In themathematical field ofknot theory, achiral knot is aknot that isnotequivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is anamphicheiral knot, also called anachiral knot. Thechirality of a knot is aknot invariant. A knot's chirality can be further classified depending on whether or not it isinvertible.

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.^[1]

The possible chirality of certain knots was suspected since 1847 whenJohann Listing asserted that thetrefoil was chiral,^[2] and this was proven byMax Dehn in 1914.P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had evencrossing number.Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphicheiral knots in the late 1910s.^[3][4] But a counterexample to Tait's conjecture, a 15-crossing amphicheiral knot, was found byJim Hoste,Morwen Thistlethwaite, andJeff Weeks in 1998.^[5] However, Tait's conjecture was proven true forprime,alternating knots.^[6]

• Both possibletrefoil knots.
• 
  The left-handed trefoil knot.
• 
  The right-handed trefoil knot.

The simplest chiral knot is thetrefoil knot, which was shown to be chiral byMax Dehn. All nontrivialtorus knots are chiral. TheAlexander polynomial cannot distinguish a knot from its mirror image, but theJones polynomial can in some cases; ifV_k(q) ≠ V_k(q⁻¹), then the knot is chiral, however the converse is not true. TheHOMFLY polynomial is even better at detecting chirality, but there is no known polynomialknot invariant that can fully detect chirality.^[7]

A chiral knot that can be smoothly deformed to itself with the opposite orientation is classified as ainvertible knot.^[8] Examples include the trefoil knot.

If a knot is not equivalent to itsinverse or its mirror image, it is a fully chiral knot, for example the9 32 knot.^[8]

An amphicheiral knot is one which has anorientation-reversing self-homeomorphism of the3-sphere, α, fixing the knot set-wise. All amphicheiralalternating knots have evencrossing number. The first amphicheiral knot with odd crossing number is a 15-crossing knot discovered byHoste et al.^[6]

If a knot isisotopic to both its reverse and its mirror image, it is fully amphicheiral. The simplest knot with this property is thefigure-eight knot.

If the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphicheiral. This is equivalent to the knot being isotopic to its mirror. No knots with crossing number smaller than twelve are positive amphicheiral and noninvertible .^[8]

If the self-homeomorphism, α, reverses the orientation of the knot, it is said to be negative amphicheiral. This is equivalent to the knot being isotopic to the reverse of its mirror image. The noninvertible knot with this property that has the fewest crossings is the knot8₁₇.^[8]

• Chiral knots and links
• Knot chirality

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