Computational Exploration of Multistable Elastic Knots
We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states th...
Saved in:
| Published in | ACM transactions on graphics Vol. 42; no. 4; pp. 1 - 16 |
|---|---|
| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York, NY, USA
ACM
01.08.2023
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation. |
|---|---|
| AbstractList | We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation. |
| ArticleNumber | 73 |
| Author | Ren, Yingying Grinspun, Eitan Pauly, Mark Panetta, Julian Vidulis, Michele |
| Author_xml | – sequence: 1 givenname: Michele orcidid: 0009-0009-2373-9581 surname: Vidulis fullname: Vidulis, Michele email: michele.vidulis@epfl.ch organization: EPFL, Lausanne, Switzerland – sequence: 2 givenname: Yingying orcidid: 0000-0001-6229-3724 surname: Ren fullname: Ren, Yingying email: yingying.ren@epfl.ch organization: EPFL, Lausanne, Switzerland – sequence: 3 givenname: Julian orcidid: 0000-0002-8331-2354 surname: Panetta fullname: Panetta, Julian email: jpanetta@ucdavis.edu organization: UC Davis, Davis, United States of America – sequence: 4 givenname: Eitan orcidid: 0000-0003-4460-7747 surname: Grinspun fullname: Grinspun, Eitan email: eitan@cs.toronto.edu organization: University of Toronto, Toronto, Canada – sequence: 5 givenname: Mark orcidid: 0000-0003-4957-4825 surname: Pauly fullname: Pauly, Mark email: mark.pauly@epfl.ch organization: EPFL, Lausanne, Switzerland |
| BookMark | eNptj01LxDAURYOMYGcU9666cxV9aZo0WUqpHzjiRtfltU2hkjYlyYD-e0c7uhBXl8c978JZk9XkJkPIOYMrxnJxzYXOuNZHJGFCFLTgUq1IAgUHChzYCVmH8AYAMs9lQmTpxnkXMQ5uQptW77N1_vtKXZ8-7WwcQsTGmrSyGOLQpo-Ti-GUHPdogzk75Ia83lYv5T3dPt89lDdbijznkTIuTCNFhxJaLnKtpM5Mpw0UxggsesaZ7pSQGltouGqwyNT-EaRSmcoR-IZcLrutdyF409ezH0b0HzWD-ku3PujuSfqHbIdFK3oc7D_8xcJjO_6O_pSfvKNe7g |
| CitedBy_id | crossref_primary_10_1073_pnas_2405744121 crossref_primary_10_1145_3687967 |
| Cites_doi | 10.1007/s00211-020-01156-6 10.1142/s0218216521500759 10.1145/1141911.1142012 10.1007/s00205-017-1100-9 10.1145/3460775 10.1142/S0218195995000064 10.1002/9780470977811 10.1103/PhysRevE.52.1176 10.1103/PhysRevLett.99.164301 10.1007/s00283-014-9472-2 10.1002/1097-0282(20001015)54:5<307::AID-BIP20>3.0.CO;2-Y 10.1103/PhysRevE.70.011803 10.1109/IM.2001.924423 10.1007/s10955-011-0164-4 10.4310/CAG.2020.v28.n2.a2 10.1016/0166-8641(93)90079-S 10.1515/9783110875911.323 10.1112/jlms/s2-30.3.512 10.1007/BF01277546 10.1111/1467-8659.00594 10.1088/0305-4470/24/23/021 10.21136/CMJ.1961.100486 10.1111/cgf.13519 10.4230/LIPIcs.SoCG.2020.25 10.48550/arXiv.math/0401051 10.1016/0370-2693(81)90545-1 10.1073/pnas.96.9.4769 10.1016/0040-9383(85)90046-1 10.1145/3450626.3459767 10.1016/B978-0-12-480440-1.50025-3 10.1145/1360612.1360662 10.1111/j.1467-8659.2008.01147.x 10.1016/S0960-0779(97)00109-4 10.1007/s00208-021-02346-9 10.1016/j.jmps.2018.03.019 10.1016/j.eml.2021.101172 10.1126/science.286.5437.11a 10.1007/978-1-4757-5362-2_15 10.1214/15-AAP1100 10.1134/S1061920814040013 10.1016/j.cma.2014.05.017 10.1023/A:1010911113919 10.1007/s41468-017-0007-8 10.1038/scientificamerican0983-18 10.1073/pnas.9.3.93 10.1073/pnas.96.10.5482 10.1016/S0006-3495(00)76277-1 10.1088/0964-1726/22/1/014005 10.1142/5229 10.1016/j.jmps.2009.05.004 10.1515/9783110571493-002 10.1007/978-1-4612-4132-4 10.1039/b509983h 10.1142/S0218216506004543 10.1080/00029890.1972.11993052 