructions of knots with the same tracehttps://zbmath.org/1496.570082022-11-17T18:59:28.764376Z"Tagami, Keiji"https://zbmath.org/authors/?q=ai:tagami.keijiThe \(m\)-trace of a knot is the 4-manifold obtained from \(B^4\) by attaching a 2-handle along the knot with \(m\)-framing. There exist infinitely many pairs of distinct knots with the same (diffeomorphic) \(m\)-trace. Two independent techniques to construct such examples are known. One is the operation \((*m)\) defined by \textit{T. Abe} et al. [Int. Math. Res. Not. 2015, No. 22, 11667--11693 (2015; Zbl 1331.57004)], the other one is a technique given by \textit{A. N. Miller} and \textit{L. Piccirillo} [J. Topol. 11, No. 1, 201--220 (2018; Zbl 1393.57010)], utilizing \textit{R. E. Gompf} and \textit{K. Miyazaki}'s dualizable pattern in [Topology Appl. 64, No. 2, 117--131 (1995; Zbl 0844.57004)].
In the paper under review, correspondences between the two constructions are studied and clarified, and a ``twisting'' operation is introduced to both techniques. It is also shown that the family of knots admitting the same 4-surgery given by \textit{M. Teragaito} [Int. Math. Res. Not. 2007, No. 9, Article ID rnm028, 16 p. (2007; Zbl 1138.57012)] can be explained by the operation \((*m)\).
Reviewer: Yuichi Yamada (Tokyo)A homogeneous presentation of symmetric quandleshttps://zbmath.org/1496.570092022-11-17T18:59:28.764376Z"Taniguchi, Yuta"https://zbmath.org/authors/?q=ai:taniguchi.yutaThe author takes a further look at the algebraic structure of symmetric quandles. An example of a symmetric quandle is constructed, and it is shown that every symmetric quandle is isomorphic to the disjoint union of such quandles. The term symmetric quandle as used by the author is according to \textit{S. Kamada} [Ser. Knots Everything 40, 101--108 (2007; Zbl 1145.57008)], that is, a symmetric quandle is a quandle with a good involution; such quandles have been used to study knottings of non-orientable surfaces.
Reviewer: Abednego Orobosa Isere (Ekpoma)The Atiyah-Patodi-Singer rho invariant and signatures of linkshttps://zbmath.org/1496.570102022-11-17T18:59:28.764376Z"Toffoli, Enrico"https://zbmath.org/authors/?q=ai:toffoli.enricoMultivariable signatures of links, defined by \textit{D. Cimasoni} and \textit{V. Florens} [Trans. Am. Math. Soc. 360, No. 3, 1223--1264 (2008; Zbl 1132.57004)] using generalized Seifert surfaces, were given a description as twisted signatures of four manifolds and also employed as concordance invariants. However, a description of them as Atiyah-Patodi-Singer rho invariants was not investigated thoroughly.
In the first part of the paper, the author fills this gap by proving a cut-and-paste formula for the Atiyah-Patodi-Singer rho invariant which allows him to manipulate manifolds in a convenient way. He gives a simple proof which does not involve rho invariants of manifolds with boundary, and which is based on Wall's non-additivity for the signature instead.
The second part of the paper is made up of applications of the main Theorem in the context of link theory. The author gives a description of the multivariable signature of a link \(L\) as the rho invariant of some closed three-manifold \(Y_L\) intrinsically associated with \(L\). Then, he studies the rho invariant of the manifolds obtained by Dehn surgery on \(L\) along integer and rational framings. Inspired by the results of \textit{A. J. Casson} and \textit{C. McA. Gordon} [in: Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 39--53 (1978; Zbl 0394.57008)] and Cimasoni and Florens [loc. cit.], he gives formulas expressing this value as a sum of the multivariable signature of \(L\) and some easy-to-compute extra terms.
