English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The (ordinary) unknotting-number of 1-dimensional knots, which is defined byusing the crossing-change, is a very basic and important invariant. It is verynatural to consider the `unknotting-number' associated with other local-moveson n-dimensional knots, where n is a natural number. In this paper we prove thefollowing facts.  For the ribbon-move on 2-knots, which is a kind of local-move on knots, wehave the following: There is a ribbon-move-unknotting-number-two 2-knot. Theribbon-move-unknotting-number of 2-knots is unbounded.  For the pass-move on 1-knots, which is a kind of local-move on knots, we havethe following: There is a pass-move-unknotting-number-two 1-knot whose(ordinary) unknotting-number is 4. For any natural number n, there is a 1-knotwhose pass-move-unknotting-number is>n and whose (ordinary) unknotting-numberis 4n.  For the high-dimensional-pass-move on high-dimensional knots, which is a kindof local-move on knots, we have the following: There is a (2k+1,2k+2)-pass-move-unknotting-number-two (4k+2)-knot. The (2k+1,2k+2)-pass-move-unknotting-number of (4k+2)-knot is unbounded. There is a(2k+1, 2k+1)-pass-move-unknotting-number-two (4k+1)-knot. The (2k+1,2k+1)-pass-move-unknotting-number of (4k+1)-knot is unbounded. There is a(4k+1)-knot whose twist-move-unknotting-number is n for any natural number n.

Ribbon-move-unknotting-number-two 2-knots, pass-move-unknotting-number-two 1-knots, and high dimensional analogue