10.1088/1751-8113/42/47/475006 10.1103/PhysRevLett.82.3003 10.1007/s00222-002-0234-y 10.1007/978-3-319-76965-3 10.1007/s00004-013-0154-8 10.1090/mcom/3633 10.2307/1969467 10.1016/S0006-3495(98)77960-3 10.1007/978-1-4615-2025-2_11 10.1145/3306346.3323040 10.1115/DETC2009-86990 10.1145/1778765.1778853 10.1007/s00466-012-0683-0 10.2312/SCA/SCA07/063-072 10.1145/3453477 10.1007/978-3-662-03315-9 10.1098/rsta.2004.1393 10.1007/978-3-319-50598-5 10.1145/3439429 10.1103/PhysRevE.75.031801 10.1115/1.4050238 10.1145/3386569.3392425 10.1038/s41586-020-03123-5 10.1021/acs.jctc.8b00169 |
| ContentType | Journal Article |
| Copyright | ACM |
| Copyright_xml | – notice: ACM |
| DBID | AAYXX CITATION |
| DOI | 10.1145/3592399 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering |
| EISSN | 1557-7368 |
| EndPage | 16 |
| ExternalDocumentID | 10_1145_3592399 3592399 |
| GrantInformation_xml | – fundername: Swiss National Science Foundation grantid: FNS 514543 / CF 1156 – fundername: Natural Sciences and Engineering Research Council of Canada grantid: RGPIN-2021-03733 funderid: http://dx.doi.org/10.13039/501100000038 |
| GroupedDBID | --Z -DZ -~X .DC 23M 2FS 4.4 5GY 5VS 6J9 85S 8US AAKMM AALFJ AAYFX ABPPZ ACGFO ACGOD ACM ADBCU ADL ADMLS ADPZR AEBYY AENEX AENSD AFWIH AFWXC AIKLT ALMA_UNASSIGNED_HOLDINGS ASPBG AVWKF BDXCO CCLIF CS3 F5P FEDTE GUFHI HGAVV I07 LHSKQ P1C P2P PQQKQ RNS ROL TWZ UHB UPT W7O WH7 XSW ZCA ~02 9M8 AAFWJ AAYOK AAYXX ABFSI AEFXT AEJOY AETEA AI. AKRVB CITATION E.L EBS EJD HF~ MVM NHB OHT UKR VH1 XJT XOL ZY4 |
| ID | FETCH-LOGICAL-a343t-135eb65da60c35498692ed9e07ee5a7f1319d8569ac0b38ba7283430688284a03 |
| ISSN | 0730-0301 |
| IngestDate | Thu Jul 03 08:19:43 EDT 2025 Thu Apr 24 22:53:21 EDT 2025 Fri Feb 21 01:26:13 EST 2025 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | physics-based simulation elastic knots multistability numerical optimization fabrication |
| Language | English |
| License | Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-a343t-135eb65da60c35498692ed9e07ee5a7f1319d8569ac0b38ba7283430688284a03 |
| ORCID | 0009-0009-2373-9581 0000-0002-8331-2354 0000-0001-6229-3724 0000-0003-4460-7747 0000-0003-4957-4825 |
| OpenAccessLink | https://infoscience.epfl.ch/record/302408/files/paper.pdf |
| PageCount | 16 |
| ParticipantIDs | crossref_primary_10_1145_3592399 crossref_citationtrail_10_1145_3592399 acm_primary_3592399 |
| PublicationCentury | 2000 |
| PublicationDate | 2023-08-01 |
| PublicationDateYYYYMMDD | 2023-08-01 |
| PublicationDate_xml | – month: 08 year: 2023 text: 2023-08-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | New York, NY, USA |
| PublicationPlace_xml | – name: New York, NY, USA |
| PublicationTitle | ACM transactions on graphics |
| PublicationTitleAbbrev | ACM TOG |
| PublicationYear | 2023 |
| Publisher | ACM |
| Publisher_xml | – name: ACM |
| References | Sergey Avvakumov and Alexey Sossinsky. 2014. On the Normal Form of Knots. Russian Journal of Mathematical Physics 21 (2014), 421--429. 10.1134/S1061920814040013 Neal Madras and Gordon Slade. 1996. The Self-Avoiding Walk. Birkhäuser Boston, Boston, MA. 10.1007/978-1-4612-4132-4 Minchen Li, Zachary Ferguson, Teseo Schneider, Timothy Langlois, Denis Zorin, Daniele Panozzo, Chenfanfu Jiang, and Danny M. Kaufman. 2020. Incremental potential contact: intersection-and inversion-free, large-deformation dynamics. ACM Trans. Graph. 