arXiv research

A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…

New findings on knot operations challenge a long-standing conjecture.

A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a gr…

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology sphere…

We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.

Table of symmetric diagrams for knots up to 10 crossings.

Spatial graphs can be unknotted with region crossing changes.

Paper classifies link diagrams on nonorientable surfaces using region crossing changes.

Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.

The paper explores when specific knot operations simplify diagrams.

Alexander polynomial condition blocks crossing changes in some knots.

We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties …

Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that a…

In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also discuss the relation between region crossing change and the Arf invariant of proper l…

Study examines how changing regions affects planar graphs.

A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due …

We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered ca…

Proves special alternating knots can't have cosmetic crossings.

Shows large unknotting number for simple knots.

Region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting numbers for a knot diagram and a knot.

Study finds only two CROSSes can have specific quaternionic structure.

New method detects changes by maximizing cross-entropy, outperforming existing techniques.

The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At …

Extends knot invariant to filtered grid complexes.

In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.

The article enumerates doubly symmetric diagrams for knots up to 18 crossings.

In a recent work of Ayaka Shimizu$^{[5]}$, she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only…

We analyze all monodromies of genus one fibered knots that possess clean or once-unclean arcs, and use this to determine all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibered knot, and classify all such generalized crossing changes between two genus on…

We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and…

New game defined on origami patterns, linking number introduced.

We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.

We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.

In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. …

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…

We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…

Graphs represent knot adjacency for n crossings.

We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.

Cross-validation pitfalls in change-point regression are addressed with new approaches.

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.

The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.

Study identifies prime strongly positive amphicheiral knots with double symmetry.

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of 4-component links were classified by Levine with enormous algebraic computations. We modify the results by using Habiro's clasper theory. The new classification…

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing cha…

Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.

We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate paper, currently in preparation, we prove that these circular Heegaard splittin…

New Boolean algebra method shows knot unknotting number is (c+1)/2.

Training accurate deep neural networks (DNNs) in the presence of noisy labels is an important and challenging task. Though a number of approaches have been proposed for learning with noisy labels, many open issues remain. In this paper, we show that DNN learning with Cross Entropy (CE) exhibits overfitting to noisy lab…