Slice genus
Encyclopedia
In mathematics
, the slice genus of a smooth knot
K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g embedded in the 4-ball D4 bounded by S3.
More precisely, if S is required to be smoothly embedded, then this integer g is the smooth slice genus of K and is often denoted gs(K) or g4(K), whereas if S is required only to be topologically locally flat
ly embedded then g is the topologically locally flat slice genus of K. (There is no point considering g if S is required only to be a topological embedding, since the cone on K is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman
says that if the Alexander polynomial
of K is 1, then the topologically locally flat slice genus of K is 0, but it can be proved in many ways (originally with gauge theory
) that for every g there exist knots K such that the Alexander polynomial of K is 1 while the genus and the smooth slice genus of K both equal g.
The (smooth) slice genus of a knot K is bounded below by a quantity involving the Thurston–Bennequin invariant of K:
The (smooth) slice genus is zero if and only if the knot is concordant to the unknot
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the slice genus of a smooth knot
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g embedded in the 4-ball D4 bounded by S3.
More precisely, if S is required to be smoothly embedded, then this integer g is the smooth slice genus of K and is often denoted gs(K) or g4(K), whereas if S is required only to be topologically locally flat
Local flatness
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension.Suppose a d dimensional manifold N is embedded in an n dimensional manifold M...
ly embedded then g is the topologically locally flat slice genus of K. (There is no point considering g if S is required only to be a topological embedding, since the cone on K is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
says that if the Alexander polynomial
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
of K is 1, then the topologically locally flat slice genus of K is 0, but it can be proved in many ways (originally with gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
) that for every g there exist knots K such that the Alexander polynomial of K is 1 while the genus and the smooth slice genus of K both equal g.
The (smooth) slice genus of a knot K is bounded below by a quantity involving the Thurston–Bennequin invariant of K:
The (smooth) slice genus is zero if and only if the knot is concordant to the unknot
Unknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
.