The dimer model and circle patterns are linked via combinatorial and geometric transformations.
problem Understanding the dimer model and its geometric counterpart.
method Established a correspondence between dimer models and circle patterns, using combinatorial and geometric transformations.
result The Miquel dynamics on circle patterns is governed by the octahedron recurrence.
New Y-systems for Miquel dynamics are Möbius invariant.
problem Miquel dynamics circle centers are not Möbius invariant.
method Introduced new Y-systems involving only intersection points.
result New Y-systems are Möbius invariant and satisfy the transformation group principle.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
A new knot move preserves pass-move equivalence and differs in count.
problem Defining and analyzing a new knot move.
method Introducing the 1-2-move and comparing it to pass-move and #-move. result Equivalence under 1-2-move for knots is equivalent to pass-move equivalence. 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 introd…
In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended ST-moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended ST-move realizes the crossing change or the …
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
The H(n)-move simplifies virtual and welded knots and links.
problem Tackling the unknotting of virtual and welded links.
method Extending the H(n)-move to virtual and welded links and showing their equivalence to Reidemeister moves.
result Virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and H(n)-moves.
We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set…
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to 2n-move and Vn-move. result Classifies welded string links up to 2n-move and Vn-move. Classifies virtual links up to a specific move.
problem Characterizing 2-component virtual links using Ξ-moves. method Refined odd writhe and linking numbers.
result Classifies 2-component virtual links up to Ξ-moves. New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
Improved estimation of VAR-Moving Average models using GLS.
problem Estimating VAR-Moving Average models efficiently.
method Use generalized least squares (GLS) for each fixed moving average polynomial.
result Likelihood function format similar to VAR models, allowing maximum likelihood estimation.
Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.
problem Finding the minimal set of moves for surfaces in 4D.
method Derived minimal generating set of spatial moves, translated into planar moves.
result Minimal generating set of spatial moves for surfaces in 4D.
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main …
New diagonal move simplifies knots and links efficiently.
problem Efficiently unknotting knots and links.
method Introduces diagonal move, proving its effectiveness for classical and welded knots.
result Diagonal move reduces any knot or link to the unknot or unlink with fewer operations.
New moves help untangle complex knots.
problem Deforming twisted knots into simpler forms.
method Finite sequences of extended Reidemeister moves and three forbidden moves.
result Any twisted knot can be simplified.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
New virtualized Δ-move simplifies virtual knots and links.
problem Simplifying virtual knots and links.
method Introducing a new local deformation called the virtualized Δ-move.
result Virtualized Δ-move is an unknotting operation for virtual knots.
Explains the difference between EMA and moving EMA, focusing on market trend indicators.
problem Understanding the difference between exponential moving average and moving exponential average.
method Explains the mathematical tools and definitions of trend indicators.
result Discusses the properties of the MACD indicator and its use in market trend analysis.
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
The Jones polynomial's divisibility is analyzed via local moves on virtual links.
problem Divisibility of the Jones polynomial under local moves on virtual links.
method Developed a general decomposition for the Jones polynomial of virtual links and analyzed divisibility conditions for various local moves.
result Succinct divisibility conditions on the Jones polynomial of virtual links differing via local moves.
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
4-move kills Alexander polynomial
problem Whether the 4-move is an unknotting operation
method showing every knot can be reduced via 4-moves and isotopies
result every knot can be reduced to one with a trivial Alexander polynomial
P-moves connect different 3-manifold decompositions.
problem Connecting different 3-manifold decompositions.
method Using P-complex and Morse 2-functions, we show P-moves between pants-block decompositions.
result Any two pants-block decompositions are related by a finite sequence of P-moves.
Complete classification of links up to specific moves.
problem Classifying links using specific local moves.
method Clasper theory.
result Complete classification of links up to clasp-pass moves.
Roseman moves are seven types of local modification for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…
In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…
We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.
In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. …
The writhe polynomial invariant is proven for virtual knots via shell moves.
problem Determining equivalence of oriented virtual knots using writhe polynomials.
method Introducing shell moves to prove equivalence of writhe polynomials.
result Two virtual knots are equivalent if they can be transformed by shell moves.
A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…
New moves prove crossing number sum for knots.
problem Proving the additivity of crossing number for knots.
method Defined restricted Reidemeister moves to prove additivity.
result Crossing number of combined knots is at least sum of individual knots.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
New contact Kirby moves complete the set for contact surgery diagrams.
problem Contact surgery diagrams and their relation to contactomorphic contact manifolds.
method Introducing lantern moves and chain moves to complete the set of contact Kirby moves.
result Two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of specific moves.
New moves for singular knots identified and described.
problem Identifying and describing moves for singular knots.
method Provided 96 generating sets of oriented singular Reidemeister moves and selected moves for Legendrian singular knots.
result Surviving moves for Legendrian singular knots were identified and described.
A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C_n-moves, under certain restrictions. Namely, we give a classification up to C_4-move…
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram
New sequences prove some link diagrams can't be transformed by specific moves.
problem Proving some link diagrams can't be transformed by specific sequences of Reidemeister moves.
method Proved the existence of I-generalized ordered sequences to create simple transformations.
result Some link diagrams cannot be transformed by sequences of moves that increase, then decrease crossings.
New move proves infinitely many non-conjugate braids for certain links.
problem Proving infinitely many non-conjugate braids for certain link types.
method Introducing a non-degenerate exchange move.
result Links have infinitely many non-conjugate braids after applying the move.
New moving average adapts weight dynamically based on polynomial and wavefunction.
problem Lagging traditional moving averages in adjusting to changes in data.
method Develops a moving average with weight as a polynomial of a wavefunction from an eigenproblem.
result Immediate 'switch' without lag, adapting to changes in data.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.