We introduce classes of restricted walks, surfaces and their generalisations. For example, self-osculating walks (SOWs) are supersets of self-avoiding walks (SAWs) where edges are still not allowed cross but may 'kiss' at a vertex. They are analogous to osculating polygons introduced in (Jensen and Guttmann, 1998) except that they are not required to be closed. By adapting the 'automata' method of (Pönitz and Tittmann, 2000), we find upper bounds for the connective constant for SOWs on the square and triangular lattices to be ≤ 2.73911 and ≤ 4.44931, respectively.

In analogy, we also introduce self-osculating surfaces (SOSs), a superset of self-avoiding surfaces (SASs) and can be generated from fixed polyominoids (XDs). We further generalise and define self-avoiding k-manifolds (SAMs) and its supersets self-osculating k-manifolds (SOMs) in the d-dim hypercubic lattice and (d, k)-XDs. By adapting the concatenation procedure (van Rensburg and Whittington, 1989), we prove that their growth constants exist, and an explicit form for their upper and lower bounds.

The upper bounds can be improved by adapting the 'twig' method, originally developed for polyominoes (Eden, 1961, Klarner and Rivest, 1973). For the cubic lattice, we find improved upper bounds for the growth constant of SASs as ≤ 17.11728.