nLab
knot invariant

Idea

A knot invariant is map from isotopy equivalence classes of knots to any kind of structure you could imagine. These are helpful because it is much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that knots are different.

Examples

• crossing number
• bridge number
• unknotting number
• colorability
• knot group= the fundamental group of the complement of the knot.
• knot genus?
• Jones polynomial
• HOMFLY polynomial
• Alexander polynomial
• Reshetikhin-Turaev invariant?s for any object in a ribbon category
• Khovanov homology
• Kauffman bracket? technically, an invariant of framed links but closely related to the Jones polynomial

Many of these extend to link invariants or have variants that depend on the knot being oriented.

category: knot theory