mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the slice genus of a smooth

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

''K'' in ''S''³ (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'' properly embedded in the 4-ball ''D''⁴ bounded by ''S''³. 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 g_s(''K'') or g₄(''K''), whereas if ''S'' is required only to be topologically locally flatly 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 Hartley Freedman (born April 21, 1951) is an American 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 4-dimensional gene ...

says that if the

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. In 1969, John Conway showed a ve ...

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 In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

) 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'': :

g_s(K) \ge ((K)+1)/2. \,

The (smooth) slice genus is zero if and only if the knot is

concordant Concordance may refer to: * Agreement (linguistics), a form of cross-reference between different parts of a sentence or phrase * Bible concordance, an alphabetical listing of terms in the Bible * Concordant coastline, in geology, where beds, or ...

to the

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

See also

Further reading