Non-classical knot theory refers to a family of extensions of classical knot theory in which one or more of the basic ingredients of the classical framework are modified. These modifications may affect the local structure of crossings, the allowed Reidemeister-type moves, the ambient space, the global form of diagrams, or the additional data carried by diagrams. Examples include skein modules of three-manifolds, pseudo knots, singular knots, stuck knots, bonded knots, tied links, virtual knots, welded knots, knotoids, and related braid-type structures. In this survey, we present an overview of several non-classical knot theories from a comparative geometric and algebraic perspective. We examine how changes in ambient topology, new crossing types, rigidity constraints, auxiliary relational data, modified isotopy relations, and open or virtual diagrammatic settings lead to generalized knot theories with distinct topological and combinatorial features. Particular emphasis is placed on braid-theoretic formulations, skein-theoretic methods, trace constructions, and extensions of classical polynomial invariants. We conclude by outlining open problems concerning generalized algebraic structures, hybrid diagrammatic theories, and the relationships among different non-classical knot theories.
We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex singularities, denoted $C_{alg}(K)$. We show that $C_{alg}(K)-h(K)$ can be arbitrarily large. We also give an upper bound for $h$ of some families of knots such as torus knots $T(2,n)$, twist knots and their mirror images.
Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsvath-Stipsicz-Szabo Upsilon function, and also introduces a new invariant of L-space knots, the formal semigroup.
We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum $\#^n K$ is at least $n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.
Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also e…
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, consequently, for bounding the topological slice genus above. As applications we give new upper bounds on the algebraic genera of torus knots and satellite knots.
It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Coch…
Torsion in the concordance group $\mathscr{C}$ of knots in $S^3$ can be studied with the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$. Here $\mathbb{F}$ is a field of characteristic $χ(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine, who also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. While the concordance group $\mathscr{C}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathscr{VC}$. It is unknown if $\mathscr{VC}$ admits non-classical finite torsion. Here we define the virtual algebraic concorda…
The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $ν^+$: we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.
Algebraic structures of Vassiliev invariants for knot families
Institute for Theoretical and Experimental Physics（理论与实验物理研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）；NRC ”Kurchatov Institute”（库尔恰托夫国家研究中心）
We prove that there are infinitely many non-homeomorphic hyperbolic knot complements $S^3\setminus K_i = \mathbb{H}^3/Γ_i$ for which $Γ_i$ contains elements whose trace is an algebraic non-integer.
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.
We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.
Stabilizing Four-Torsion in Classical Knot Concordance
设$M_K$是纽结的2重分支覆盖$K in $ S^3 $. If $ H_1（M_K）= { \bf Z}_3 \oplus { \bf Z}_{3^{2 i}} \oplus G $ where 3 does not divide the order of $ G $ then $ K$在协调群中不是4阶的.这个障碍检测无限新的家庭的节点，代表元素的顺序4的代数和谐组，而不是顺序4的和谐。
This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.
