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With the idea of an eventual classification of 3-bridge links, we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of edges is a link. Several examples are given. We prove that every link can be represented in this way (butterfly representation). We define the butterfly number of a link, and we show that the butterfly number and the bridge number of a link coincide. This is done by defining a move on the butterfly diagram. We give an example of two different butterflies with minimal butterfly number representing the knot $8_{20}.$ This raises the problem of finding a set of moves on a butterfly diagram connecting diagrams representing the same link. This is left as an open problem.
In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction $left[ a_{1},a_{2},cdots a_{m}right],$ $a_{i}inmathbb{N}$, and the two bridge link of type $p/q,$ where $p/q$ is the rational given
The following is an amalgamation of four preprints and some computer programs which together represent the current state of our investigations of higher order links. This investigation was motivated by questions discussed and raised in the first auth
In the present work, we realize the space of string 2-links $mathcal{L}$ as a free algebra over a colored operad denoted $mathcal{SCL}$ (for Swiss-Cheese for links). This result extends works of Burke and Koytcheff about the quotient of $mathcal{L}$
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive brai
We study configuration space integral formulas for Milnors homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of