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Kim, Seonhwa
기하학 수리물리 연구단
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FORMALITY OF FLOER COMPLEX OF THE IDEAL BOUNDARY OF HYPERBOLIC KNOT COMPLEMENT

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Title
FORMALITY OF FLOER COMPLEX OF THE IDEAL BOUNDARY OF HYPERBOLIC KNOT COMPLEMENT
Author(s)
YOUNGJIN BAE; SEONHWA KIM; YONG-GEUN OH
Publication Date
2021-02
Journal
ASIAN JOURNAL OF MATHEMATICS, v.25, no.1, pp.117 - 176
Publisher
INT PRESS BOSTON, INC
Abstract
This is a sequel to the authors' article [BKO]. We consider a hyperbolic knot K in a closed 3-manifold M and the cotangent bundle of its complement M \ K. We equip M \ K with a hyperbolic metric h and its cotangent bundle T* (M\K) with the induced kinetic energy Hamiltonian H-h = 1/2 vertical bar p vertical bar 2/h and Sasakian almost complex structure J(h), and associate a wrapped Fukaya category to T* (M \ K) whose wrapping is given by H-h. We then consider the conormal nu*T of a horo-torus T as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus T, and so that the structure maps satisfy (m) over tilde (k) = 0 unless k not equal 2 and an A(infinity)-algebra associated to nu*T is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology HW (nu*T; H-h) with respect to H-h is well-defined and isomorphic to the Knot Floer cohomology HW (partial derivative(infinity)(M / K)) that was introduced in [BKO] for arbitrary knot K subset of M. We also define a reduced cohomology, denoted by (HW) over tilde (d) (partial derivative(infinity)(M / K)), by modding out constant chords and prove that if (HW) over tilde (d) (partial derivative(infinity)(M / K)) not equal 0 for some d >= 1, then K cannot be hyperbolic. On the other hand, we prove that all torus knots have (HW) over tilde (1) (partial derivative(infinity)(M / K)) not equal 0.
URI
https://pr.ibs.re.kr/handle/8788114/10811
ISSN
1093-6106
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > 1. Journal Papers (저널논문)
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