Unknotting number
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings.[1] The unknotting number of a knot is always less than half of its crossing number.[2]
Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots:
-
Trefoil knot
unknotting number 1 -
Figure-eight knot
unknotting number 1 -
Cinquefoil knot
unknotting number 2 -
Three-twist knot
unknotting number 1 -
Stevedore knot
unknotting number 1 -
6₂ knot
unknotting number 1 -
6₃ knot
unknotting number 1 -
7₁ knot
unknotting number 3
In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:
- The unknotting number of a nontrivial twist knot is always equal to one.
- The unknotting number of a -torus knot is equal to .
- The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[3] (The unknotting number of the 1011 prime knot is unknown.)
Other numerical knot invariants
See also
References
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found..
- ↑ Weisstein, Eric W., "Unknotting Number", MathWorld.
External links
<templatestyles src="Asbox/styles.css"></templatestyles>