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Braid group and leveling of a knot
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2020-11-21, DOI: 10.1142/s1793525321500114
Sangbum Cho, Yuya Koda, Arim Seo

Any knot K in genus-11-bridge position can be moved by isotopy to lie in a union of n parallel tori tubed by n1 tubes so that K intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal n for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1,1)-length. We show that the (1,1)-length equals the level number. We then find braid descriptions for (1,1)-positions of all 2-bridge knots providing upper bounds for their level numbers and also show that the (2,3,7)-pretzel knot has level number two.