Reviewer: Leila Ben Abdelghani (Monastir)A numerical invariant of oriented linkshttps://zbmath.org/1496.570112022-11-17T18:59:28.764376Z"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kai"Yang, Zhiqing"https://zbmath.org/authors/?q=ai:yang.zhiqing\(G\)-family polynomialshttps://zbmath.org/1496.570122022-11-17T18:59:28.764376Z"Brown, Madeline"https://zbmath.org/authors/?q=ai:brown.madeline"Nelson, Sam"https://zbmath.org/authors/?q=ai:nelson.samGiven a group \(G\) and a set \(X\), a \(G\)-family of quandles is a family of quandle structures \(\triangleleft^g\) on \(X\), for \(g\in G\), that are compatible with the group structure of \(G\). The paper under review defines a two-variable polynomial \(\phi_{(G,X)}\), associated with \((G,X)\), with coefficients in the group ring over \(G\).
Given a diagram \(D\) of an oriented trivalent spatial graph \(K\), a \((G,X)\)-coloring assigns to each arc in \(D\) a pair \((g,x)\in G\times X\) so that certain equations are satisfied at each crossing and trivalent vertex.
For each \((G,X)\)-coloring \(f\) of \(K\), the paper considers the \(G\)-subfamily \((G,\mathrm{Im}(f))\) of quandles of \((G,X)\) defined as the smallest \(G\)-subfamily of \((G,X)\) containing all the pairs \((g,x)\) in the coloring \(f\), and then defines the \(G\)-family subquandle polynomial invariant of \(K\) with respect to \((G,X)\) to be the multiset of polynomials
\[
\Phi_{(G,X)}(K):=\{\phi_{(G,\mathrm{Im}(f))}\mid f \text{ is a coloring of \(D\)}\},
\]
The main result states that the invariant is invariant under neighborhood equivalence of \(K\), and thus gives an invariant of handlebody-links. The author demonstrates by examples that the handlebody-link invariant is stronger than counting the number of \((G,X)\)-colorings.
A similar polynomial invariant for handlebody-links, using the associated quandle of a \(G\)-family of quandles, is also discussed and computed.
Reviewer: Yi-Sheng Wang (Taipei)HOMFLYPT homology for links in handlebodies via type \textsf{A} Soergel bimoduleshttps://zbmath.org/1496.570132022-11-17T18:59:28.764376Z"Rose, David E. V."https://zbmath.org/authors/?q=ai:rose.david-e-v"Tubbenhauer, Daniel"https://zbmath.org/authors/?q=ai:tubbenhauer.danielSummary: We define a triply-graded invariant of links in a genus \(g\) handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-sphere. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.Concordance maps in \(\mathrm{HFK}^-\)https://zbmath.org/1496.570142022-11-17T18:59:28.764376Z"Tovstopyat-Nelip, Lev"https://zbmath.org/authors/?q=ai:tovstopyat-nelip.levKnot Floer homology is a knot invariant defined by \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)] and \textit{J. Rasmussen} [Floer homology and knot complements. Harvard University (PhD Thesis) (2003)] which categorifies the Alexander polynomial. It was proven by \textit{A. Juhász} [Adv. Math. 299, 940--1038 (2016; Zbl 1358.57021)] that the simplest version of knot Floer homology \(\widehat{\mathrm{HFK}}\) is functorial with respect to decorated knot cobordisms, and that the induced map \(F_{\mathcal C} \colon \widehat{\mathrm{HFK}}(-S^3, K_0) \longrightarrow \widehat{\mathrm{HFK}}(-S^3,K_1)\) preserves the Alexander and absolute \(\mathbb{Q}\)-Maslov gradings.
The paper under review concerns the more powerful flavor of knot Floer homology, denoted by \(\mathrm{HFK}^-\), which is a finitely generated \(\mathbb{Z}_2[U]\)-module. The main result is that a decorated knot concordance \(\mathcal C\) from \(K_0\) to \(K_1\) induces a \(\mathbb{Z}_2[U]\)-module morphism
\[
G_{\mathcal C} \colon \mathrm{HFK}^-(-S^3, K_0) \longrightarrow \mathrm{HFK}^-(-S^3,K-1),
\]
which preserves the Alexander and absolute \(\mathbb{Z}_2\)-Maslov gradings. Furthermore there is a commutative diagram
\[
\begin{tikzcd}\mathrm{HFK}^-(-S^3,K_0) \arrow[r,"G_{\mathcal C}"] \arrow[d,"p_\ast"] & \mathrm{HFK}^-(-S^3,K_1) \arrow[d,"p_\ast"] \\
\widehat{\mathrm{HFK}}(-S^3,K_0) \arrow[r,"F_{\mathcal C}"] & \widehat{\mathrm{HFK}}(-S^3,K_1) \end{tikzcd}
\]
where the map \(p_\ast\) is the induced map in homology of the map which is setting \(U = 0\) on the chain level.
