Celebratio Mathematica — Hass — Complete Bibliography

[1] J. Hass: Embedded minimal surfaces in three and four dimensional manifolds. Ph.D. thesis, University of California, Berkeley, 1981. Advised by R. Kirby. MR 2631519 phdthesis

Recent results of Schoen–Yau and Sacks–Uhlenbeck have shown the existence of immersed minimal surfaces of positive genus in 3-manifolds, but said nothing about the self-intersections of these surfaces. Conjectures about the existence of area minimizing embeddings homotopic to these surfaces were made by Meeks and by Uhlenbeck. These conjectures are shown to be false as originally stated and a modified conjecture is presented. It is demonstrated that in many cases, the area minimizing surface homotopic to an incompressible embedding is itself imbedded. In addition, the idea of using minimal surfaces to study foliations is examined. A new proof of Novikov’s theorem about the existence of compact leaves in foliations of certain 3-manifolds is obtained, with the hypothesis of Novikov’s theorem weakened somewhat. The nature of the intersection of minimal surfaces and totally geodesic submanifolds of 4-manifolds is used to prove an analogue of the sphere theorem for certain 4-manifolds. Finally, minimal disks bounded by curves on the unit 3-sphere in \( R^4 \) are examined. This topic is shown to be closely related to the slice-ribbon problem of knot theory.

[2] article M. Freedman, J. Hass, and P. Scott: “Closed geodesics on surfaces,” Bull. London Math. Soc. 14 : 5 (1982), pp. 385–391. MR 0671777 Zbl 0476.53026

Let \( M^2 \) be a closed Riemannian 2-manifold, and let \( \alpha \) denote a non-trivial element of \( \pi_1(M) \). The set of all loops in \( M \) which represent a has a shortest element \( f:\mathbb{S}^1 \rightarrow M \), which can be assumed smooth and which will be a closed geodesic. (We say a loop represents \( \alpha \) when it represents any conjugate of \( \alpha \). Such a loop need not pass through the base point of \( M \).) The map \( f \) cannot be unique, because \( f \) is not necessarily parametrised by arc length and because there is no base point. In general, even the image set of a shortest loop is not unique. In this note, we prove the following result.

Let \( M^2 \) be a closed, Riemannian 2-manifold and let \( \alpha \) denote a non-trivial element of \( \pi_1M \) which is represented by a two-sided embedded loop \( C \). Then any shortest loop \( f:\mathbb{S}^1 \rightarrow M \) representing \( \alpha \) is either an embedding or a double cover of a one-sided embedded curve \( K \). In the second case, \( C \) bounds a Moebius band in \( M \) and \( K \) is isotopic to the centre of this band.

M. H. Freedman	Related	Volume
Peter Scott	Related	Volume

[3] article M. Freedman, J. Hass, and P. Scott: “Least area incompressible surfaces in 3-manifolds,” Invent. Math. 71 : 3 (1983), pp. 609–642. MR 0695910 Zbl 0482.53045

Let \( M \) be a Riemannian manifold and let \( F \) be a closed surface. A map \( f:F\rightarrow M \) is called least area if the area of \( f \) is less than the area of any homotopic map from \( F \) to \( M \). Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function.

In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly.

M. H. Freedman	Related	Volume
Peter Scott	Related	Volume

[4] J. Hass: “The geometry of the slice-ribbon problem,” Math. Proc. Cambridge Philos. Soc. 94 : 1 (July 1983), pp. 101–108. MR 704804 Zbl 0535.57004 article

[5] J. Hass: “Complete area minimizing minimal surfaces which are not totally geodesic,” Pac. J. Math. 111 : 1 (1984), pp. 35–38. MR 732056 Zbl 0534.53004 article

[6] J. Hass: “Minimal surfaces in Seifert fiber spaces,” Topology Appl. 18 : 2–3 (1984), pp. 145–151. MR 769287 Zbl 0559.57005 article

[7] J. Hass and J. Hughes: “Immersions of surfaces in 3-manifolds,” Topology 24 : 1 (1985), pp. 97–112. MR 790679 Zbl 0527.57020 article

[8] J. Hass and P. Scott: “Intersections of curves on surfaces,” Israel J. Math. 51 : 1–2 (1985), pp. 90–120. MR 804478 Zbl 0576.57009 article

The authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which has excess self-intersection. Then there is a singular 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.

