ScholarLens Back to search
2010•Journal of Knot Theory and Its RamificationsRequires access

ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS

Marko Stošić

Open publisher page 0 citations

Abstract

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

About this research paper

What this paper is about

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

Why it matters

A significance statement is not available in the OpenAlex record.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

Key concepts: Khovanov homology, Mathematics, Braid, Torus, Knot invariant, Knot theory, Conjecture, Tricolorability

Related papers

Related paper 1Related paper 2Related paper 3
Back to paper searchBrowse research topicsOriginal source
ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS — Research Paper | ScholarLens