Peaked or visor caps usually have a loose-fitting strap that sits on top of the bill. This strap is attached at each side with a button. For officers, this strap can be a gold or silver "rope" and usually doubled.
The visor cap evolved from earlier forms of military hats. All hats had a chin strap to help hold the hat firmly on the head. Officer's hats evolved to have a brass strap or over-lapping plates. Also, when not in use, the straps could be tied or buckled so that it was worn above the bill. Some even attached to the very top of the hat.
A close comparison is the German pickelhaulbe (or spiked) helmet, which was a leather helmet first worn in the early 1800's. These were usually worn with the chin strap resting on the bill of the helmet. Enlisted men's helmets had leather chin straps and officer or elite troops(cavalry) had brass chin straps.
So, the golden threads are actually chin straps. And they still function, some what, as an adjustable strap. The knot at each end will actually slide such that the strap will gown wide enough to place under the chin. But of course, there is no way to secute it snugly and it is not practical nor does it look cool, either.
Another hat that also hat gold or colored cords were the Slouch hats worn by US Army. These resemble the common "cowboy hat". They were common wear during American Civil War and out West. A tan version was issued during the Spanish American War and later evolved into the Campaign Hat that Army drill instructors wore. These had a double cord that had two ends that hung loose. These were colored to indicate branch of service. I think this evolved to the colored piping worn on the Overseas caps used in WW2.
British hat from Waterloo with white braided cord
British hat from Waterloo era with chin strap tied up to peak of hat.
French Napoleonic hats
British shako (post Napoleonic) with 1-piece chin strap attached across front of hat.
US Slouch hats and Campaign hats and other hats from various times.
WW2 USAAF 'crusher' caps
Soviet Visor caps
The modern visor cap has the chin strap usually worn on top of the bill. Officers are rope cords and enlistedmen's are leather straps. Both are made of two strands, one attached at each button on the side of the hat. The cord wraps around the front of the hat and has a loop or knot at the other end that slips over the mating cord. This arrangement allows the chin strap to be loosened or tightened. This double cord design allows for this adjustment of the strap.
The slouch hats also have double cords probably for the same reason.