[9]J. Hass and J. H. Rubinstein: “One-sided closed geodesics on surfaces,” Mich. Math. J. 33 : 2 (1986), pp. 155–168. MR 837574 Zbl 0614.53035 article

[10] J. Hass: “Minimal surfaces in foliated manifolds,” Comment. Math. Helv. 61 : 1 (1986), pp. 1–32. MR 847517 Zbl 0601.53024 article

[11]J. Hass, H. Rubinstein, and P. Scott: “Covering spaces of 3-manifolds,” Bull. Am. Math. Soc., New Ser. 16 : 1 (January 1987), pp. 117–119. MR 866028 Zbl 0624.57016 article

J. Hyam Rubinstein	Related	Volume
Peter Scott	Related	Volume

[12] J. Hass: “Minimal surfaces in manifolds with \( S^1 \) actions and the simple loop conjecture for Seifert fibered spaces,” Proc. Am. Math. Soc. 99 : 2 (February 1987), pp. 383–388. MR 870806 Zbl 0627.57008 article

[13] J. Hass and P. Scott: “The existence of least area surfaces in 3-manifolds,” Trans. Am. Math. Soc. 310 : 1 (November 1988), pp. 87–114. MR 965747 Zbl 0711.53008 article

[14] J. Hass: “Surfaces minimizing area in their homology class and group actions on 3-manifolds,” Math. Z. 199 : 4 (1988), pp. 501–509. MR 968316 Zbl 0715.57005 article

[15]J. Hass, H. Rubinstein, and P. Scott: “Compactifying coverings of closed 3-manifolds,” J. Differ. Geom. 30 : 3 (1989), pp. 817–832. MR 1021374 Zbl 0693.57011 article

J. Hyam Rubinstein	Related	Volume
Peter Scott	Related	Volume

[16] C. Frohman and J. Hass: “Unstable minimal surfaces and Heegaard splittings,” Invent. Math. 95 : 3 (1989), pp. 529–540. MR 979363 Zbl 0678.57009 article

[17] J. Hass and A. Thompson: “A necessary and sufficient condition for a 3-manifold to have Heegaard genus one,” Proc. Am. Math. Soc. 107 : 4 (December 1989), pp. 1107–1110. MR 984792 Zbl 0694.57006 article

[18] J. Hass: “Singular curves and the Plateau problem,” Int. J. Math. 2 : 1 (1991), pp. 1–16. MR 1082833 Zbl 0737.49029 article

[19] J. Hass: “Genus two Heegaard splittings,” Proc. Am. Math. Soc. 114 : 2 (1992), pp. 565–570. MR 1070519 Zbl 0746.57005 article

[20] J. Hass: “Intersections of least area surfaces,” Pac. J. Math. 152 : 1 (1992), pp. 119–123. MR 1139976 Zbl 0747.53047 article

[21] J. Hass and P. Scott: “Homotopy equivalence and homeomorphism of 3-manifolds,” Topology 31 : 3 (July 1992), pp. 493–517. MR 1174254 Zbl 0771.57007 article

[22]J. Hass, J. T. Pitts, and J. H. Rubinstein: “Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces,” pp. 147–162 in Differential geometry (Los Angeles, 8–28 July 1990), part 1: Partial differential equations on manifolds. Edited by R. E. Green and S.-T. Yau. Proceedings of Symposia in Pure Mathematics 54. American Mathematical Society (Providence, RI), 1993. MR 1216582 Zbl 0798.53009 incollection

J. Hyam Rubinstein	Related	Volume
Robert Everist Greene	Related
Shing-Tung Yau	Related
Jon T. Pitts	Related

[23] J. Hass and P. Scott: “Curve flows on surfaces and intersections of curves,” pp. 415–421 in Differential geometry (Los Angeles, 8–28 July 1990), Part 3: Riemannian geometry. Edited by R. Greene and S.-T. Yau. Proceedings of Symposia in Pure Mathematics 54. Amererican Mathematical Society (Providence, RI), 1993. MR 1216633 Zbl 0793.53006 incollection

[24] J. Hass and W. Menasco: “Topologically rigid non-Haken 3-manifolds,” J. Austral. Math. Soc. Ser. A 55 : 1 (August 1993), pp. 60–71. MR 1231694 Zbl 0806.57006 article

[25] J. Hass and P. Scott: “Homotopy and isotopy in dimension three,” Comment. Math. Helv. 68 : 3 (1993), pp. 341–364. MR 1236759 Zbl 0805.57008 article

[26] J. Hass and P. Scott: “Shortening curves on surfaces,” Topology 33 : 1 (January 1994), pp. 25–43. MR 1259513 Zbl 0798.58019 article

[27] J. Hass: “Metrics on manifolds with convex or concave boundary,” pp. 41–46 in Geometric topology: Joint US-Israel workshop on geometric topology (Haifa, Israel, 10–16 June 1992). Edited by C. Gordon, Y. Moriah, and B. Wajnryb. Contemporary Mathematics 164. American Mathematical Society (Providence, RI), 1994. MR 1282754 Zbl 0822.53024 incollection

Cameron McAllan Gordon	Related	Volume
Dov Bronislaw Wajnryb	Related
Yoav Moriah	Related

[28] J. Hass: “Bounded 3-manifolds admit negatively curved metrics with concave boundary,” J. Diff. Geom. 40 : 3 (1994), pp. 449–459. MR 1305977 Zbl 0821.53035 article

