Cabling conjecture

Picture a closed knotted loop with a thick tube around it. Remove the tube and glue in a replacement solid torus, a solid doughnut shape, choosing which boundary curve now bounds a disk inside the replacement. This is Dehn surgery. For a nontrivial knot in the 3-sphere, can such a replacement produce a space split by a sphere into nontrivial pieces only when the knot winds around a companion knot as a cable? The conjecture also specifies the gluing direction: it must be the cable’s annulus slope.[1] [2]

Remove a thick tube around a nontrivial knot and refill it by a solid torus, a solid tube bent into a loop. The gluing direction determines which boundary curve becomes the boundary of a disk in the replacement. The question is whether a resulting space that splits into nontrivial pieces must reveal a cable structure in the original knot, together with its specific cabling-annulus direction. This is a claim about knots in the 3-sphere, not about arbitrary knots in arbitrary ambient spaces.

For a nontrivial $K\subset S^3$ and a nonmeridional surgery slope r, reducibility means $S^3_K(r)$ contains an essential sphere: an embedded 2-sphere not bounding an embedded 3-ball. A cable winds on a companion’s tubular boundary; its cabling annulus is an annular surface in the knot exterior whose boundary determines a slope. The claim is that K is a cable, including the torus-knot case, and r is this boundary slope. Stating the conclusion through the annulus avoids competing p/q conventions.[1] [2]

Many knot families are already covered, including all thin knots. What remains is ruling out an unexpected reducible surgery for general hyperbolic knots. The splitting sphere must be essential; an ordinary tiny sphere exists in every 3-manifold and proves nothing.[1] [2]

DeYeso’s revised Introduction credits the complete thin-class result of Bodish–DeYeso, rather than leaving the older thin hyperbolic L-space exception open. General reducing-slope integrality and hyperbolic bounds such as 1<|r|≤2 g(K)−1 restrict candidates; they are not a universal obstruction to reducible surgery.[1] [2]

Dehn surgery builds closed three-dimensional spaces from knot exteriors. Whether the result splits into simpler pieces is a global feature of that space, although the construction changes only a tube around one knot. Understanding this link would connect a way of building spaces with the geometric structure responsible for their decomposition.[1] [2]

A connected-sum decomposition separates a closed three-dimensional space into constituent pieces, while surgery reconstructs a space from its boundary gluing data. Relating these descriptions explains how a local construction can create a global decomposition. Floer homology offers a concrete connection: its data for a connected sum satisfy a product rule, so the algebra can obstruct proposed decompositions. Recognizing the geometric cause of every reducible filling would connect those algebraic restrictions to the original knot’s structure.[1] [2]