Knot invariant

A knot invariant isuseful toolknot theory. It isquantity (inbroad sense - someindeed numbers, somepolynomialssome aresimple yes/no) definedeach knot. Their usefulnessin distinguishing knots from one another oroutlining other propertiesknots.

Some knot invariantsworked out fromdiagram,which casemust be unchanged (that issay, invariant) underReidemeister moves. Knot polynomialsexamplesthis. These areinvariants most usefuldistinguishing knots from one another, though attimewriting itnot known whether anythese distinguishes all knots from each other or even justunknot from all other knots.

Other invariantsdefined by choosingparticular diagram,example, many takeminimum value over all possible diagrams ofknot. This category includes crossing number, which isminimum numbercrossingsany diagram ofknot.

Finally, some invariantsmore or less unrelateddiagrams ofknotneedbe worked outother ways. For example,genus ofknot.

• More needed. Suggestions:
  • n-colourabilty
  • the twist (I think that'swrong term - I meanone where you add togethercrossings including signs - writhe, maybe)
  • for genus, we obviously need Seifert sufaces, which could really dosome pictures.
  • braid index - probably needs an article on braids
  • unknotting number
  • linking number? Reallylink invariant
  • relationships between invariants
  • Alexander Polynomial
  • Jones Polynomialgeneralisations
  • homologyknots?