arXiv research

Study theta-curves on torus in 3-sphere, classifying them.

Determinants of theta curves and symmetric graphs are studied.

Proves prime theta-curves for knots on minimal genus surfaces.

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U …

The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.

New findings on prime theta-curves with simple tangles.

Spatial graphs can be unknotted with region crossing changes.

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results. *Smallness of Montesinos tangles with lengt…

Brunnian theta curves in 3D spheres have hyperbolic exteriors.

There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handl…

Study on knot properties, showing relation between unknotting and crossing numbers.

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. An approach to estimating $c(M^3)$ from below for total spaces o…

We study the minimal crossing number $c(K_{1}\# K_{2})$ of composite knots $K_{1}\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $θ_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n…

The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order $\leq k-1$. We see that the set of $C_k$-…

The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.

Unique arcs create slopes in genus two handlebodies.