Segment number of knots

TL;DR

A new numerical invariant of knots and links made from the partitioned diagrams is introduced that measures the complexity of Knots and links.

Abstract

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number.

Figures (5)

• Figure 1: A 3-segment presentation $D$ of the trefoil knot and the digraph $G(D)$ corresponding to $D$
• Figure 2: A 2-partitioned diagram that is NOT a 2-segment presentation because the crossing relations are mixed.
• Figure 3: Reducing the segment number by merging $e_i$ and $e_{i+1}$.
• Figure 4: Merging segments when orientation constraints are not violated.
• Figure 5: Examples of knots with $s(K)=3$ and varying crossing numbers.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• proof
• Corollary 3.5