3-bridge Links Research Articles

Meridional rank and bridge number for a class of links

We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and meridional ranks of links. To prove this, we also show that the meridional rank of any satellite knot is at least 4.

Characterization of 3-bridge links with infinitely many 3-bridge spheres

In an earlier paper, the author constructed an infinite family of 3-bridge links each of which admits infinitely many 3-bridge spheres up to isotopy. In this paper, we prove that if a prime, unsplittable link L in S 3 admits infinitely many 3-bridge spheres up to isotopy then L belongs to the family.

The fourth Skein Module and the Montesinos-Nakanishi Conjecture for 3-algebraic links

We study the fourth skein module of 3-manifolds, based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L(1) = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We show how elements of the skein module behave under the n-move and we compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using mutants and rotors, we find different links which represent the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a by product we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). For links in S3, the structure of our skein module suggests the existence of three new polynomial invariants of unoriented framed (or unframed) links. One of them would generalize the Kauffman polynomial of links and another one could be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the conclusion, we speculate that our new knot invariants are related to a deformation of the symplectic quotient of braid groups.

COVERING PROPERTIES OF SMALL VOLUME HYPERBOLIC 3-MANIFOLDS

Closed hyperbolic 3-manifolds obtained by Dehn surgeries on the Whitehead link yield interesting examples of manifolds of small volume. In the present paper these manifolds are described as 2-fold coverings of the 3-sphere branched over 3-bridge links. As a corollary, maximally symmetric [Formula: see text]-manifolds of small volume are obtained.

Special representations forn-bridge links

We describe a simple algorithm to obtain a catalogue of 3-bridge links by computer, depending upon 6-tuples of positive integers. This permits us to represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, we prove some results about the topological structure of these manifolds and extend the combinatorial representation ton-bridge links

The Splittability and Triviality of 3-Bridge Links

The Splittability and Triviality of 3-Bridge Links

The splittability and triviality of 3-bridge links

A method to simplify 3 3 -bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of 3 3 -bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every 3 3 -bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of 2 2 -fold branched coverings of S 3 {S^3} , it follows that every genus 2 2 Heegaard diagram of S 2 × S 2 # L ( p , q ) {S^2} \times {S^2}\# L(p,q) or S 3 {S^3} can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.

The minimum crossing of 3-bridge links

The minimum crossing of 3-bridge links