[29] J. Hass: “Acylindrical surfaces in 3-manifolds,” Michigan Math. J. 42 : 2 (1995), pp. 357–365. MR 1342495 Zbl 0862.57011 article

[30] J. Hass, M. Hutchings, and R. Schlafly: “The double bubble conjecture,” Electron. Res. Announc. Am. Math. Soc. 1 : 3 (1995), pp. 98–102. MR 1369639 Zbl 0864.53007 article

Michael Lounsbery Hutchings	Related
Roger Sherwood Schlafly	Related

[31] J. Hass and R. Schlafly: “Bubbles and double bubbles,” Amer. Sci. 84 : 5 (September–October 1996), pp. 462–467. A French version was published in La Recherche 303 (1997). article

[32] J. Hass and F. Morgan: “Geodesic nets on the 2-sphere,” Proc. Am. Math. Soc. 124 : 12 (1996), pp. 3843–3850. MR 1343696 Zbl 0871.53038 article

[33] J. Hass and F. Morgan: “Geodesics and soap bubbles in surfaces,” Math. Z. 223 : 2 (1996), pp. 185–196. MR 1417428 Zbl 0865.53009 article

[34] J. Hass, J. C. Lagarias, and N. Pippenger: “The computational complexity of knot and link problems,” pp. 172–181 in Proceedings: 38th annual symposium on the foundations of computer science (Miami Beach, FL, 20–22 October 1997). IEEE Computer Society Press (Los Alamitos, CA), 1997. preliminary report. preliminary report for an article eventually published in J. ACM 46:2 (1999). incollection

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted (that is, whether it is capable of being continuously deformed without self-intersection so that it lies in a plane). We show that this problem, UNKNOTTING PROBLEM, is in NP. We also consider the problem, SPLITTING PROBLEM, of determining whether two or more such polygons can be split (that is, whether they are capable of being continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it), and show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE.

[35] J. Hass and R. Schlafly: “Histoires de bulles et de double bulles” [Stories of bubbles and double bubbles], La Recherche 303 (November 1997), pp. 42–47. French version of an article originally published in Am. Sci. 84:5 (1996). article

[36] J. Hass and A. Thompson: “Neon bulbs and the unknotting of arcs in manifolds,” J. Knot Theor. Ramif. 6 : 2 (April 1997), pp. 235–242. MR 1452439 Zbl 0886.57003 article

[37] C. Adams, J. Hass, and A. Thompson: How to ace calculus: The streetwise guide. W. H. Freeman (New York), 1998. book

Abigail A. Thompson	Related	Volume
Colin C. Adams	Related

[38] J. Hass: “Algorithms for recognizing knots and 3-manifolds,” pp. 569–581 in Knot theory and its applications: Expository articles on current research, published as Chaos Solitons Fractals 9 : 4–5. Issue edited by M. S. El Naschie. April–May 1998. MR 1628743 Zbl 0935.57014 incollection

Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. It is important to know if other problems fall into this category. Secondly, the discovery of a reasonably efficient algorithm can lead to a computer program which can be used to examine interesting examples. In this paper we will survey some topological algorithms, in particular those that relate to distinguishing knots. Our approach is somewhat informal, with many details omitted, but references are given to sources which develop these ideas in full depth.

[39] C. Adams, J. Hass, and P. Scott: “Simple closed geodesics in hyperbolic 3-manifolds,” Bull. London Math. Soc. 31 : 1 (January 1999), pp. 81–86. MR 1650997 Zbl 0955.53025 ArXiv math/9801071 article

Colin C. Adams	Related
Peter Scott	Related	Volume

[40] J. Hass, J. C. Lagarias, and N. Pippenger: “The computational complexity of knot and link problems,” J. ACM 46 : 2 (March 1999), pp. 185–211. A preliminary report was published in Proceedings: 38th annual symposium on the foundations of computer science (1997). MR 1693203 Zbl 1065.68667 article

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, i.e., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

[41]Proceedings of the Kirbyfest (Berkeley, CA, June 22–26, 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. Geometry & Topology Publications (Coventry), 1999. MR 1734398

[42] J. Hass and P. Scott: “Configurations of curves and geodesics on surfaces,” pp. 201–213 in Proceedings of the Kirbyfest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. G. Scharlemann. Geometry & Topology Monographs 2. Geometry & Topology Publications (Coventry, UK), 1999. MR 1734409 Zbl 1035.53053 ArXiv math/9903130 incollection

Robion C. Kirby	Related	Volume
Peter Scott	Related	Volume

[43] J. Hass, J. H. Rubinstein, and S. Wang: “Boundary slopes of immersed surfaces in 3-manifolds,” J. Diff. Geom. 52 : 2 (1999), pp. 303–325. MR 1758298 Zbl 0978.57016 ArXiv math/9911072 article

