Seifert circles, crossing number and the braid index of generalized knots and links

TL;DR

This work extends classical and virtual knot inequalities to generalized knots and links by introducing generalized crossings and Reidemeister-type moves, and by proving that Seifert graphs of all generalized diagrams are planar and bipartite. Using graph-theoretic bounds on edges and the Seifert-circle count, it derives the inequality $tc(L) \ge 2(gb(L) - 1)$ for diagrams with no nugatory crossings under an index compatibility condition, thereby bounding the generalized braid index via total crossings. The results specialize to virtual singular links, giving $tc(L) \ge 2(vsb(L) - 1)$ when $ind(D) = ind_0(D)$. This unifies and broadens Ohyama’s and Takeda’s inequalities across classical, virtual, and generalized knot theories through the lens of Seifert graphs and generalized braid representations.

Abstract

For classical links Ohyama proved an inequality involving the minimal crossing number and the braid index, then motivated from this Takeda showed an analogous inequality for virtual links. In this paper, we are interested in studying properties of links independent of the type of crossings, and for this reason, we introduce generalized crossings for diagrams and generalized Reidemeister-type moves. The aim of this work is to prove the same type of inequality mentioned above but now involving the total crossing number and the braid index of generalized knots and links. In particular, we show that the result holds for virtual singular links.

Figures (20)

• Figure 1: Negative and positive real crossings, respectively.
• Figure 2: Reidemeister moves.
• Figure 3: Flat, virtual and singular crossings.
• Figure 4: $R_{1}$-type moves.
• Figure 5: $R_{2}$-type moves.

Theorems & Definitions (28)

• Remark 1
• Definition 3
• Theorem 4: book2
• Corollary 5: article3
• Theorem 6: article1
• Example 7
• Example 8
• Remark 9
• Lemma 10
• proof