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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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How to show whether a given knot and its mirror image are the same or not?

Rigorous answer to this question is to use the Hemion's algorithm. See http://ukcatalogue.oup.com/product/9780198596974.do But it is complicated.
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Introductory article of knot Heegaard Floer Homology

I am looking for some article that gives an introduction to Heegaard Floer homology of knot. I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any intro …
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