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Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [ABSTRACT FROM AUTHOR]

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained. [ABSTRACT FROM AUTHOR]

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π -orbifold group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank 2 are 2 -bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

Ichihara and Matsudo introduced the notions of -colorable links and the minimal coloring number for -colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable -colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable -colorable link is exactly 4. [ABSTRACT FROM AUTHOR]

We construct an algebra of nontrivial homological operations on Khovanov homology with coefficients in generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the even and odd Khovanov homology combined. The case of reduced Khovanov homology is also considered. [ABSTRACT FROM AUTHOR]

We prove that every Legendrian knot in the tight contact structure of the 3 -sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3 -sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given. [ABSTRACT FROM AUTHOR]

We study knots of order in the grope filtration and the solvable filtration of the knot concordance group. We show that, for any integer , there are knots generating a subgroup of . Considering the solvable filtration, our knots generate a subgroup of distinct from the subgroup generated by the previously known -torsion knots of Cochran, Harvey, and Leidy. We also present a result on the -torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration. [ABSTRACT FROM AUTHOR]

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots. [ABSTRACT FROM AUTHOR]