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We investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology ℝP3 are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in L–spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than 1 in the Brieskorn sphere Σ(2,3,7) are also determined by their complements.
Keywords: Heegaard Floer homology
Gainullin, Fyodor 1
@article{10_2140_agt_2018_18_69,
author = {Gainullin, Fyodor},
title = {Heegaard {Floer} homology and knots determined by their complements},
journal = {Algebraic and Geometric Topology},
pages = {69--109},
publisher = {mathdoc},
volume = {18},
number = {1},
year = {2018},
doi = {10.2140/agt.2018.18.69},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.69/}
}
TY - JOUR AU - Gainullin, Fyodor TI - Heegaard Floer homology and knots determined by their complements JO - Algebraic and Geometric Topology PY - 2018 SP - 69 EP - 109 VL - 18 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.69/ DO - 10.2140/agt.2018.18.69 ID - 10_2140_agt_2018_18_69 ER -
Gainullin, Fyodor. Heegaard Floer homology and knots determined by their complements. Algebraic and Geometric Topology, Tome 18 (2018) no. 1, pp. 69-109. doi: 10.2140/agt.2018.18.69
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