I am trying to find how do enginners build roads especially highway interchanges. Are they relying on any topology theories such as braids and knots theory? Is there any connection or link between topology and designing interchanges ? I can't find any article explaining it. Thanks a lot.
It's much closer to graph theory. First, traffic projections: basing on known and projected urban data - habitation, production, recreation, trade zones, transit, etc, the general transition patterns are predicted: how many cars per hour, from which origins to which destinations. Knowing how many people want to go from where to where allows projecting road profiles - numbers of lanes required to allow given throughput at given speed on given graph edge. The road layout is at this point completely absent - there are just nodes of the graph - sources/sinks for the cars - and edges - roads; given required throughput.
Knowing the general numbers you can then match your available budget against common solutions - the generalized 'generic' interchanges that fit given local conditions and are 'doable' within the budget. Usually the more efficient the solution the more expensive it is; sprawling, wide roads, long multi-level viaducts, gentle curves, the entire interchange covering a large (expensive) land area. If you tighten things, you can save up money but reduce throughput. At this point the general model is decided. Engineers don't design these. This is an academic domain. Engineers just know these, apply them, and...
...customize. Obviously one size doesn't fit all. Local conditions have their demands. So the generic model of the interchange needs to be modified - evolved to match the local conditions. Inclines, angles, matching road profiles to predicted throughputs - fitting the graph edges into actual roads and adjusting width and profile (curve radii, inclines) to the individual predicted traffic.
Occasionally a design bureau with a particularly ambitious and talented (or at least reputed to be talented) engineer will create an entirely new pattern from scratch, for specific situation. If it passes the scrutiny of interested parties (one way or another) it may be implemented and either enters the canon, as a pattern for future derivatives, or becomes an eternal source of shame and mockery for that engineer.
Somekind of theory like that, I reckon. I've often wondered this myself! When I was a kid I became fascinated by this & had a phase of 'designing' such junctions - just drawing them freehand on sheets of paper - I must've done dozens of them!
One thing I could mention concretely (haha! see what I did there!?) in this connection, that your question has 'popped' out of my memory, is that the optimum curve for a road-bend is a Cornu spiral, which is simply the curve on which curvature is proportional to arc-length; but the representation of which in cartesian co-ordinates broaches some fascinating mathematics - Fresnel integrals which can in turn be expressed as Bessel functions.
I think it's distinctly possible that knot theory (and that rather than braid theory) has been used. Not (ha ... no ... leave it!) ... not so much the profound depths of knot theory, but at least maybe upto the point of checking some gallery of knots to find whether some configuration is equivalent to some other more-economical-to-build one ... which can be an amazingly subtle question: have you read the story of the so-called 'culprit' - a knot that was only perceived to be the same as some other by a sort of fluke - a guy who was actually a lawyer just decided to browse a bit of knot-theory, and just somehow 'saw' that they were the same after no-one else had before!