The construction of the concordance map \(G_{\mathcal C}\) is via a certain relationship between \(\mathrm{HFK}^-\) and sutured Floer homology. More precisely, it is proven by Etnyre-Vela-Vick-Zarev that \(\mathrm{HFK}^-\) is isomorphic to a direct limit of maps between sutured Floer homology groups associated to the knot complement [\textit{J. B. Etnyre} et al., Geom. Topol. 21, No. 3, 1469--1582 (2017; Zbl 1420.57035)]. The construction of the map \(F_{\mathcal C}\) also exploits a relationship between knot Floer homology and sutured Floer homology, and the map \(G_{\mathcal C}\) is defined as a direct limit of maps which are defined in the same way as \(F_{\mathcal C}\) (Proposition 5.1).
Similar results using different approaches have been obtained by \textit{I. Zemke} [J. Topol. 12, No. 1, 94--220 (2019; Zbl 1455.57020)] and \textit{A. Alishahi} and \textit{E. Eftekhary} [ibid. 13, No. 4, 1582--1657 (2020; Zbl 07262229)].
Reviewer: Johan Asplund (New York)Categorifying biquandle bracketshttps://zbmath.org/1496.570152022-11-17T18:59:28.764376Z"Vengal, Adu"https://zbmath.org/authors/?q=ai:vengal.adu"Winstein, Vilas"https://zbmath.org/authors/?q=ai:winstein.vilasKhovanov homology is the categorification of the Jones polynomial. The biquandle bracket is a generalization of the Jones polynomial. In this paper, the authors provide a construction of what seems to be the most natural step from Khovanov homology towards a categorification of biquandle brackets. The invariant they obtain generalizes Khovanov homology, but it is not a true categorification of all biquandle brackets: in some cases, the biquandle bracket cannot be recovered from their invariant via the graded Euler characteristic. Nevertheless, the invariant does lead to a way of assigning a biquandle 2-cocycle to any given biquandle bracket, and the relationship and power of the invariant associated with this new biquandle 2-cocycle may be of interest. To this end, the authors provide some Mathematica packages that can be used to experiment with biquandles, biquandle brackets, and biquandle 2-cocycles, including this new canonical biquandle 2-cocycle associated with a biquandle bracket. This work is a continuation of work done in [\textit{W. Hoffer} et al., J. Knot Theory Ramifications 29, No. 6, Article ID 2050042, 13 p. (2020; Zbl 1448.57014)].
Reviewer: Meili Zhang (Dalian)Khovanov homology for links in \(\#^r (S^2\times S^1)\)https://zbmath.org/1496.570162022-11-17T18:59:28.764376Z"Willis, Michael"https://zbmath.org/authors/?q=ai:willis.michaelIn this paper the author generalises Khovanov homology to links in a \(3\)-sphere with handles attached. When \(r=0\), the manifold \(M^r = S^3\) and the author recovers classical Khovanov homology. When \(r=1\), \(M^r = S^2\times S^1\) and the author recovers the Khovanov homology for links in \(S^2\times S^1\) of \textit{L. Rozansky} [``A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in \(S^2\times S^1\)'', Preprint, \url{arXiv:1011.1958}].
The \(3\)-sphere with \(r\) handles \(M^r\) can be visualised as the complement in \(S^3\) of \(r\) pairs of balls, where the boundary of each ball is identified with the boundary of the other ball in its pair. Now fix a segment in the boundary of each ball, so that the two segments in each pair of balls are identified. Fix also an embedding of a square \([0,1]^2\) in \(S^3\) (without removing the balls) that isotopes each segment into its paired segment.
A link in \(M^r\) can be represented as a tangle in that complement disjoint from the embedded square whose intersection with the boundary of each ball lies in the segment and has the same cardinality as the intersection with the boundary of the other ball in the pair.