J. Hyam Rubinstein	Related	Volume
Shicheng Wang	Related

[44] J. Hass: “General double bubble conjecture in \( \mathbb{R}^3 \) solved,” MAA Focus 20 : 5 (May–June 2000), pp. 4–5. article

[45] J. Hass and R. Schlafly: “Double bubbles minimize,” Ann. Math. (2) 151 : 2 (March 2000), pp. 459–515. MR 1765704 Zbl 0970.53008 article

[46] C. Adams, J. Hass, and A. Thompson: How to ace the rest of calculus: The streetwise guide. W. H. Freeman (New York), 2001. book

Abigail A. Thompson	Related	Volume
Colin C. Adams	Related

[47] J. Hass and J. C. Lagarias: “The number of Reidemeister moves needed for unknotting,” J. Am. Math. Soc. 14 : 2 (2001), pp. 399–428. MR 1815217 Zbl 0964.57005 article

There is a positive constant \( c_1 \) such that for any diagram \( \mathcal{D} \) representing the unknot, there is a sequence of at most \( 2^{c_1 n} \) Reidemeister moves that will convert it to a trivial knot diagram, where \( n \) is the number of crossings in \( \mathcal{D} \). A similar result holds for elementary moves on a polygonal knot \( K \) embedded in the 1-skeleton of the interior of a compact, orientable, triangulated \( PL \) 3-manifold \( M \). There is a positive constant \( c_2 \) such that for each \( t \geq 1 \), if \( M \) consists of \( t \) tetrahedra and \( K \) is unknotted, then there is a sequence of at most \( 2^{c_2 t} \) elementary moves in \( M \) which transforms \( K \) to a triangle contained inside one tetrahedron of \( M \). We obtain explicit values for \( c_1 \) and \( c_2 \).

[48] J. Hass, S. Wang, and Q. Zhou: “On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds,” Proc. Am. Math. Soc. 130 : 6 (2002), pp. 1851–1857. MR 1887034 Zbl 0993.57007 article

Qing Zhou	Related
Shicheng Wang	Related

[49] I. Agol, J. Hass, and W. Thurston: “3-manifold knot genus is NP-complete,” pp. 761–766 in Proceedings of the thirty-fourth annual ACM symposium on theory of computing (Montreal, 19–21 May 2002). Edited by J. H. Reif. ACM (New York), 2002. MR 2121524 Zbl 1192.68305 incollection

Ian Agol	Related
William Paul Thurston	Related

[50] J. Hass, J. Snoeyink, and W. P. Thurston: “The size of spanning disks for polygonal curves,” Discrete Comput. Geom. 29 : 1 (2003), pp. 1–17. MR 1946790 Zbl 1015.57008 article

Jack Scott Snoeyink	Related
William Paul Thurston	Related

[51]J. Hass, P. Norbury, and J. H. Rubinstein: “Minimal spheres of arbitrarily high Morse index,” Commun. Anal. Geom. 11 : 3 (2003), pp. 425–439. MR 2015753 Zbl 1104.53055 ArXiv 0206286 article

J. Hyam Rubinstein	Related	Volume
Paul Timothy Norbury	Related

[52] J. Hass and J. C. Lagarias: “The minimal number of triangles needed to span a polygon embedded in \( \mathbb{R}^d \),” pp. 509–526 in Discrete and computational geometry: The Goodman–Pollack Festschrift. Edited by B. Aronov, S. Basu, J. Pach, and M. Sharir. Algorithms and Combinatorics 25. Springer (Berlin), 2003. MR 2038489 Zbl 1103.52013 incollection

Given a closed polygon \( P \) having \( n \) edges, embedded in \( \mathbb{R}^d \), we give upper and lower bounds for the minimal number of triangles \( t \) needed to form a triangulated PL surface embedded in \( \mathbb{R}^d \) having \( P \) as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having \( P \) as its boundary which is immersed in \( \mathbb{R}^d \) and whose interior is disjoint from \( P \). The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in \( \mathbb{R}^3 \) there exists an embedded orientable triangulated PL surface having at most \( 7n^2 \) triangles, whose boundary is a subdivision of \( P \). We complement this with a construction of families of polygons with \( n \) vertices for which any such embedded surface requires at least \( \frac{1}{2}n^2 - O(n) \) triangles. We also exhibit families of polygons in \( \mathbb{R}^3 \) for which \( \Omega(n^2) \) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions \( d\geq 5 \) there always exists an embedded locally flat PL disk having \( P \) as boundary that contains at most \( n \) triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most \( 3n \) triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an \( O(n^2) \) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between \( 1/2 \) and 7.

János Pach	Related
Jacob E. Goodman	Related
Richard Pollack	Related

[53] X. Song, T. W. Sederberg, J. Zheng, R. T. Farouki, and J. Hass: “Linear perturbation methods for topologically consistent representations of free-form surface intersections,” Comput. Aided Geom. Design 21 : 3 (March 2004), pp. 303–319. A corrigendum to this article was published in Comput. Aided Geom. Design 21:3 (2004). MR 2042022 Zbl 1069.65567 article

By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in \( \mathbb{R}^3 \) of parameter-domain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. Moreover, the control point perturbations require only the solution of a linear system for their determination. The basic principles of this approach to topologically consistent surface trimming curves are described, and example results from the implementation of a simple instance of the method are presented.

