this is my mental picture for how they travel without a medium, how (like water waves) some can't stay still, why they have wave and particle properties, energy/mass equivalence, conservation, etc. it might capture uncertainty too -- i've heard that all waves have an uncertainty relation (say in their power spectrum), but i don't get why -- it seems like we can discuss waves with absolute precision.
Actually that's not too far off the mark, although I'm not sure "knot" or "kink" is the best word. Quantum field theory, the best theory we currently have to describe particles, says that particles correspond to excitations of a field, which are kind of like waves in water; you could consider the surface of a pond "excited" whenever it's not flat. Just as with water waves, there's an infinite variety of "shapes" you can have for these excitations. For example, you could have a repeating wave, in which the surface of the water cycles up and down over a large area, or you could have just one wave front that just propagates across the water without spreading out very much. The former case is pretty typical for things like light waves, and the latter case is pretty typical for particles of matter, although technically any kind of field (whether it's the electromagnetic field for light, or a quark field in matter, or whatever) can have any of the different types of excitations.
By the way, according to special (and general) relativity, even an object that is standing still is moving through time. So all of these excitations move through spacetime in one way or another. But only certain ones (the excitations in fields corresponding to massless particles) can move through space in such a way that they appear to us to be traveling at the speed of light.
particles are ordinary quanta of the corresponding quantum fields - without any knots or other topologically nontrivial features. (You have to get used to the wave-particle duality, probabilities, and the uncertainty principle - they're fundamental features of the world around us.)
However, this is only true for "weakly coupled particles" that are directly described by a Lagrangian composed of "free fields" plus "weak interactions". On the other hand, there can be many other objects that are not just quanta of waves but they have "knots" on them.
We say that they are topological defects. Most typically, one can take the original fields and write down a classical solution of the equations of motion that is topologically nontrivial. The resulting configuration of fields has to exist in the quantum theory, too. It is preserved by the nontrivial topology.
In 1 spatial dimension, a topologically nontrivial configuration is one that drives a scalar field from 1 local minimum of the potential to another, as you move from the left to the right; it is known as the "kink".
In 2 spatial dimensions, there are solutions known as "vortices" (or one "vortex") in which the circular round trip around the center of the solution sees a field to go around a noncontractible curve in the configuration space.
In 3 spatial dimensions, we similarly find magnetic "monopoles" which are also some kinds of "knots" on the fields. More generally, the topological defects that correspond to real objects are known as "solitons". This is contrasted with the "instantons" which are objects localized in the Euclidean spacetime; they don't describe static objects but rather special histories that contribute to various probability amplitudes for processes that appear at one "instant" of time.
Another class of topologically nontrivial configurations are skyrmions, and if you want to see a representative that is really close to your "knots", see a paper about knitted fivebranes in M-theory:
In the quantum theory, objects such as solitons - especially monopoles - may behave indistinguishably from the original quanta. There is sometimes a full symmetry between them - the most famous example is the S-duality of gauge theories in 4 dimensions. Electrically charged particles such as gauginos are exchanged with the magnetic monopoles - solitons - as the coupling $g$ is changed to $1/g$.
All these things, when done properly, still satisfy all the postulates of quantum mechanics such as the nonzero commutators between observables (uncertainty principle). In string theory, topological defects include objects such as D-branes, NS5-branes, and in extra-dimensional theories in general, one also has special "knitted" configurations of the metric such as the Kaluza-Klein monopoles (magnetic monopoles extended in extra dimensions).
Best wishes Lubos
The short answer is no. Particles obey a very important property called "cluster decomposition principle", while those type of lumps that you mentioned do not.
What that means is that if you have a particle as an excitation of a theory, you might perfectly well have a two particle state as a solution when their separation goes to infinity. This is a essential ingredient to calculate S matrices and cross sections (that's almost everything that is useful in particle collision experiments). This is not true, in general, for these extended solutions.
Further details can be found in that excellent book Aspects of Symmetry, by Coleman. Check section 6.2.3: Lumps are like particles (almost).