To each such representation of a link \(L\) one can associate a sequence of links \(\{L(k)_{k\geq 0}\}\) in \(S^3\) as follows: \(L(0)\) is the result of closing the tangle by gluing parallel strands in each square. \(L(k)\) is the result of performing \(k\) full twists to the parallel strands in each square of \(L(0)\).
After appropriate shiftings and homotopy replacements whose definition takes two fifths of the paper, the Khovanov chain complexes of the links \(\{L(k)\}_{k\geq 0}\) form a directed system and the Khovanov chain complex of the link \(L\) is its colimit. The Khovanov homology of the link \(L\) is the homology of this colimit, which is shown to be a link invariant and functorial with respect to cobordisms up to sign. These constructions apply only if the intersection number of the link with each of the belt spheres of the handles of \(M^r\) is even, otherwise the Khovanov homology of the link is defined to be trivial.
The graded Euler characteristic of the Khovanov homology of a link in \(M^r\) recovers the Witten-Reshetikhin-Turaev invariant of the link. The relation with the skein module of \(M^r\) of [\textit{J. Hoste} and \textit{J. H. Przytycki}, in: Knots 90. Proceedings of the international conference on knot theory and related topics, held in Osaka, Japan, August 15-19, 1990. Berlin etc.: Walter de Gruyter. 363--379 (1992; Zbl 0772.57022)] is also discused.
Finally, there is a construction that associates to each isotopy class of links in \(S^3\) an isotopy class of knots in \(M^r\) called \textit{knotification}. The author also provides a direct method for computing the Khovanov homology of the knotification of any link in a single step.
Reviewer: Federico Cantero Morán (Louvain-la-Neuve)Automorphisms of contact graphs of CAT(0) cube complexeshttps://zbmath.org/1496.570172022-11-17T18:59:28.764376Z"Fioravanti, Elia"https://zbmath.org/authors/?q=ai:fioravanti.eliaIt was shown by \textit{N. V. Ivanov} [Int. Math. Res. Not. 1997, No. 14, 651--666 (1997; Zbl 0890.57018)] that (apart from exceptional cases) the extended mapping class group of a surface is isomorphic to the automorphism group of that surface's curve complex. Mapping class groups are hierarchically hyperbolic, with the curve complex serving as the underlying hyperbolic space. Many CAT(0) cube complexes are also hierarchically hyperbolic [\textit{J. Behrstock} et al., Geom. Topol. 21, No. 3, 1731--1804 (2017; Zbl 1439.20043)], with the role of the curve complex played by the contact graph. The author asks, and partially answers, the subsequently natural question: when is the automorphism group of a CAT(0) cube complex isomorphic to that of the complex's contact graph? The author provides a sufficient criterion, based on extremal vertices, for those cube complexes for which there is such an isomorphism. This result holds in particular for the universal covers of Salvetti complexes, but not for the universal covers of Davis complexes.
Reviewer: Anschel Schaffer-Cohen (Morelia)A complexity of compact \(3\)-manifolds via immersed surfaceshttps://zbmath.org/1496.570182022-11-17T18:59:28.764376Z"Amendola, Gennaro"https://zbmath.org/authors/?q=ai:amendola.gennaroThe complexity of a \(3\)-manifold is an important topic in low dimensional topology. It measures how ``complicated'' a manifold is. For compact 3-manifolds, many complexities have been found. For example, the Heegaard genus, the Gromov norm, the Matveev complexity, the surface-complexity and the Hempel distance.
In the paper under review, the author generalises the definition of surface-complexity to the compact case and the properties holding in the closed case, and gives bounds as follows:
\textbf{Theorem 3}.
\begin{itemize}
\item[--] Each (connected and compact) \(3\)-manifold has a filling Dehn surface.
\item[--] If \(\Sigma_{1}\) and \(\Sigma_{2}\) are homeomorphic filling Dehn surfaces of (connected and compact) \(3\)-manifolds \(M_{1}\) and \(M_{2}\), respectively, with the same number of spherical boundary components, then \(M_{1}\) and \(M_{2}\) are also homeomorphic.
\end{itemize}
\textbf{Theorem 6}. Let \(M\) be a (connected and compact) \(P^{2}\)-irreducible and boundary-irreducible \(3\)-manifold without essential annuli and Möbius strips.
\begin{itemize}
\item[--] If \(sc(M) = 0\), the manifold \(M\) is the sphere \(S^{3}\), the ball \(B^{3}\), the projective space \(\mathbb RP^{3}\) or the lens space \(L_{4,1}\).