39, 4 (2020), 1--20. 10.1145/3386569.3392425 Sotero Alvarado, Jorge Alberto Calvo, and Kenneth C. Millett. 2011. The Generation of Random Equilateral Polygons. Journal of Statistical Physics 143, 1 (2011), 102--138. 10.1007/s10955-011-0164-4 Minchen Li, Danny M. Kaufman, and Chenfanfu Jiang. 2021. Codimensional incremental potential contact. ACM Trans. Graph. 40, 4 (2021), 1--24. 10.1145/3450626.3459767 Tian Chen, Mark Pauly, and Pedro M. Reis. 2021. A reprogrammable mechanical metamaterial with stable memory. Nature 589, 7842 (2021), 386--390. 10.1038/s41586-020-03123-5 Dmitri Kozlov. 2013. Structures of Periodical Knots and Links as Geometric Models of Complex Surfaces for Designing. Nexus Network Journal 15, 2 (2013), 241--255. 10.1007/s00004-013-0154-8 Florence Bertails, Basile Audoly, Marie-Paule Cani, Bernard Querleux, Frédéric Leroy, and Jean-Luc Lévëque. 2006. Super-Helices for Predicting the Dynamics of Natural Hair. ACM Trans. Graph. 25, 3 (2006), 1180--1187. 10.1145/1141911.1142012 James W. Alexander. 1923. A Lemma on Systems of Knotted Curves. Proceedings of the National Academy of Sciences of the United States of America 9, 3 (1923), 93--95. Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete Elastic Rods. ACM Trans. Graph. 27, 3 (2008), 1--12. 10.1145/1360612.1360662 Ryan Blair, Alexandra Kjuchukova, Roman Velazquez, and Paul Villanueva. 2020. Wirtinger systems of generators of knot groups. Communications in Analysis and Geometry 28, 2 (2020), 243--262. 10.4310/CAG.2020.v28.n2.a2 Bolun Wang, Zachary Ferguson, Teseo Schneider, Xin Jiang, Marco Attene, and Daniele Panozzo. 2021. A Large-scale Benchmark and an Inclusion-based Algorithm for Continuous Collision Detection. ACM Trans. Graph. 40, 5 (2021), 1--16. 10.1145/3460775 E. J. Janse van Rensburg and S. G. Whittington. 1991. The BFACF algorithm and knotted polygons. Journal of Physics A: Mathematical and General 24, 23 (1991), 5553--5567. 10.1088/0305-4470/24/23/021 Peter Guthrie Tait. 1884. On Knots, Part II. Transactions of the Royal Society of Edinburgh 32 (1884), 318--334. G. M. Guisewite. 1995. Network Problems. Springer US, Boston, MA, 609--648. 10.1007/978-1-4615-2025-2_11 Jonas Spillmann and Matthias Teschner. 2008. An Adaptive Contact Model for the Robust Simulation of Knots. Computer Graphics Forum 27, 2 (2008), 497--506. 10.1111/j.1467-8659.2008.01147.x Thomas A. Gittings. 2004. Minimum Braids: A Complete Invariant of Knots and Links. (2004). 10.48550/arXiv.math/0401051 Henryk Gerlach, Philipp Reiter, and Heiko von der Mosel. 2017. The elastic trefoil is the twice covered circle. Archive for Rational Mechanics and Analysis 225, 1 (2017), 89--139. 10.1007/s00205-017-1100-9 Benjamin A. Burton. 2020. The Next 350 Million Knots. In 36th International Symposium on Computational Geometry (SoCG 2020), Vol. 164. 25:1--25:17. 10.4230/LIPIcs.SoCG.2020.25 Alexandra Gilsbach, Philipp Reiter, and Heiko von der Mosel. 2021. Symmetric elastic knots. Math. Ann. 385 (2021), 811--844. 10.1007/s00208-021-02346-9 Gregory Buck and Eric J. Rawdon. 2004. Role of flexibility in entanglement. Physical Review E 70, 1 (2004), 011803. 10.1103/PhysRevE.70.011803 Konstantin Klenin and Jörg Langowski. 2000. Computation of writhe in modeling of supercoiled DNA. Biopolymers 54, 5 (2000), 307--317. Adam Liwo, Jooyoung Lee, Daniel R Ripoll, Jaroslaw Pillardy, and Harold A Scheraga. 