Related papers: Milnor Invariants for Spatial Graphs

Graph theory is a branch of mathematics in which pair-wise relations between objects are studied. My PhD thesis, supervised by David R. Wood, introduces and investigates a new family of graphs, called link graphs, that generalises the…

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…

We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse…

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the…

Several possible presentations for the homotopy theory of (non-hypercomplete) $\infty$-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists,…

We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links…

We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite…

Given a link map f into a manifold of the form Q = N \times \Bbb R, when can it be deformed to an unlinked position (in some sense, e.g. where its components map to disjoint \Bbb R-levels) ? Using the language of normal bordism theory as…

The slope is an isotopy invariant of colored links with a distinguished component, initially introduced by the authors to describe an extra correction term in the computation of the signature of the splice. It appeared to be closely related…

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the…

In the paper we prove that the primitive part of the Sinha homology spectral sequence E^2-term for the space of long knots is rationally isomorphic to the homotopy E^2-term. We also define natural graph-complexes computing the rational…

Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self crossing changes. Milnor introduced invariants under link homotopy called $\bar{\mu}$. Nanophrases, introduced by…

We affirmatively address the question of whether the proposed link homotopy invariant $\omega$ of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant $\tau$ of Schneiderman-Teichner to a link homotopy…

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…

For each positive integer $n$, we define the divisibility relation graph $D_n$ whose vertex set is the set of divisors of $n$, and in which two vertices are adjacent if one is a divisor of the other. This type of graph is a special case of…

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral…

Let $K_n$ be a complete graph with $n$ vertices. An embedding of $K_n$ in $S^3$ is called a spatial $K_n$-graph. Knots in a spatial $K_n$-graph corresponding to simple cycles of $K_n$ are said to be constituent knots. We consider the case…

In 2003, Ozsv\'ath and Szab\'o defined the concordance invariant $\tau$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$ and a…

In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion…

In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with…