Singular knot theory extends classical knot theory by allowing transverse double points without over/under information, together with singular Reidemeister moves of types IV and V. A central open problem in this theory is to determine the minimal generating sets of oriented singular Reidemeister moves. In this paper, we completely solve this problem. In addition, we establish independence results for singular Reidemeister moves by introducing an invariant that provides obstructions and lower bounds for generating sets, including the independence of type III from types I, II, IV, and V. More precisely, starting from a minimal generating set of ordinary Reidemeister moves of types I--III, we prove that the singular moves admit exactly $96$ distinct inclusion-minimal generating sets, and that these exhaust all possibilities. Our proof introduces a new invariant for singular links, constructed via a projection to self-singular links, which detects the distinction between the two families of type IV moves and provides an obstruction for generating type V moves from types I--IV. We also determine the unoriented case, where the classification collapses to exactly $8$ minimal generating se
In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.
We introduce five novel types of Monte Carlo (MC) moves that brings the number of moves of ensemble MC calculations from three to eight. So far such calculations have relied on affine invariant stretch moves that were originally introduced by Christen (2007), 'walk' moves by Goodman and Weare (2010) and quadratic moves by Militzer (2023). Ensemble MC methods have been very popular because they harness information about the fitness landscape from a population of walkers rather than relying on expert knowledge. Here we modified the affine method and employed a simplex of points to set the stretch direction. We adopt the simplex concept to quadratic moves. We also generalize quadratic moves to arbitrary order. Finally, we introduce directed moves that employ the values of the probability density while all other types of moves rely solely on the location of the walkers. We apply all algorithms to the Rosenbrock density in 2 and 20 dimensions and to the ring potential in 12 and 24 dimensions. We evaluate their efficiency by comparing error bars, autocorrelation time, travel time, and the level of cohesion that measures whether any walkers were left behind. Our code is open source.
Head moves are a type of rearrangement moves for phylogenetic networks. They have mostly been studied as part of more encompassing types of moves, such as rSPR moves. Here, we study head moves as a type of moves on themselves. We show that the tiers ($k>0$) of phylogenetic network space are connected by local head moves. Then we show tail moves and head moves are closely related: sequences of tail moves can be converted to sequences of head moves and vice versa, changing the length by at most a constant factor. Because the tiers of network space are connected by rSPR moves, this gives a second proof of the connectivity of these tiers. Furthermore, we show that these tiers have small diameter by reproving the connectivity a third time. As the head move neighbourhood is in general small, this makes head moves a good candidate for local search heuristics. Finally we prove that finding the shortest sequence of head moves between two networks is NP-hard.
Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2, and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak's work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these twelve sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains ten moves.
Understanding what makes tutoring effective requires methods for systematically analyzing tutors' instructional actions during learning interactions. This paper presents a tutor move taxonomy designed to support large-scale analysis of tutoring dialogue within the National Tutoring Observatory. The taxonomy provides a structured annotation framework for labeling tutors' instructional moves during one-on-one tutoring sessions. We developed the taxonomy through a hybrid deductive-inductive process. First, we synthesized research from cognitive science, the learning sciences, classroom discourse analysis, and intelligent tutoring systems to construct a preliminary framework of tutoring moves. We then refined the taxonomy through iterative coding of authentic tutoring transcripts conducted by expert annotators with extensive instructional and qualitative research experience. The resulting taxonomy organizes tutoring behaviors into four categories: tutoring support, learning support, social-emotional and motivational support, and logistical support. Learning support moves are further organized along a spectrum of student engagement, distinguishing between moves that elicit student reaso
In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush.
Rotational tangle diagrams have been proven to be extremely important in the study of quantum invariants, as they provide a natural passage between topology and quantum algebra. In this paper, we give a detailed description of several generating sets of rotational Reidemeister moves for rotational diagrams of both unframed and framed tangles. In particular, we prove that the minimal number of moves needed to generate all oriented unframed (resp. framed) rotational Reidemeister moves is 8 (resp. 9). The latter implies that a minimal generating set of Reidemeister moves for oriented, framed links contains 5 moves.
The Matveev-Piergallini (MP) moves on spines of $3$-manifolds are well-known for their correspondence to the Pachner $2$-$3$ moves in dual ideal triangulations. Benedetti and Petronio introduced combinatorial descriptions of closed $3$-manifolds and combed $3$-manifolds by using branched spines and their equivalence relations, which involve MP moves with 16 distinct patterns of branchings. In this paper, we demonstrate that these 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses. Consequently, we obtain simpler combinatorial descriptions for closed $3$-manifolds and combed $3$-manifolds. Furthermore, we extend these results to framed $3$-manifolds and spin $3$-manifolds. These descriptions are advantageous, particularly when constructing and studying quantum invariants of links and $3$-manifolds. In various constructions of quantum invariants using (ideal) triangulations, branching structures naturally arise to facilitate the assignment of non-symmetric algebraic objects to tetrahedra. In these frameworks, the primary MP move precisely corresponds to certain algebraic pentagon relations, such as the pentagon relation of the ca
We give a complete classification of links up to clasp-pass moves, which coincides with Habiro's $C_3$-equivalence. We also classify links up to band-pass and band-# moves, which are versions of the usual pass- and #-move, respectively, where each pair of parallel strands belong to the same component. This recovers and generalizes widely a number of partial results in the study of these local moves. The proofs make use of clasper theory.
