A395644 Number of fibered prime knots with n crossings.

1, 1, 1, 2, 3, 12, 23, 74, 256, 873, 4151

Offset

3,4

Comments

A knot is fibered if its complement in the 3-sphere fibers over the circle. Equivalently, a knot is fibered iff its Alexander polynomial is monic and its degree equals 2*g, where g is the three-genus.

Counts include all prime knots (alternating and non-alternating, rational and non-rational). The Perko duplicate (10_161 = 10_162) is excluded; counts are over distinct unoriented prime knots up to mirror image.

Fiberedness is a mirror-invariant property, so this count does not depend on chirality conventions.

Subsumes A051449 (rational fibered knots): every rational knot is prime, but not conversely. The sequences agree at c = 3, 4, 5, 6, 7 (all prime knots through c = 7 are rational) and diverge at c = 8 (A051449(8) = 4 fibered rational vs. 12 fibered prime).

Links

Table of n, a(n) for n=3..13.

KnotInfo, Table of Knot Invariants.

S. Oehms, database_knotinfo: KnotInfo and LinkInfo databases as Python package, GitHub.

Alex Towell, Joint distributions of knot invariants for prime knots through 11 crossings, Zenodo, 2026.

Example

a(7) = 3: the 3 fibered prime knots with 7 crossings are 7_1, 7_6, 7_7. These are also the 3 fibered rational knots with 7 crossings (cf. A051449), since all prime knots with c <= 7 are rational.
a(8) = 12: of the 21 prime knots with 8 crossings, 12 are fibered. Of these, 4 are rational (cf. A051449(8) = 4) and 8 are not.

Crossrefs

Cf. A002863 (number of prime knots with n crossings).

Cf. A051449 (number of fibered rational knots with n crossings).

Cf. A002864 (alternating prime knots), A051763 (nonalternating prime knots).

Sequence in context: A323126 A323127 A323111 * A302545 A320173 A107928

Adjacent sequences: A395641 A395642 A395643 * A395645 A395646 A395647

Keyword

nonn,hard,more

Author

Alex Towell, May 01 2026

Status

approved