Thomas W. Sederberg	Related
Xiaowen Song	Related
Jian Min Zheng	Related
Rida T. Farouki	Related

[54] X. Song, T. W. Sederberg, J. Zheng, R. T. Farouki, and J. Hass: “Corrigendum to ‘Linear perturbation methods for topologically consistent representations of free-form surface intersections’,” Comput. Aided Geom. Design 21 : 3 (2004), pp. 321. corrigendum to an article published in Comput. Aided Geom. Design 21:3 (2004). MR 2042023 Zbl 1069.65568 article

Thomas W. Sederberg	Related
Xiaowen Song	Related
Jian Min Zheng	Related
Rida T. Farouki	Related

[55] R. T. Farouki, C. Y. Han, J. Hass, and T. W. Sederberg: “Topologically consistent trimmed surface approximations based on triangular patches,” Comput. Aided Geom. Design 21 : 5 (2004), pp. 459–478. MR 2058392 Zbl 1069.65552 article

A scheme to approximate the trimmed surfaces defined by two tensor-product surface patches, intersecting in a smooth curve segment that extends between diametrically opposite patch corners, is formulated. The trimmed surface approximations are specified by triangular Bézier patches, whose tangent planes agree precisely with those of the tensor-product surfaces along the two sides where they coincide. Topological consistency of the two trimmed surfaces is achieved by requiring the “hypotenuse” sides of the triangular patches to be coincident. In the case of bicubic tensor-product patches and quintic triangular trimmed surface approximations, enforcing these conditions entails the solution of a linear system of 30 equations in 32 unknowns. The two remaining scalar freedoms, together with four additional free control points, are employed to enhance the accuracy and/or smoothness properties of the intersection curve and trimmed surface approximations. By means of an appropriate subdivision preprocessing, the trimmed surface scheme may be used on models described by arbitrary bicubic surface patches.

Thomas W. Sederberg	Related
Chang Yong Han	Related
Rida T. Farouki	Related

[56] J. Hass, J. C. Lagarias, and W. P. Thurston: “Area inequalities for embedded disks spanning unknotted curves,” J. Diff. Geom. 68 : 1 (2004), pp. 1–29. MR 2152907 Zbl 1104.53006 ArXiv math/0306313 article

We show that a smooth unknotted curve in \( \mathbb{R}^3 \) satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length \( L \) of the curve, and the thickness \( r \) (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in \( 1/r^2 \). In the direction of lower bounds, we give a sequence of length one curves with \( r \to 0 \) for which the area of any spanning disk is bounded from below by a function that grows exponentially with \( 1/r \). In particular, given any constant \( A \), there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than \( A \).

[57] J. Hass: “Minimal surfaces and the topology of three-manifolds,” pp. 705–724 in Global theory of minimal surfaces (Berkeley, CA, 25 June–27 July 2001). Edited by D. Hoffman. Clay Mathematical Proceedings 2. American Mathematical Society (Providence, RI), 2005. MR 2167285 Zbl 1100.57021 incollection

[58] R. T. Farouki, C. Y. Han, and J. Hass: “Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves,” Numer. Algorithms 40 : 3 (2005), pp. 251–283. MR 2189407 Zbl 1087.65011 article

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in \( \mathbb{C} \). This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space \( \mathbb{R}^n \) whose coordinates are boundary parameters for the \( n \) given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in \( \mathbb{R}^n \) are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.

Chang Yong Han	Related
Rida T. Farouki	Related

[59] F. R. Giordano, M. D. Weir, and J. Hass: Thomas’ calculus: Early transcendentals, 11th edition. Addison-Wesley (Boston, MA), 2005. Hass also contributed to the 12th edition (2009). book

Maurice D. Weir	Related
Frank R. Giordano	Related

[60] I. Agol, J. Hass, and W. Thurston: “The computational complexity of knot genus and spanning area,” Trans. Am. Math. Soc. 358 : 9 (2006), pp. 3821–3850. MR 2219001 Zbl 1098.57003 article

Ian Agol	Related
William Paul Thurston	Related

[61] J. Hass, R. T. Farouki, C. Y. Han, X. Song, and T. W. Sederberg: “Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme,” Adv. Comput. Math. 27 : 1 (2007), pp. 1–26. MR 2317919 Zbl 1118.65011 article

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve \( F(x,y) = 0 \) in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces \( \mathbf{p}(s,t) \) and \( \mathbf{q}(u,v) \) defined on \( (s,t)\in [0,1]^2 \) and \( (u,v)\in [0,1]^2 \). The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given.

