Concordance Crosscap Number of a Knot
نویسنده
چکیده
We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap number is smaller than or equal to the concordance crosscap number. We construct two infinite sequences of knots to explain the gap between the two. In particular, the knot 74 is one of the examples.
منابع مشابه
Bounds on the Crosscap Number of Torus Knots
For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K): c(K) ≤ ⌊(g(K) + 9)/6⌋ and c(K) ≤ ⌊(n(K)+16)/12⌋. The (6n− 2, 3) torus knots show that these bounds are sharp.
متن کاملKnot Floer Homology and the Four-ball Genus
We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots...
متن کاملCrosscap Numbers of Two-component Links
We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, nonorientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.
متن کامل2nd Order Algebraic Concordance and Twisted Blanchfield Forms
In my first year report [16] I described how to write down explicitly the chain complex C∗(X̃) for the universal cover of the knot exterior X, given a knot diagram. In this report we describe work to fit this chain complex into an algebraic group which we construct which measures a “2nd order slice-ness obstruction,” in the sense that it obstructs the concordance class of a knot lying in the lev...
متن کاملKnot Mutation: 4–genus of Knots and Algebraic Concordance
Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4–genus 1 and has a mutant of 4–genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4–genus m with a mutant of 4–genus n. A second result of this paper is a crossing change form...
متن کامل