Higher-dimensional invariants in 6D super Yang-Mills theory

We exploit the 6D,$$ \mathcal{N} $$ N = (1, 0) and $$ \mathcal{N} $$ N = (1, 1) harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension d = 12 in 6D,$$ \mathcal{N} $$ N = (1, 1) supersymmetric Yang-Mills (SYM) theory. Both single- and double-trace invariants are derived. Only four single-trace and two double-trace invariants prove to be independent. The invariants constructed can provide the possible counterterms of $$ \mathcal{N} $$ N = (1, 1) SYM theory at four-loop order, where the first double-trace divergences are expected to appear. We explicitly exhibit the gauge sector of all invariants in terms of $$ \mathcal{N} $$ N = (1, 0) gauge superfields and find the absence of $$ \mathcal{N} $$ N = (1, 1) supercompletion of the F6 term in the abelian limit.

PLANAR AND SPHERICAL STICK INDICES OF KNOTS

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants, we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.

FOUR-DIMENSIONAL INVARIANTS OF LINKS AND THE ADJUNCTION FORMULA

We compute the slice euler characteristic of certain links by using the so-called generalized adjunction formula, which is proved by Kronheimer, Mrowka, Morgan, Szabó and Taubes. Furthermore, for links obtained from such links by band surgery, we estimate the unknotting numbers, the 4-dimensional clasp numbers, and the slice euler characteristic.