\item[--] If \(sc(M) > 0\), the manifold \(M\) has a minimal filling Dehn surface.
\end{itemize}
\textbf{Corollary 7}. For any integer \(c\) there exists only a finite number of (connected and compact) \(P^{2}\)-irreducible and boundary-irreducible \(3\)-manifolds without essential annuli and Möbius strips that have surface-complexity \(c\).
\textbf{Corollary 8}. The surface-complexity of a (connected and compact) \(P^{2}\)-irreducible and boundary-irreducible \(3\)-manifold without essential annuli and Möbius strips, different from \(S^{3}\), \(B^{3}\), \(\mathbb RP^{3}\) and \(L_{4,1}\), is equal to the minimal number of cubes in an ideal cubulation of \(M\).
\textbf{Theorem 10}. Let \(\Sigma\) be a minimal quasi-filling Dehn surface of a (connected and compact) \(P^{2}\)-irreducible and boundary-irreducible \(3\)-manifold \(M\) without essential annuli and Möbius strips.
\begin{itemize}
\item[--] If \(sc(M) = 0\), one of the following holds:
\item[--] \(M\) is the sphere \(S^{3}\) or the ball \(B^{3}\), and \(\Sigma\) is derived from the sphere \(S^{2}\);
\item[--] \(M\) is the projective space \(\mathbb RP^{3}\), and \(\Sigma\) is derived from the projective plane \(\mathbb RP^{2}\) or from the double projective plane \(2\times \mathbb RP^{2}\);
\item[--] \(M\) is the lens space \(L_{4,1}\), and \(\Sigma\) is derived from the four-hat.
\item[--] If \(sc(M) > 0\), the Dehn surface \(\Sigma\) is derived from a minimal filling Dehn surface of \(M\).
\end{itemize}
\textbf{Theorem 11}. The surface-complexity of the connected sum and of the boundary connected sum of (connected and compact) \(3\)-manifolds is less than or equal to the sum of their surface-complexity.
\textbf{Theorem 12}. If a (connected and compact) \(3\)-manifold \(M\) has an ideal triangulation with \(n\) tetrahedra, the inequality \(sc(M)\leq 4n\) holds.
\textbf{Theorem 13}. Let \(M\) be a (connected and compact) \(P^{2}\)-irreducible and boundary-irreducible \(3\)-manifold without essential annuli and Möbius strips, different from the lens spaces \(L_{3,1}\) and \(L_{4,1}\); then the inequalities \(sc(M)\leq 4c(M)\) and \(c(M)\leq 8sc(M)\) hold. Moreover, we have \(c(L_{3,1}) = 0\), \(sc(L_{3,1}) > 0\), \(c(L_{4,1}) > 0\) and \(sc(L_{4,1}) = 0\).
Reviewer: Kun Du (Lanzhou)On Heegaard splittings of mapping torus and induced by open book decompositionshttps://zbmath.org/1496.570192022-11-17T18:59:28.764376Z"Du, Kun"https://zbmath.org/authors/?q=ai:du.kun.2|du.kun|du.kun.1In this paper, the author discusses two cases of Heegaard splittings. One is the Heegaard splitting of a mapping torus, the other one is the Heegaard splitting induced by an open book decomposition. For the former, the author gives a sufficient condition for when it is keen weakly reducible and unstabilized. In the latter case, the author also gives a sufficient condition for when it is irreducible, unstabilized and strongly irreducible. These concepts about Heegaard splittings can be described uniformly using the distance related to the curve complex of the Heegaard surface. Here ``keen'' means that the distance is realized by a unique disk pair up to isotopy.
All these sufficient conditions are related to the translation distance of a homeomorphism of a surface (the former case is closed and the latter case with boundary). The author gets the conclusion under the condition that the translation distance is large. The proofs are combinatorial and technical.
