Journal of Knot Theory and Its Ramifications

We fill a gap in the proof that the proposed link homotopy invariant $\omega$ of Li is well defined. It is also shown that if the homotopy invariant $\tau$ of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with $\omega$.

A 2-dimensional braid over an oriented surface-knot $F$ is presented by a graph called a chart on a surface diagram of $F$. We consider 2-dimensional braids obtained by an addition of 1-handles equipped with chart loops. We introduce moves of 1-handles with chart loops, called 1-handle moves, and we investigate how much we can simplify a 2-dimensional braid by using 1-handle moves. Further, we show that an addition of 1-handles with chart loops is an unbraiding operation.

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at $q=-1$, are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of $M(0,4)$, Math. Proc. Cambridge Philos. Soc. 141(3) (2006) 477–488] by extending their original argument for the sphere with four marked points to our more general case.

The Tait conjecture states that reduced alternating diagrams of links in $S^3$ have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper, we prove an analogous result for alternating links in $S^1\times S^2$ giving a complete answer to this problem. In $S^1\times S^2$ we find a dichotomy: the appropriate version of the statement is true for ${\mathbb{Z}}_2$-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for ${\mathbb{Z}}_2$-homologically non-trivial links, for which the Jones polynomial vanishes.

In this paper, we prove three simplicial complexes associate to a handlebody, which are separating disk complex, half disk complex and disk pants graph, are connected.

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

In this paper and its sequel, we give a necessary and sufficient condition for two essential simple loops on a $2$-bridge sphere in an even Heckoid orbifold for a $2$-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a $2$-bridge sphere in an even Heckoid orbifold for a $2$-bridge link to be peripheral or torsion in the orbifold. This paper treats the case, when the $2$-bridge link is a $(2,p)$-torus link, and its sequel will treat the remaining cases.

In this paper and its prequel, we give a necessary and sufficient condition for two essential simple loops on a $2$-bridge sphere in an even Heckoid orbifold for a $2$-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a $2$-bridge sphere in an even Heckoid orbifold for a $2$-bridge link to be peripheral or torsion in the orbifold. The prequel treated the case when the $2$-bridge link is a $(2,p)$-torus link, and this paper treats the remaining cases.

In the present paper, we develop the parity theory invented in [V. O. Manturov, Mathematics 201(5) (2010) 693733]; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, where parities were previously applied.