The Conway knot is not slice

Abstract: A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

The Conway Knot and Its Sliceness: An Examination

The paper by Lisa Piccirillo addresses a significant question in low-dimensional topology regarding the sliceness of the Conway knot. The problem of determining whether a specific knot is slice has been an enduring area of inquiry within knot theory, with implications for understanding 4-manifolds and concordance. This research contributes to the classification of slice knots under 13 crossings and provides insight into the broader relationship between 3-dimensional knot theory and 4-dimensional topology.

Overview of Key Concepts

The notion of a slice knot is fundamental to this paper. A knot is considered slice if it bounds a smooth, properly embedded disk in a 4-dimensional ball, denoted ( B^4 ). This contrasts with a topologically slice knot, which bounds a locally flat disk in ( B^4 ). The distinction between these concepts is crucial, as not all topologically slice knots are smoothly slice, highlighting the nuanced landscape of knot concordance in four dimensions.

This paper proves the non-sliceness of the Conway knot (listed as ( 11n34 ) in the Rolfsen tables), which has eluded detection by many known invariants preserved under mutation and had previously defied classification despite extensive study. The Conway knot is notable not only for its intricate structure but also because it is a positive mutant of a slice knot, the Kinoshita-Terasaka knot. This relationship complicates the use of traditional invariants to assess its sliceness.

Methodology and Results

To establish that the Conway knot is not slice, the author utilizes a strategic construction of another knot ( K' ) with properties that relate directly to the sliceness of the Conway knot. The paper introduces the concept of a knot trace, ( X(K) ), a 4-manifold formed by attaching a 0-framed 2-handle to ( B^4 ) using ( K ) as the attaching sphere. The critical lemma posits that a knot is slice if and only if its trace ( X(K) ) smoothly embeds in ( S^4 ).

By constructing a knot ( K' ) that shares a trace with the Conway knot, the paper avoids direct calculations for the Conway knot while facilitating the use of powerful obstructions. Specifically, the paper calculates the Rasmussen's ( s )-invariant for ( K' ) and demonstrates that ( s(K') \neq 0 ). Given that ( s(K') = 0 ) must hold for slice knots, it follows that ( K' ), and consequently the Conway knot, is not slice.

Implications and Future Directions

This finding completes the classification of slice knots with fewer than 13 crossings, resolving a longstanding question and enhancing the understanding of knot concordance. This example also provides a foundation for further exploration of sophisticated invariants that are not mutation invariant—highlighting the potential for novel obstruction techniques that bypass limitations inherent in classical knot theory tools.

Ultimately, this research stimulates further inquiry into the nature of 4-dimensional spaces and the subtle interplays of knot properties therein. It prompts mathematicians to consider how newer theoretical approaches can unearth characteristics previously undetectable through conventional metrics, potentially influencing future explorations in both theoretical and algebraic topology.