Full twists and stability of knots and quivers

TL;DR

This paper explores the relationship between knot invariants and quiver stability, demonstrating stable growth behaviors under twisting and linking operations, and conjecturing a direct knot-quiver correspondence.

Contribution

It establishes a connection between knot twisting stability and quiver linking stability, proposing a conjecture linking full twists to quiver unlinking/linking, advancing the knot-quiver correspondence.

Findings

Confirmed stable growth of HOMFLY-PT polynomials under full twists.

Showed symmetric quivers exhibit analogous stable growth under linking/unlinking.

Validated the conjecture for various classes of knots, including twist and torus knots.

Abstract

We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region.