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:

 [1] 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 · doi:10.1142/S0218216502001731 [2] Artin E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4, 47–72 (1925) · doi:10.1007/BF02950718 [3] Fabricius-Bjerre Fr.: On the double tangents of plane closed curves. Math. Scand. 11, 113–116 (1962) · Zbl 0173.50501 [4] 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 [5] Freedman M.H., He Z., Wang Z.: Mobius energy of knots and unknots. Ann. Math. 139(1), 1–50 (1994) · Zbl 0817.57011 · doi:10.2307/2946626 [6] Gabai D.: Foliations and the topology of 3-manifolds III. J. Diff. Geometry 26, 479–536 (1987) · Zbl 0639.57008 [7] Kuiper N.H.: A new knot invariant. Math. Ann. 278(1–4), 193–209 (1987) · Zbl 0632.57006 · doi:10.1007/BF01458068 [8] Langevin R., O’Hara J.: Conformal geometric viewpoints for knots and links I. Contemp. Math. 304, 187–194 (2002) · Zbl 1014.57008 [9] Milnor J.W.: On the total curvature of knots. Ann. Math. 52(2), 248–257 (1950) · Zbl 0037.38904 · doi:10.2307/1969467 [10] Murasugi K.: On invariants of graphs with applications to knot theory. Trans. Am. Math. Soc. 314, 1–49 (1989) · Zbl 0726.05051 · doi:10.1090/S0002-9947-1989-0930077-6 [11] Schubert H.: Uber eine numerische Knoteninvariante. Math. Z. 61, 245–288 (1954) (German) · Zbl 0058.17403 · doi:10.1007/BF01181346 [12] Stasiak A., Katritchx A., Bednar J., Michoud D., Dubochet J.: Electrophoretic mobility of DNA knots. Nature 384, 122 (1996) · doi:10.1038/384122a0 [13] Sumners D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Not. AMS 42, 528–537 (1995) · Zbl 1003.92515 [14] 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) · doi:10.1529/biophysj.105.070128
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