Narrow Results

We give a lower bound for the alternation number of a knot by using the Rasmussen s-invariant and the signature of a knot. Then, we determine the torus knots with alternation number one and show that many torus knots are "far" from the alternating knots. As an application, we determine the almost alternating torus knots, solving a conjecture due to Adams et al.

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

For each positive integer $n$ we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to $n$, braid index equal to $n+3$ and Turaev genus equal to $n$.

In this paper, we construct, by using the Alexander polynomial, infinite families of nonalternating prime knots, which have alternation number equal to one. More specifically these knots after one crossing change yield a 2-bridge knot or the trivial knot. In particular, we display two infinite families of nonalternating knots and their Alexander polynomials.

Moreover, we give formulae to obtain the Conway and Alexander polynomials of oriented 3-tangles and the links formed from their closure with a specific orientation. In particular, we propose a construction to form families of links for which their Alexander polynomials can be obtained by nonrecursive formulae.

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links.

We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.