Kirby Problem 1.82

Definitions and mathematical statement

An oriented knot or link is positive if it admits a regular diagram in which each crossing has positive oriented sign under the usual right-hand convention. This is an existence condition on diagrams: another diagram of the same link may contain negative crossings. The exterior of a link $L$ is $X_L=S^3\setminus\operatorname{int}\nu L$, together with its meridians and longitudes as peripheral data.

The first question seeks an intrinsic characterization of positive knots. Give necessary and sufficient conditions phrased in the topology or geometry of the knot and its exterior, such as suitable spanning surfaces or group and peripheral structure, that do not merely repeat the existence of a positive diagram.

The second question asks for a simple explicit calculus relating positive diagrams of the same oriented knot or link. Specify moves that preserve the link type and positivity, and prove that any two positive diagrams of that type are connected by finitely many such moves and planar isotopies. A complete calculus must allow positive diagrams with different numbers of crossings; the problem does not assume that every positive diagram is crossing-minimal.

Short English statement

Characterize positive knots without choosing a positive diagram, and find moves connecting all positive diagrams of a given link.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2741 (KP-1.82), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.82. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.