Milnor conjecture (topology)
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Short description: Theorem that the slice genus of the (p, q) torus knot is (p-1)(q-1)/2
In knot theory, the Milnor conjecture says that the slice genus of the [math]\displaystyle{ (p, q) }[/math] torus knot is
- [math]\displaystyle{ (p-1)(q-1)/2. }[/math]
It is in a similar vein to the Thom conjecture.
It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka.[1] Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant.[2]
References
- ↑ Kronheimer, P. B.; Mrowka, T. S. (1993), "Gauge theory for embedded surfaces, I", Topology 32 (4): 773–826, doi:10.1016/0040-9383(93)90051-V, http://www.math.harvard.edu/~kronheim/thom1.pdf.
- ↑ Rasmussen, Jacob A. (2004). "Khovanov homology and the slice genus". arXiv:math.GT/0402131..
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