Popular methods for exploring the space of rooted phylogenetic trees use rearrangement moves such as rNNI (rooted Nearest Neighbour Interchange) and rSPR (rooted Subtree Prune and Regraft). Recently, these moves were generalized to rooted phylogenetic networks, which are a more suitable representation of reticulate evolutionary histories, and it was shown that any two rooted phylogenetic networks of the same complexity are connected by a sequence of either rSPR or rNNI moves. Here, we show that this is possible using only tail moves, which are a restricted version of rSPR moves on networks that are more closely related to rSPR moves on trees. The connectedness still holds even when we restrict to distance-1 tail moves (a localized version of tail-moves). Moreover, we give bounds on the number of (distance-1) tail moves necessary to turn one network into another, which in turn yield new bounds for rSPR, rNNI and SPR (i.e. the equivalent of rSPR on unrooted networks). The upper bounds are constructive, meaning that we can actually find a sequence with at most this length for any pair of networks. Finally, we show that finding a shortest sequence of tail or rSPR moves is NP-hard.
The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different vertices cannot communicate to co-ordinate their strategy. In this paper, we extend the deduction game to allow each searcher to make $k$ moves, where $k$ is a fixed positive integer. We consider the value of the $k$-move deduction number on several classes of graphs including paths, cycles, complete graphs, complete bipartite graphs, and Cartesian and strong products of paths.
Combinatorial Game Theory typically studies sequential rulesets with perfect information where two players alternate moves. There are rulesets with {\em entailing moves} that break the alternating play axiom and/or restrict the other player's options within the disjunctive sum components. Although some examples have been analyzed in the classical work Winning Ways, such rulesets usually fall outside the scope of the established normal play mathematical theory. At the first Combinatorial Games Workshop at MSRI, John H. Conway proposed that an effort should be made to devise some nontrivial ruleset with entailing moves that had a complete analysis. Recently, Larsson, Nowakowski, and Santos proposed a more general theory, {\em affine impartial}, which facilitates the mathematical analysis of impartial rulesets with entailing moves. Here, by using this theory, we present a complete solution for a nontrivial ruleset with entailing moves.
We define Bergman presentations and Bergman algebras associated to Bergman presentations. These algebras embrace various generalisations of Leavitt path algebras. A Bergman presentation can be visualised by a Bergman graph, which is a finite bicoloured hypergraph satisfying two conditions. We define several moves for Bergman graphs and prove that they preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Bergman algebra. One recovers the well-known results, that in the context of finite directed graphs the shift move, outsplitting, insplitting, source elimination and collapsing preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Leavitt path algebra. Moreover, we mention some connections between Tietze transformations and the moves for Bergman graphs defined in this paper.
It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introduce a minimal generating set of four oriented Reidemeister moves, which includes two O1 moves, one O2 move, and one O3 move. We then study which other sets of up to 5 oriented moves generate all moves, and show that only few of them do. Some commonly considered sets are shown not to be generating. An unexpected non-equivalence of different O3 moves is discussed.
It is a natural question to ask whether two links are equivalent by the following moves -- parallel parts of a link are changed to k-times half-twisted parts and if they are, how many moves are needed to go from one link to the other. In particular if k=2 and the second link is a trivial link it is the question about the unknotting number. The new polynomial invariants of links often allow us to answer the above questions. Also the first homology groups of cyclic branch covers over links provide some interesting information. In the first part of the paper we apply the Jones-Conway (Homflypt) and Kauffman polynomials to find whether two links are not t_k equivalent and if they are, to gain some information how many moves are needed to go from one link to the other. In the second part we describe the Fox congruence classes and their relations with t_k moves. We use the Fox method to analyze relations between t_k moves and the first homology groups of branched cyclic covers of links.In the third part we consider the influence of t_k moves on the Goeritz and Seifert matrices and analyze Lickorish-Millett and Murakami formulas from the point of view of t_k moves and illustrate them by v
Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $8$ directed Polyak moves $\{ Ω{1a}^\uparrow, Ω{1a}^\downarrow, Ω{1b}^\uparrow, Ω{1b}^\downarrow, Ω{2a}^\uparrow, Ω{2a}^\downarrow, Ω{3a}^\uparrow, Ω{3a}^\downarrow \}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least $2$ components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\Omega3$.
In many games, moves consist of several decisions made by the player. These decisions can be viewed as separate moves, which is already a common practice in multi-action games for efficiency reasons. Such division of a player move into a sequence of simpler / lower level moves is called \emph{splitting}. So far, split moves have been applied only in forementioned straightforward cases, and furthermore, there was almost no study revealing its impact on agents' playing strength. Taking the knowledge-free perspective, we aim to answer how to effectively use split moves within Monte-Carlo Tree Search (MCTS) and what is the practical impact of split design on agents' strength. This paper proposes a generalization of MCTS that works with arbitrarily split moves. We design several variations of the algorithm and try to measure the impact of split moves separately on efficiency, quality of MCTS, simulations, and action-based heuristics. The tests are carried out on a set of board games and performed using the Regular Boardgames General Game Playing formalism, where split strategies of different granularity can be automatically derived based on an abstract description of the game. The resul
In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We define four generalized moves based on the Reidemeister Moves. These moves have the capability of untangling knots which must be made more complicated before they can be simplified. With these moves, we construct a computer program which reads the two-dimensional projection of a knot in its Gauss Code notation and untangles it to the fewest possible number of crossings. Due to the properties of the Gauss Code notation, the program runs efficiently with minimal computation time, compared to currently existing untangling programs. We have tested it on all possible Gauss Codes of up to 50 crossings, and the program successfully determines a simplest possible projection of each knot. The results are consistent with our predictions about how the moves work in untangling a knot.
In the Tower of Hanoi problem, there is six types of moves between the three pegs. The main purpose of the present paper is to find out the number of each of these six elementary moves in the optimal sequence of moves. We present a recursive function based on indicator functions, which counts the number of each elementary move, we investigate some of its properties including combinatorial identities, recursive formulas and generating functions. Also we found and interesting sequence that is strongly related to counting each type of these elementary moves that we'll establish some if its properties as well.