[Paper Review] A Sliceness Criterion for Odd Free Knots

D. A. Fedoseev, Vassily Olegovich Manturov|arXiv (Cornell University)|Jul 16, 2017

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper establishes a complete sliceness criterion for odd free knots using chord pairing in their chord diagrams. It proves that an odd free knot is slice if and only if its chords can be paired without intersections, providing a finite, algorithmic test for sliceness based on parity and combinatorial smoothing of double lines in spanning complexes.

ABSTRACT

The main goal of this paper is to prove that for odd free knots - that is free knots with all odd crossings - the problem of sliceness (the existence of a spanning disc) has an explicit answer based on the pairing of the knot diagram chords.

Motivation & Objective

• To determine a complete, algorithmic criterion for sliceness in odd free knots.
• To extend the use of parity invariants to detect spanning discs in free knot theory.
• To characterize when a free knot diagram with all odd crossings bounds a spanning disc in 4-dimensional space.
• To provide a finite procedure to decide sliceness using chord pairing and topological smoothing.

Proposed method

• Define a Gaussian parity for double lines in 2-complexes representing free knots, ensuring consistency with Reidemeister moves.
• Apply a combinatorial smoothing procedure to all interior double lines, preserving the number of connected components and reducing triple points and double lines.
• Use the resulting complex with only boundary-attached double lines to analyze chord pairings in the corresponding chord diagram.
• Prove that the absence of cusps and triple points in the smoothed complex implies a non-intersecting chord pairing.
• Establish that such a non-intersecting pairing corresponds to a spanning disc via gluing along paired chords.
• Leverage the fact that odd crossings force all double lines to be odd, which prevents cusps and triple points in the final complex.

Experimental results

Research questions

• RQ1 Can the sliceness of an odd free knot be determined by a purely combinatorial condition on its chord diagram?
• RQ2 What role does Gaussian parity play in ensuring the existence of a spanning disc for odd free knots?
• RQ3 Under what conditions can a chord diagram with only odd chords be paired without intersections to yield a spanning disc?
• RQ4 How does the smoothing of double lines in a 2-complex relate to the sliceness of the underlying free knot?

Key findings

• An odd free knot is slice if and only if its chords can be paired such that the resulting chords in the pairing diagram are pairwise unlinked (no intersections).
• The smoothing of all interior double lines in a spanning complex leads to a disc-like complex with no cusps or triple points, provided the original knot is odd.
• The absence of cusps and triple points in the final complex is guaranteed by the oddness of all crossings and the properties of Gaussian parity.
• The existence of a non-intersecting chord pairing is both necessary and sufficient for sliceness in odd free knots, making the criterion complete and decidable.
• The method provides a finite algorithm to test sliceness, as the number of possible pairings is finite and each can be checked for intersection.