Thomas W. Sederberg	Related
Xiaowen Song	Related
Rida T. Farouki	Related
Chang Yong Han	Related

[62] R. T. Farouki and J. Hass: “Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions,” J. Comput. Appl. Math. 209 : 2 (2007), pp. 246–266. MR 2387129 Zbl 1140.65020 article

Algorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in \( R^3 \), and other forms of geometrical convolution, are briefly discussed.

[63] J. Hass and T. Nowik: “Invariants of knot diagrams,” Math. Ann. 342 : 1 (2008), pp. 125–137. MR 2415317 Zbl 1161.57002 article

[64] J. Hass, A. Thompson, and W. Thurston: “Stabilization of Heegaard splittings,” Geom. Topol. 13 : 4 (2009), pp. 2029–2050. MR 2507114 Zbl 1177.57018 ArXiv 0802.2145 article

Abigail A. Thompson	Related	Volume
William Paul Thurston	Related

[65] J. Hass, J. H. Rubinstein, and A. Thompson: “Knots and \( k \)-width,” Geom. Dedicata 143 : 7 (December 2009), pp. 7–18. MR 2576289 Zbl 1189.57005 ArXiv math/0604256 article

J. Hyam Rubinstein	Related	Volume
Abigail A. Thompson	Related	Volume

[66] M. D. Weir, G. B. Thomas, and J. Hass: Thomas’ calculus: Early transcendentals, 12th edition. Addison-Wesley (Boston, MA), 2009. An abridged German translation of this was published as Basisbuch {Analysis} (2013). Hass also contributed to the 11th edition (2005). book

Maurice D. Weir	Related
George Brinton Thomas, Jr.	Related

[67] J. R. Hass, M. D. Weir, and G. B. Thomas, Jr.: University calculus: Elements with early transcendentals. Pearson (Boston, MA), 2009. single variable. A second, revised edition was published (with slightly different title) in 2012. Zbl 1271.00017 book

Maurice D. Weir	Related
George Brinton Thomas, Jr.	Related

[68] P. Francis-Lyon, S. Gu, J. Hass, N. Amenta, and P. Koehl: “Sampling the conformation of protein surface residues for flexible protein docking,” BMC Bioinform. 11 (2010), pp. 575–588. article

Nina Amenta	Related
Patricia Francis-Lyon	Related
Shengyin Gu	Related
Patrice Koehl	Related

[69] J. Hass and T. Nowik: “Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle,” Discrete Comput. Geom. 44 : 1 (2010), pp. 91–95. MR 2639820 Zbl 1191.57006 article

[70] J. Hass and A. Thompson: “Is it knotted?,” pp. 129–135 in Expeditions in mathematics. Edited by T. Shubin, D. F. Hayes, and G. L. Alexanderson. MAA Spectrum 68. Mathematical Association of America (Washington, DC), 2011. incollection

Tatiana Shubin	Related
Gerald Lee Alexanderson	Related
Abigail A. Thompson	Related	Volume
David Frank Hayes	Related

[71] S. Gu, P. Koehl, J. Hass, and N. Amenta: “Surface-histogram: A new shape descriptor for protein-protein docking,” Proteins Struct. Funct. Bioinf. 80 : 1 (January 2012), pp. 221–238. article

Determining the structure of protein-protein complexes remains a difficult and lengthy process, either by NMR or by X-ray crystallography. Several computational methods based on docking have been developed to support and even serve as possible alternatives to these experimental methods. In this article, we introduce a new protein-protein docking algorithm, shDock, based on shape complementarity. We characterize the local geometry on each protein surface with a new shape descriptor, the surface-histogram. We measure the complementarity between two surface-histograms, one on each protein, using a modified Manhattan distance. When a match is found between two local protein surfaces, a model is generated for the protein complex, which is then scored by checking for collision between the two proteins. We have tested our algorithm on Version 3 of the ZDOCK protein-protein docking benchmark. We found that for 110 out of the 124 test cases of bound docking in the benchmark, our algorithm was able to generate a model in the top 3600 candidates for the protein complex within an root-mean-square deviation of \( 2.5\,Å \) from its native structure. For unbound docking predictions, we found a model within \( 2.5\,Å \) in the top 3600 models in 54 out of 124 test cases. A comparison with other shape-based docking algorithms demonstrates that our approach gives significantly improved performance for both bound and unbound docking test cases.

