arXiv research

We study comparison formulas for $ζ$-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus $g\geq 2$. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly.

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …

Knot invariants from XC-structures on Sweedler algebra are trivially determined.

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…

New approach to symmetries in teleparallel geometries with non-trivial isotropy groups.

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperb…

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…

We show that if a split link is obtained from a split link $L$ in $S^3$ by $1/n$-Dehn surgery along a trivial knot $C$, then the link $L\cup C$ is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…

We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjectu…

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…

The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

Characterizes G2-structures on Lie algebras with non-trivial center.

Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.

The study determines fiber homotopy trivial bundles and their impact on curvature.

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discus…

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

For a lattice $Γ$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $Γ$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under…

We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hy…

The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.

It is an open problem whether Kirk's $σ$ invariant is the complete obstruction to a link map $S^2\cup S^2\to S^4$ being link homotopically trivial. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $ω$ that is defined on the kernel of $σ$ and also obstructs link nullhomotopy. We …

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link.…

Algebraic structure of the group of pseudo-isotopy classes of diffeomorphisms of the trivial disk bundle over the standard sphere which restrict to the identity map on the boundary is determined.

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a coro…

This paper improves bounds on how many Delta-moves are needed to trivialize a link.

In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used …

We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…

For $n \geq 2$ we describe an $O(l^3n)$-time algorithm that determines if a length $l$ virtual braid word in the standard presentation of the virtual braid group ${\mathcal VB}_n$ represents the trivial virtual braid.

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.

Study determines homotopy types of specific 6-manifolds.

Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit versi…

In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $Υ$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $Υ$ of $(n,1)$-cable…

A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinan…

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.

We show that for $n>2$ a compact locally conformally Kähler manifold $(M^{2n},g,J)$ carrying a non-trivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by some compact Kähler manifold of dimension $2n-2$.

In this paper, we prove the existence of certain symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4--manifolds, which generalizes Smith, Thomas and Yau's examples of symplectic conifold transitions on trivial $CP^{1}$-bundles over Kähler surfaces. Our main result is to determine the diffeomorphis…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…

The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.

Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.

The paper defines new homotopy relations on knot projections and classifies certain knot types.