arXiv research

New polynomials defined for quandle structures, enhancing graph invariants.

New method constructs multiple group racks, differing from known constructions.

We introduce the notion of a $G$-family of quandles which is an algebraic system whose axioms are motivated by handlebody-knot theory, and use it to construct invariants for handlebody-knots. Our invariant can detect the chiralities of some handlebody-knots including unknown ones.

New singquandles help distinguish certain types of links.

We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-k…

We define a functor $\mathcal{Q}$ from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle $X$, there is a one-to-one correspondence between the set of $X$-colorings and that of $\mathcal{Q}(X)$-colorings diagrammatically for any …

The paper explores new quandle systems for handlebody-links and spatial graphs.

We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a "dual" notion to the tunnel number in the handlebody-knot theory. We provide necessary conditions for constituent handlebody-knots by using $G$-family of quandles colorings. The above …

Generalizes quandle constructions and defines a multiplication that results in an abelian group.

Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.

The paper studies invariants of surfaces in the 3-sphere using handlebody-links.

New algorithms improve vascular flow simulations in aortic aneurysms.