One of my favorite research papers is Involutive Heegaard Floer homology by Hendricks and Manolescu. Inspired by Ciprian's groundbreaking work on Pin(2)-equivariant Seiberg-Witten Floer homology, they developed a similar invariant in Heegaard Floer theory. The j-action is replaced here by the conjugation action of Heegaard diagrams. I love this paper because it represents a basic approach of studying math: find analogue in different branches and explore new phenomena, which I believe that is the most feasible way for non-genius people like me. Its formalism has also led to interesting topological applications (e.g. The (2,1)-cable of the figure-eight knot is not smoothly slice). Recently, Daren and I defined an analog in Khovanov homology; see here. More or less surprisingly, it turns out to not provide any new information (at least for the "hat" version). It hence reflects some mysterious difference between Khovanov homology and Floer theory.
My favorite open problem is the (categorified) Wrapping Conjecture. Proposed by Eli Grigsby, it predicts that the third grading on annular Khovanov homology can tell you the wrapping number of a knot in the thickened annulus, i.e. the minimal intersection number of the knot with a meridional disk. I learned this problem when I was writing my first paper. I found it interesting because it has the same flavor as many "classical" applications of Floer homology (e.g. knot Floer homology detects Seifert genus and fiberness) and is stated in an equally succint way, but it seems very difficult to attack using "traditional" strategy, argued by Gage Martin. It therefore may also allude some foundamental difference between two theories (or maybe it's just wrong!). Currently I don't have anything interesting to say about this conjecture, but I enjoy thinking about it in my leisure.
Judson had an insightful comment on low-dimensional topologists: "There are two kinds of topologists. Some of them do math for doing topology, and others do topology for doing math". Apparently I am in the latter. To be honest, I don't really care about topological applications that only apply to specific examples (disclaimer: I understand some people do consider them important); instead, I believe that (at least part of ) the vitality of low-dimensional topology is rooted in its connections with other rapidly developing fields in mathematics.
香蕉空间 is a Chinese mathematics website in the style of nLab. You may find it especially helpful if you can read Chinese and are interested in some abstract mathematics (in this case, there is a high chance you already know this website).