Related papers: Braid equivalence in 3-manifolds with rational sur…
For a knot $K$ in a homology $3$-sphere $\Sigma$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a…
We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid…
For a knot $K$ with $\Delta_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies…
We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a Dehn surgery on a twist knot.
It is well known that any three-manifold can be obtained by surgery on a framed link in $S^3$. Lickorish gave an elementary proof for the existence of the three-manifold invariants of Witten using a framed link description of the manifold…
We take advantage of the correspondence between fibered links, open book decompositions and contact structures on a closed connected 3-dimensional manifold to determine a mixed link diagram presentation for a particular fibered link $L$ in…
Alexander's and Markov's theorems state that any link type in $R^3$ is represented by a closed braid and that such representations are related by some elementary operations called Markov moves. We generalize the notion of a braid to that in…
In this work we present a complete (no misses, no duplicates) census for closed, connected, orientable and prime 3-manifolds induced by plane graphs with a bipartition of its edge set (blinks) up to $k=9$ edges. Blinks form a universal…
Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing.…
Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the…
We determine the adjoint Reidemeister torsion of a $3$-manifold obtained by some Dehn surgery along $K$, where $K$ is either the figure-eight knot or the $5_2$-knot. As in a vanishing conjecture, we consider a similar conjecture and show…
Kirby proved that two framed links in S^3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke…
In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a…
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…
It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is…
This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein…
Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton's conjectured inequality and…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…
This is an expository article on diagrammatic representations of knots and links in various settings via braids.