## Knots and $$k$$-width.(English)Zbl 1189.57005

For a generic closed curve $$\gamma$$ in $$\mathbb{R}^3$$, the bridge number $$b(\gamma)$$ is defined to be the number of maxima with respect to the $$z$$-axis, and the bridge number of a knot in $$\mathbb{R}^3$$ is the minimum of the bridge numbers of closed curves that represent the knot.
Let $$\mathcal{S}_1$$ be the set of all flat planes in $$\mathbb{R}^3$$ that are parallel to the $$xy$$-plane. Then $$\mathcal{S}_1$$ is identified with a line. Given a generic closed curve $$\gamma$$, let $$\mathcal{T}_1(\gamma)\subset\mathcal{S}_1$$ be the set of planes transverse to $$\gamma$$. It forms an open subset in $$\mathcal{S}_1$$. If we define the $$1$$-bridge number $$b_1(\gamma)$$ to be the number of connected components of $$\mathcal{T}_1(\gamma)$$, then $$b_1(\gamma)=2b(\gamma)+1$$.
Similarly, let $$\mathcal{S}_2$$ be the set of all flat planes in $$\mathbb{R}^3$$ that are parallel to the $$z$$-axis, $$\mathcal{S}_3$$ the set of all flat planes in $$\mathbb{R}^3$$, and $$\mathcal{S}_4$$ the set of all flat planes and round two-spheres in $$\mathbb{R}^3$$. For a generic (in an appropriate sense) closed curve $$\gamma$$, let $$\mathcal{T}_k(\gamma)$$ be the open set in $$\mathcal{S}_k$$ that consists of planes or spheres transverse to $$\gamma$$.
The authors define the $$k$$-bridge number $$b_k(\gamma)$$ to be the number of connected components of $$\mathcal{T}_k(\gamma)$$. For a knot, the $$k$$-bridge number is defined similarly.
The authors also generalize the width of a knot using $$\mathcal{S}_k$$. For a generic closed curve $$\gamma$$, define $$w_k(\gamma):=\sum_{P_i\in\mathcal{T}_k(\gamma)}\sharp(P_i\cap\gamma)$$. Note that $$w_1(\gamma)$$ is the width introduced by D. Gabai [J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)].
In the paper under review the authors study $$k$$-bridge number and $$k$$-width.
They show that for a given integer $$n$$, the number of knots with $$2$$-bridge number $$n$$ or less is finite. They also show that the $$2$$-width has the same property. In particular, it is shown that if a knot has $$2$$-bridge number less than $$7$$ or $$2$$-width less than $$11$$, then it is either the unknot or the trefoil.
Lower bounds for $$b_2$$ and $$w_2$$ are given in terms of the total curvature of the plane curve projected to the $$xy$$-plane, contrary to the bridge number; one can construct a closed curve with a fixed bridge number with arbitrarily large total curvature.
Some observations for $$b_3$$, $$w_3$$, $$b_4$$ and $$w_4$$ are also given.

### MSC:

 57M25 Knots and links in the $$3$$-sphere (MSC2010)

Zbl 0639.57008
Full Text:

### References:

  Adams C., Othmer J., Stier A., Lefever C., Pahk S., Tripp J.: An introduction to the supercrossing index of knots and the crossing map. J. Knot Theory Ramif. 11(3), 445–459 (2002) · Zbl 1003.57008  Artin E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4, 47–72 (1925) · JFM 51.0450.01  Fabricius-Bjerre Fr.: On the double tangents of plane closed curves. Math. Scand. 11, 113–116 (1962) · Zbl 0173.50501  Fáry I.: Sur la courbure totale d’une courbe gauche faisant un noeud. Bull. Soc. Math. Fr. 77, 128–138 (1949) (French) · Zbl 0037.23604  Freedman M.H., He Z., Wang Z.: Mobius energy of knots and unknots. Ann. Math. 139(1), 1–50 (1994) · Zbl 0817.57011  Gabai D.: Foliations and the topology of 3-manifolds III. J. Diff. Geometry 26, 479–536 (1987) · Zbl 0639.57008  Kuiper N.H.: A new knot invariant. Math. Ann. 278(1–4), 193–209 (1987) · Zbl 0632.57006  Langevin R., O’Hara J.: Conformal geometric viewpoints for knots and links I. Contemp. Math. 304, 187–194 (2002) · Zbl 1014.57008  Milnor J.W.: On the total curvature of knots. Ann. Math. 52(2), 248–257 (1950) · Zbl 0037.38904  Murasugi K.: On invariants of graphs with applications to knot theory. Trans. Am. Math. Soc. 314, 1–49 (1989) · Zbl 0726.05051  Schubert H.: Uber eine numerische Knoteninvariante. Math. Z. 61, 245–288 (1954) (German) · Zbl 0058.17403  Stasiak A., Katritchx A., Bednar J., Michoud D., Dubochet J.: Electrophoretic mobility of DNA knots. Nature 384, 122 (1996)  Sumners D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Not. AMS 42, 528–537 (1995) · Zbl 1003.92515  Weber C., Stasiak A., Los D., Dietler D.G.: Numerical simulation of gel electrophoresis of dna knots in weak and strong electric fields. Biophys. J. 90(9), 3100–3105 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.