A knot invariant is a function from the set of all knots to any other set such that the function does not change as the knot is changed (up to isotopy). In other words, a knot invariant always assigns the same value to equivalent knots (although different knots may have the same knot invariant). Standard knot invariants include the fundamental group of the knot complement, numerical knot invariants (such as Vassiliev invariants), polynomial invariants (knot polynomials such as the Alexander polynomial, Jones polynomial, Kauffman polynomial F, and Kauffman polynomial X), and torsion invariants (such as the torsion number).

# Knot Invariant

## See also

Arf Invariant, Knot, Knot Polynomial, Link Invariant, Torsion Number, Vassiliev Invariant## Explore with Wolfram|Alpha

## References

Aneziris, C. N. "The Knot Invariants." Ch. 5 in*The Mystery of Knots: Computer Programming for Knot Tabulation.*Singapore: World Scientific, pp. 35-42, 1999.

## Referenced on Wolfram|Alpha

Knot Invariant## Cite this as:

Weisstein, Eric W. "Knot Invariant." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/KnotInvariant.html