On the neighborhood of knots

Abstract

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the knot type via its knot-type knotoids. We prove that the pure knotoids in the knotoid spectra of a knot, which are individually agnostic of the knot type, can distinguish knots of Gordian distance greater than one. We also prove that the neighborhood of some embeddings of the unknot can be distinguished from any embedding of any non-trivial knot that satisfies the cosmetic crossing conjecture. Topological invariants of knots can be extended to their open curve neighborhood to define continuous functions in the neighborhood of knots. We discuss their properties and prove that invariants in the neighborhood of knots may be able to distinguish more knots than their application to the knots themselves. For example, we prove that an invariant of knots that fails to distinguish mutant knots (and mutant knotoids), can distinguish them by their neighborhoods, unless it also fails to distinguish non-mutant pure knotoids in their spectra. Studying the neighborhood of knots opens the possibility of answering questions, such as if an invariant can detect the unknot, via examining possibly easier questions, such as whether it can distinguish height one knotoids from the trivial knotoid.

Figures (5)

• Figure 1: (Left) Examples of (polygonal) knotoids (open simple arc diagrams). Notice that knotoids refer to projections of open chains, while knots refer to closed chains in 3-space. (Right) Forbidden moves on knotoids. Knotoids are classified via Reidemeister moves and the forbidden moves.
• Figure 2: Generalised Reidemeister moves on virtual knots/links : the classical Reidemeister moves, $R_1$, $R_2$, $R_3$; the virtual moves, $V_1$, $V_2$, $V_3$; and the semi-virtual move $SV$. (Figure from Barkataki2024.)
• Figure 3: Examples of open curves in the neighborhood of an embedding of the trefoil, $K$, based at different points, $x_1, x_2$ along the knot. Each open curve (shown in black), we denote $l_{x_1}, l_{x_2}$, gives rise to a knotoid spectrum (examples of knotoid diagrams shown in blue). The knotoids shown belong to the based neighborhoods of $K$ at $x_1$ and $x_2$, $N_{h,x_1}$ and $N_{h,x_2}$, respectively. The union of the knotoids from both belong to the neighborhood of $K$, $N_h(K)$.
• Figure 4: (Left) An alternating quadtisecant that connects the points $a_{i-1}a_ia_{i+1}a_{i+2}$ on a piecewise linear knot embedding in 3-space (only 4 edges of the knot shown, with orientation). (Right) A part of a projection of an open curve in the based neighborhood of the corresponding knot based at $a_{i-1}$, in the direction of the quadrisecant of the knot. For simplicity, we denote by $a_{i-1}$ both endpoints of the edge incident to the point $a_{i-1}$. The virtual arc closure of the knotoid shown. The neighborhood of the knot based at $a_{i-1}$ gives rise to knotoids of diagrammatic height 3.
• Figure 5: Parts of projections of two mutant open curves in the neighborhood of two mutant knot embeddings. Outside the circle, both projections are identical. Inside the circle, the two diagrams may not be related by rotation.

Theorems & Definitions (59)

• Definition 2.1
• Definition 2.2: knotoids, knot-type knotoids, pure knotoids
• Definition 2.3
• Definition 2.4: height, virtual crossing number
• Remark 2.1
• Remark 2.2
• Definition 2.5: Essential and strongly essential secant, n-secant
• Definition 2.6
• Conjecture 2.1: Cosmetic crossing conjecture
• Lemma 2.1