Geodesic
Bridge distance and plat projections
Johnson, Jesse1 ; Moriah, Yoav2
1Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, 39 Chilton St #2, Cambridge, MA 02138, United States
2Department of Mathematics, Technion, 32000 Haifa, Israel
Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3361-3384
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Every knot or link K ⊂ S3 can be put in a bridge position with respect to a 2–sphere for some bridge number m ≥ m0, where m0 is the bridge number for K. Such m–bridge positions determine 2m–plat projections for the knot. We show that if m ≥ 3 and the underlying braid of the plat has n − 1 rows of twists and all the twisting coefficients have absolute values greater than or equal to three then the distance of the bridge sphere is exactly ⌈n∕(2(m − 2))⌉, where ⌈x⌉ is the smallest integer greater than or equal to x. As a corollary, we conclude that if such a diagram has n > 4m(m − 2) rows then the bridge sphere defining the plat projection is the unique, up to isotopy, minimal bridge sphere for the knot or link. This is a crucial step towards proving a canonical (thus a classifying) form for knots that are “highly twisted” in the sense we define.

DOI : 10.2140/agt.2016.16.3361
Classification : 57M27
Keywords: Heegaard splittings, bridge sphere, plats, bridge distance, train tracks

Johnson, Jesse  1   ; Moriah, Yoav  2

1 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, 39 Chilton St #2, Cambridge, MA 02138, United States
2 Department of Mathematics, Technion, 32000 Haifa, Israel 
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Johnson, Jesse; Moriah, Yoav. Bridge distance and plat projections. Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3361-3384. doi: 10.2140/agt.2016.16.3361

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