Nina Amenta	Related
Shengyin Gu	Related
Patrice Koehl	Related

[72] J. Hass and G. Kuperberg: “The complexity of recognizing the 3-sphere,” pp. 1425–1426 in Triangulations, published as OWR 9 : 2. Issue edited by G.-M. Greuel. EMS (Zürich), 2012. nonrefereed extended abstract. incollection

Gert-Martin Greuel	Related
Gregory John Kuperberg	Related

[73] J. Hass: “What is an almost normal surface?,” pp. 1–13 in Geometry and topology down under: A conference in honour of Hyam Rubinstein (Melbourne, 11–22 July 2011). Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillmann. Contemporary Mathematics 597. American Mathematical Society (Providence, RI), 2013. MR 3186667 Zbl 1279.57002 incollection

J. Hyam Rubinstein	Related	Volume
Martin Scharlemann	Related	Volume
William H. Jaco	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

[74] A. Tsui, D. Fenton, P. Vuong, J. Hass, P. Koehl, N. Amenta, D. Coeurjolly, C. DeCarli, and O. Carmichael: “Globally optimal cortical surface matching with exact landmark correspondence,” pp. 487–498 in Information processing in medical imaging: 23rd international conference, IPMI 2013 (Asilomar, CA, 28 June–3 July 2013). Edited by J. C. Gee, S. Joshi, K. M. Pohl, W. M. Wells, and L. Zöllei. Lecture Notes in Computer Science 7917. Springer (Berlin), 2013. incollection

We present a method for establishing correspondences between human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an objective function that quantifies how far the mapping deviates from conformality. On each surface, a conformal transformation is applied to the Euclidean distance metric, resulting in a hyperbolic metric with isolated cone point singularities at the landmarks. Equivalently, each surface is mapped to a hyperbolic orbifold: a pillow-like surface with each point landmark corresponding to a pillow corner. An initial surface-to-surface mapping exactly aligns the landmarks, and gradient descent is used to find the single, global minimum of the Dirichlet energy of the remainder of the mapping. Using a population of real MRI-based cortical surfaces with manually labeled sulcus endpoints as landmarks, we evaluate the approach by how much it distorts surfaces and by its biological plausibility: how well it aligns previously-unseen anatomical landmarks and by how well it promotes expected associations between cortical thickness and age. We show that, compared to a painstakingly-tuned approach that balances a tradeoff between minimizing landmark mismatch and Dirichlet energy, our method has similar biological plausibility, superior surface distortion, a better theoretical foundation, and fewer arbitrary parameters to tune. We also compare to conformal mapper in the spherical domain to show that sacrificing exact conformality of the mapping does not cause noticeable reductions in biological plausibility.

Padmini Joshi	Related
Alex Tsui	Related
Phong Vuong	Related
Nina Amenta	Related
Owen Carmichael	Related
David Coeurjolly	Related
Charles DeCarli	Related
David Fenton	Related
Patrice Koehl	Related

@incollection {key76348359,
	AUTHOR = {Tsui, Alex and Fenton, Devin and Vuong,
		 Phong and Hass, Joel and Koehl, Patrice
		 and Amenta, Nina and Coeurjolly, David
		 and DeCarli, Charles and Carmichael,
		 Owen},
	TITLE = {Globally optimal cortical surface matching
		 with exact landmark correspondence},
	BOOKTITLE = {Information processing in medical imaging:
		 23rd international conference, {IPMI}
		 2013},
	EDITOR = {Gee, James C. and Joshi, Sarang and
		 Pohl, Kilian M. and Wells, William M.
		 and Z\"ollei, Lilla},
	SERIES = {Lecture Notes in Computer Science},
	NUMBER = {7917},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {2013},
	PAGES = {487--498},
	DOI = {10.1007/978-3-642-38868-2_41},
	NOTE = {(Asilomar, CA, 28 June--3 July 2013).},
	ISSN = {0302-9743},
	ISBN = {9783642388675},
}

[75] G. B. Thomas, M. D. Weir, and J. Hass: Analysis 1: Lehr- und Übungsbuch [Analysis 1: Teaching and practice book]. Pearson Studium (München), 2013. Part 2 was published in 2013. Zbl 1270.00022 book

Maurice D. Weir	Related
George Brinton Thomas, Jr.	Related

[76] G. B. Thomas, M. D. Weir, and J. Hass: Basisbuch Analysis [Basic book of analysis]. Pearson Higher Eucation (München), 2013. abridged German translation of 12th edition of Thomas’ calculus (2009). Zbl 1283.26001 book

Maurice D. Weir	Related
George Brinton Thomas, Jr.	Related

[77] P. Koehl and J. Hass: “Automatic alignment of genus-zero surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 36 : 3 (March 2014), pp. 466–478. article

A new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation, as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero. The algorithm takes as input a pair of surfaces that are topological 2-spheres, each surface given by a distinct triangulation. The algorithm then constructs a map \( f \) between the two surfaces. First, each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm. The two mappings are then composed with a Mobius transformation to generate the function \( f \). The Mobius transformation is chosen by minimizing an energy that measures the distance of \( f \) from an isometry. We illustrate our approach using several “real life” data sets. We show first that the algorithm allows for accurate, automatic, and landmark-free nonrigid registration of brain surfaces. We then validate our approach by comparing shapes of proteins. We provide numerical experiments to demonstrate that the distances computed with our algorithm between low-resolution, surface-based representations of proteins are highly correlated with the corresponding distances computed between high-resolution, atomistic models for the same proteins.

