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By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers. [ABSTRACT FROM AUTHOR]

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring R using virtual crossings as smoothings, these invariants take the form of multisets of elements of R and can be written in a "polynomial" form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition. [ABSTRACT FROM AUTHOR]

The warping sum e (K) of a knot K is the minimal value of the sum of the warping degrees of a minimal diagram of K with both orientations. In this paper, knots K with e (K) ≤ 3 are characterized, and some knots K with e (K) = 4 are given. [ABSTRACT FROM AUTHOR]

An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given. [ABSTRACT FROM AUTHOR]

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves. [ABSTRACT FROM AUTHOR]

Region crossing change at a region of a knot, link or spatial-graph diagram is a local transformation which changes all the crossings on the boundary of the region. In this paper, we show that we can make any crossing change by a finite number of region crossing changes on any diagram of a connected spatial graph which has no cutting circles. [ABSTRACT FROM AUTHOR]

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number. [ABSTRACT FROM AUTHOR]

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number is greater than or equal to , and less than or equal to . In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with . Moreover, we construct a link L whose unlinking number is greater than . [Copyright &y& Elsevier]

For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover, the equality holds if and only if D is an alternating diagram. For a knot K, we also estimate the minimum of d(D) + d(-D) for all diagrams D of K with c(D) = c(K). [ABSTRACT FROM AUTHOR]