What is the maximum shortening of a string after tying an overhand knot?

[Loren Booda]

Given a flexible, inelastic and incompressible string of radius R, how much at most would it shorten from tying an overhand knot in it?

 

[sneutron]

The following is not the answer to the question you asked, but try this for starters:

The http://en.wikipedia.org/wiki/Overhand_knot" .

http://en.wikipedia.org/wiki/Ropelength" is expressed as a ratio to thickness and depends heavily (for simple knots) on crossing number (3 for trefoil).

The absolute minimum length/thickness for any type of knot was shown to be 15.66, and for the trefoil specifically the answer is 16.372.

 

[Loren Booda]

These are great lines, sneutron! Where did you get them?

Could projections of a knot's perpendicular surfaces (while the linear string rotates) give any insight?

 

[sneutron]

Wiki, Loren, just follow the links in my post.

 

[blue_raver22]

The efficiency of an overhand knot can be determined by calculating the amount of shortening in the string after tying the knot. This can be done by considering the properties of the string, including its flexibility, inelasticity, and incompressibility.

The maximum shortening of the string can be determined by considering the geometry of the overhand knot. The knot essentially forms a loop in the string, with the two ends passing through the loop. The amount of shortening in the string will depend on the radius of the loop, which is equal to the radius of the string, and the thickness of the string itself.

Using mathematical calculations, it can be determined that the maximum shortening in the string would be equal to the circumference of the loop formed by the overhand knot. This can be expressed as 2πR, where R is the radius of the string. This means that the string would shorten by a maximum of 2πR when an overhand knot is tied in it.

It is important to note that this calculation assumes ideal conditions, where the string is perfectly flexible, inelastic, and incompressible. In reality, there may be slight variations in the properties of the string, which could affect the amount of shortening in the string after tying an overhand knot.

In conclusion, the efficiency of an overhand knot can be determined by calculating the maximum shortening in the string after tying the knot. This is dependent on the properties of the string, such as its flexibility, inelasticity, and incompressibility, and can be approximated by considering the geometry of the knot.