[78] G. B. Thomas, M. D. Weir, and J. Hass: Analysis 2: Lehr- und Übungsbuch [Analysis 2: Teaching and practice book], 12th updated edition. Pearson Studium (München), 2014. Part 1 was published in 2013. Zbl 1296.26005 book

Maurice D. Weir	Related
George Brinton Thomas, Jr.	Related

[79] J. Hass and P. Koehl: “How round is a protein? Exploring protein structures for globularity using conformal mapping,” Front. Mol. Biosci. 1 (December 2014), pp. 1–26. article

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map \( f \) between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy \( E_S(f) \) that measures the distance of \( f \) from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that \( E_S(f) \) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy \( E_S(f) \) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that \( E_S(f) \) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry.

[80] J. Hass: Minimal fibrations of hyperbolic 3-manifolds. Preprint, 12 2015. ArXiv 1512.04145 techreport

[81] J. Hass and P. Koehl: A metric for genus-zero surfaces. Preprint, July 2015. ArXiv 1507.00798 techreport

[82] P. Koehl and J. Hass: “Landmark-free geometric methods in biological shape analysis,” J. R. Soc. Interface 12 : 113 (2015), pp. 1–11. article

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map \( f \) between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy \( E_{sd}(f) \) which we introduce. This measures the distance of \( f \) from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

[83] A. Coward and J. Hass: “Topological and physical link theory are distinct,” Pac. J. Math. 276 : 2 (2015), pp. 387–400. MR 3374064 Zbl 1326.53008 article

[84] J. Hass and P. Scott: “Simplicial energy and simplicial harmonic maps,” Asian J. Math. 19 : 4 (2015), pp. 593–636. MR 3423736 Zbl 1332.57024 ArXiv 1206.2574 article

[85] J. Hass: Isoperimetric regions in nonpositively curved manifolds. Preprint, 11 2016. ArXiv 1604.02768 techreport

[86] J. Hass, C. Even-Zohar, N. Linial, and T. Nowik: Invariants of random knots and links. Preprint, June 2016. ArXiv 1411.3308 techreport

We study random knots and links in \( \mathbb{R}^3 \) using the Petaluma model, which is based on the petal projections developed in [Adams et al. 2015]. In this model we obtain a formula for the limiting distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant \( v_3 \). These are the first precise formulas given for the distributions and higher moments of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.

Nathan Linial	Related
Tahl Nowik	Related
Chaim Even-Zohar	Related

[87] J. Hass, A. Thompson, and A. Tsvietkova: “The number of surfaces of fixed genus in an alternating link complement,” Int. Math. Res. Not. 2017 : 6 (March 2017), pp. 1611–1622. MR 3658176 ArXiv 1508.03680 article

Abigail A. Thompson	Related	Volume
Anastasiia Tsvietkova	Related

[88] J. Hass and P. Koehl: “Comparing shapes of genus-zero surfaces,” J. Appl. Comput. Topol. 1 : 1 (2017), pp. 57–87. MR 3975549 article

[89] C. Even-Zohar, J. Hass, N. Linial, and T. Nowik: “The distribution of knots in the Petaluma model,” Algebr. Geom. Topol. 18 : 6 (2018), pp. 3647–3667. MR 3868230 article

The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random \( n \)-petal model the probability of obtaining every specific knot type decays to zero as \( n \), the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the \( n \)-petal model represents at least exponentially many distinct knots.

Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question.

[90] M. Bell, J. Hass, J. H. Rubinstein, and S. Tillmann: “Computing trisections of 4-manifolds,” Proc. Natl. Acad. Sci. USA 115 : 43 (2018), pp. 10901–10907. MR 3871794 article

[91] C. Even-Zohar, J. Hass, N. Linial, and T. Nowik: “Universal knot diagrams,” J. Knot Theory Ramifications 28 : 7 (2019), pp. 1950031, 30. MR 3975570 article

[92] J. Hass, A. Thompson, and A. Tsvietkova: “Alternating links have at most polynomially many Seifert surfaces of fixed genus,” Indiana Univ. Math. J. 70 : 2 (2021), pp. 525–534. MR 4257618 article

Abigail A. Thompson	Related	Volume
Anastasiia Tsvietkova	Related

[93] J. Hass, A. Thompson, and A. Tsvietkova: “Tangle decompositions of alternating link complements,” Illinois J. Math. 65 : 3 (2021), pp. 533–545. MR 4312193 article

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish; Lickorish proved that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles, each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists, either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.

Abigail A. Thompson	Related	Volume
Anastasiia Tsvietkova	Related

[94] C. Even-Zohar and J. Hass: “Random colorings in manifolds,” Israel J. Math. 256 : 1 (2023), pp. 153–211. MR 4652937 article

We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel–Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of \( k \) colors in \( d \)-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.