A note on the unknotting number and the region unknotting number of weaving knots

TL;DR

The paper develops warping degree as a diagrammatic tool to bound the unknotting number $u(K)$ and region unknotting number $u_R(K)$ for weaving knots $W(p,q)$. It defines warping degree for link and braid diagrams, derives general bounds, and proves them by decomposing braid words $B_W(p,np+r)$ into blocks $B_W(p,p)$ and $B_W(p,r)$, with $d(B_W(p,p))$ taking the values $(p^2-1)/4$ for odd $p$ and $p(p-1)/2$ for even $p$. The results yield concrete upper bounds such as $u(W(p,q)) \le (p-1)q/2-1$ and $u_R(W(p,q)) \le (p-1)q/2 + 1/2$, and provide sharper bounds for specific families, along with a discussion of minimal diagrams on $S^2$, region crossing changes, and region unlinking numbers. The work also connects warping degree to lower bounds via signatures in some even-$p$ cases and offers constructions where the bounds are sharp, contributing to a finer combinatorial understanding of weaving knots' unknotting properties.

Abstract

A weaving knot is an alternating knot whose minimal diagram is a closed braid of a lattice-like pattern. In this paper, the warping degree of a braid diagram is defined, and upper bounds of the unknotting number and the region unknotting number for some families of weaving knots are given by diagrammatical and combinatorial examination of the warping degree of weaving knot diagrams.

Figures (14)

• Figure 1: A flype on an alternating diagram. Assume that the two crossings have visually same over/under information.
• Figure 2: Flyping can be applied at the circle.
• Figure 3: Trivial flyping at the circle.
• Figure 4: The braid diagram $B_W(7,7)$ has warping degree 12 with the sequence of base points $\mathbf{b}'=(b_1, b_3, b_5, b_7, b_2, b_4, b_6)$.
• Figure 5: $\rho (1,2,3,4,5)=(3,5,1,2,4)$.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1
• proof
• Theorem 2.2: WM
• Proposition 2.3
• proof