Reviewer: Junhua Wang (Shanghai)Chern-Simons theory on a general Seifert 3-manifoldhttps://zbmath.org/1496.570202022-11-17T18:59:28.764376Z"Blau, Matthias"https://zbmath.org/authors/?q=ai:blau.matthias"Keita, Kaniba Mady"https://zbmath.org/authors/?q=ai:keita.kaniba-mady"Narain, K. S."https://zbmath.org/authors/?q=ai:narain.kumar-s"Thompson, George"https://zbmath.org/authors/?q=ai:thompson.georgeSummary: The path integral for the partition function of Chern-Simons gauge theory with a compact gauge group is evaluated on a general Seifert 3-manifold. This extends previous results and relies on abelianisation, a background field method and local application of the Kawasaki Index theorem.Contact geometry and the mapping class grouphttps://zbmath.org/1496.570212022-11-17T18:59:28.764376Z"Licata, Joan E."https://zbmath.org/authors/?q=ai:licata.joan-eThis is a very short exposition of cutting and gluing of 3-manifolds, mapping class groups, contact structures and their interrelations.Transverse invariants and exotic surfaces in the \(4\)-ballhttps://zbmath.org/1496.570222022-11-17T18:59:28.764376Z"Juhász, András"https://zbmath.org/authors/?q=ai:juhasz.andras"Miller, Maggie"https://zbmath.org/authors/?q=ai:miller.maggie"Zemke, Ian"https://zbmath.org/authors/?q=ai:zemke.ianTwo smooth surfaces \(S\) and \(S'\) in a smooth \(4\)-manifold \(X\) are an \textit{exotic pair} if \(S\) and \(S'\) are topologically, but not smoothly, isotopic. The main result in the paper under review is that there are infinitely many knots in \(\mathbb{S}^3\) each bounding countably many properly embedded, compact, orientable, smooth surfaces in the \(4\)-dimensional disc \(\mathbb{D}^4\) which are pairwise topologically isotopic but which cannot be sent one into the other via a diffeomorphism of \(\mathbb{D}^4\). In particular, one obtains infinitely many exotic pairs of properly embedded, smooth, orientable surfaces in \(\mathbb{D}^4\).
The construction of these surfaces is based on \(1\)-twist rim surgery. This technique was introduced by \textit{H. J. Kim} [Geom. Topol. 10, 27--56 (2006; Zbl 1104.57018)] to produce compact oriented surfaces in \(\mathbb{CP}^2\) which are smoothly knotted, but topologically unknotted. Roughly speaking, twist rim surgery combines Zeeman twist-spinning construction of \(2\)-knots [\textit{E. C. Zeeman}, Trans. Am. Math. Soc. 115, 471--495 (1965; Zbl 0134.42902)] and \textit{R. Fintushel} and \textit{R. J. Stern}'s rim surgery [Math. Res. Lett. 4, No. 6, 907--914 (1997; Zbl 0894.57014)]. Twist rim surgery allows the author to produce surfaces which are topologically isotopic, potentially not smoothly isotopic, without conditions on the fundamental group of their complement.
To obstruct two surfaces being sent one to the other via a diffeomorphism of \(\mathbb{D}^4\), the authors introduce a numerical invariant of properly embedded surfaces
\[
\Omega(S)\in \mathbb{Z} \cup \{ -\infty \},\quad S\subset \mathbb{D}^4.
\]
This invariant is defined in terms of induced maps in Heegaard-Floer homology. The invariant \(\Omega\) behaves in a controlled way under twist rim surgery, and vanishes for quasi-positive surfaces pushed in the \(4\)-disk.
A key point in the proof of the main theorem is the non-vanishing of certain maps in Heegaard-Floer homology. To this end the authors prove that the (transverse version of the) LOSS invariant (see [\textit{P. Lisca} et al., J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307--1363 (2009; Zbl 1232.57017)] and [\textit{J. A. Baldwin} et al., Geom. Topol. 17, No. 2, 925--974 (2013; Zbl 1285.57005)]) is preserved under the maps induced by some link cobordisms which are ascending surfaces in Weinstein manifolds. This is an interesting result on its own, and fits into a family of similar results due to various authors, e.g. \textit{P. Ozsváth} and \textit{Z. Szabó} [Duke Math. J. 129, No. 1, 39--61 (2005; Zbl 1083.57042)], \textit{A. Juhász} [Adv. Math. 299, 940--1038 (2016; Zbl 1358.57021)], \textit{J. A. Baldwin} and \textit{S. Sivek} [J. Symplectic Geom. 16, No. 4, 959--1000 (2018; Zbl 1411.57019); Geom. Topol. 25, No. 3, 1087--1164 (2021; Zbl 1479.53080)].
