Narrow Results

Applying the concept of braiding sequences and the inequality between the signature and number of roots of the Alexander polynomial on the unit circle, we prove that only finitely many special alternating knots are (even algebraically) concordant, in that their concordance class determines their Alexander polynomial. We discuss some extensions of this result to positive and almost positive knots, and links.

Using the new approach of braiding sequences we give a proof of the Lin-Wing conjecture, stating that a Vassiliev invariant ν of degree k has a value O_ν (c(K)^k) on a knot K, where c(K) is the crossing number of K and O_ν depends on ν only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only.

We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.

We give a family of linear inequalities which strictly estimate relation among the coefficients of the Alexander polynomials of alternating knots of genus two. We also give such families for positive knots of genus two, and for homogeneous knots of genus two. As an application, we determine the alternating knots of genus two such that the leading coefficients of them are less than or equal to three.

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.

In this note we show that the rank of the trip matrix of a positive knot diagram is exactly twice the genus of the associated positive knot. From this, we give a quick proof of the following result of Murasugi: The term of lowest degree in the Jones polynomial of a positive knot is 1 · t^g, where g is the genus of the knot.

We prove some estimates on the crossing number of satellites of adequate, and partly of semiadequate knots. We mostly deal with classical cables, obtaining the first examples where the asymptotic cable crossing number is shown to be equal to the companion crossing number. Our approach can also be applied to Whitehead and Bing doubles and positive braid cables. For some patterns one can use it to determine their geometric degree.