Reviewer: Carlo Collari (Abu Dhabi)2-knot homology and Roseman movehttps://zbmath.org/1496.570232022-11-17T18:59:28.764376Z"Matsuda, Hiroshi"https://zbmath.org/authors/?q=ai:matsuda.hiroshi|matsuda.hiroshi.1A surface-knot is a closed connected oriented surface embedded locally flatly in \(\mathbb{R}^4\). For a surface-knot \(F\), a diagram of \(F\) is the image \(\pi(F)\) by a generic projection \(\pi: \mathbb{R}^4 \to \mathbb{R}^3\) equipped with a height information with respect to \(\pi\). Two surface-knots are equivalent if and only if their diagrams are related by a finite sequence of local moves called Roseman moves.
In this paper, for a surface-knot \(F\), the author introduces 4 differential graded algebras \((CR^{--}(D), \partial)\), \((CR^{-+}(D), \partial)\), \((CR^{+-}(D), \partial)\) and \((CR^{++}(D), \partial)\) associated with a diagram \(D\) of \(F\); where the difference is the differential \(\partial\) on generators involving triple points of \(D\). The author defines a stably tame isomorphism on \((CR^{\epsilon \delta}(D), \partial)\) \((\text{where }\epsilon, \delta \in \{+, -\})\). The main result is that the stably tame isomorphism class of the differential graded algebra \((CR^{\epsilon \delta}(D), \partial)\) is an invariant of the surface-knot \(F\). This implies that the homology of \((CR^{\epsilon \delta}(D), \partial)\), called Roseman homology \(HR^{\epsilon \delta}(F)\), is an invariant of \(F\), where again \(\epsilon, \delta \in \{+, -\}\). As a corollary, the author shows that the tuple of 4 differential graded algebras \((CR^{--}(D), \partial)\), \((CR^{-+}(D), \partial)\), \((CR^{+-}(D), \partial)\), \((CR^{++}(D), \partial)\) distinguishes the spun-trefoil from the 2-twist spun trefoil. In the construction of \((CR^{\varepsilon \delta}(D), \partial)\), the author extends the construction of classical knot invariants due to Ng (see e.g. [\textit{L. Ng}, Geom. Topol. 9, 247--297 (2005; Zbl 1111.57011), ibid. 9, 1603--1637 (2005; Zbl 1112.57001)] and other references given in the paper), and shows that for diagrams \(D_1\) and \(D_2\) related by each Roseman move, \((CR^{--}(D_i), \partial)\) \((i=1,2)\) are stably tame isomorphic.
Reviewer: Inasa Nakamura (Kanazawa)Erratum to: ``A note on the McShane's identity for Hecke groups''https://zbmath.org/1496.570242022-11-17T18:59:28.764376Z"Farooq, K."https://zbmath.org/authors/?q=ai:farooq.kErratum to the author's paper [ibid. 52, No. 3, 915--931 (2021; Zbl 1489.57015)].Unknotted strand routings of triangulated mesheshttps://zbmath.org/1496.920732022-11-17T18:59:28.764376Z"Mohammed, Abdulmelik"https://zbmath.org/authors/?q=ai:mohammed.abdulmelik"Hajij, Mustafa"https://zbmath.org/authors/?q=ai:hajij.mustafaSummary: In molecular self-assembly such as DNA origami, a circular strand's topological routing determines the feasibility of a design to assemble to a target. In this regard, the Chinese Postman DNA scaffold routings of \textit{E. Benson} et al. [``DNA rendering of polyhedral meshes at the nanoscale'', Nature 523, No. 7561, 441--444 (2015; \url{doi:10.1038/nature14586})] only ensure the unknottedness of the scaffold strand for triangulated topological spheres. In this paper, we present a cubic-time \(\frac{5}{3}-\)approximation algorithm to compute unknotted Chinese Postman scaffold routings on triangulated orientable surfaces of higher genus. Our algorithm guarantees every edge is routed at most twice, hence permitting low-packed designs suitable for physiological conditions.
For the entire collection see [Zbl 1369.68012].