1999. Protein structure prediction by global optimization of a potential energy function. Proceedings of the National Academy of Sciences 96, 10 (1999), 5482--5485. Michalewicz Zbigniew. 1996. Genetic algorithms + data structures = evolution programs. In Computational Statistics. Springer-Verlag, 372--373. 10.1007/978-3-662-03315-9 Jun O'Hara. 2003. Energy of knots and conformal geometry. Series on knots and everything, Vol. 33. World Scientific Pub. 10.1142/5229 Alain C. Vaucher and Markus Reiher. 2018. Minimum Energy Paths and Transition States by Curve Optimization. Journal of Chemical Theory and Computation 14, 6 (2018), 3091--3099. 10.1021/acs.jctc.8b00169 Zachary Ferguson et al. 2020. IPC Toolkit. https://ipc-sim.github.io/ipc-toolkit Dinesh K. Pai. 2002. STRANDS: Interactive Simulation of Thin Solids using Cosserat Models. Computer Graphics Forum 21, 3 (2002), 347--352. 10.1111/1467-8659.00594 Robert G. Scharein. 1998. Interactive Topological Drawing. Ph. D. Dissertation. Department of Computer Science, The University of British Columbia. Paweł Strzelecki and Heiko von der Mosel. 2017. Geometric curvature energies: facts, trends, and open problems. In New Directions in Geometric and Applied Knot Theory. 8--35. 10.1515/9783110571493-002 Christoph Meier, Alexander Popp, and Wolfgang A. Wall. 2014. An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Computer Methods in Applied Mechanics and Engineering 278 (2014), 445--478. 10.1016/j.cma.2014.05.017 Richard A. Hord. 1972. Torsion at an inflection point of a space curve. The American Mathematical Monthly 79, 4 (1972), 371--374. Joel Hass. 1998. Algorithms for recognizing knots and 3-manifolds. Chaos, Solitons & Fractals 9, 4--5 (1998), 569--581. 10.1016/S0960-0779(97)00109-4 C. Livingston and A. H. Moore. 2022. KnotInfo: Table of Knot Invariants. https://knotinfo.math.indiana.edu/index.php Chengzi Liang and Kurt Mislow. 1994. On amphicheiral knots. Journal of Mathematical Chemistry 15, 1 (1994), 1--34. 10.1007/BF01277546 Andrew Choi, Dezhong Tong, Mohammad K. Jawed, and Jungseock Joo. 2021. Implicit Contact Model for Discrete Elastic Rods in Knot Tying. Journal of Applied Mechanics 88, 5 (2021), 051010. 10.1115/1.4050238 Yuanan Diao, Claus Ernst, and Philipp Reiter. 2021. Knots with Equal Bridge Index and Braid Index. Journal of Knot Theory and Its Ramifications (2021). 10.1142/s0218216521500759 Paul Johanns, Paul Grandgeorge, Changyeob Baek, Tomohiko G. Sano, John H. Maddocks, and Pedro M. Reis. 2021. The shapes of physical trefoil knots. Extreme Mechanics Letters 43 (2021), 101172. 10.1016/j.eml.2021.101172 Wolfgang Wenzel and Kay Hamacher. 1999. Stochastic tunneling approach for global minimization of complex potential energy landscapes. Physical Review Letters 82, 15 (1999), 3003. Miklós Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2010. Discrete Viscous Threads. ACM Trans. Graph. 29, 4 (2010), 1--10. 10.1145/1778765.1778853 Sören Bartels and Philipp Reiter. 2021. Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves. Math. Comp. 90, 330 (2021), 1499--1526. 10.1090/mcom/3633 Patrick B. Furrer, Robert S. Manning, and John H. Maddocks. 2000. DNA Rings with Multiple Energy Minima. Biophysical Journal 79, 1 (2000), 116--136. 10.1016/S0006-3495(00)76277-1 K. Kodama and M. Sakuma. 1992. Symmetry groups of prime knots up to 10 crossings. In Knot 90, Proceedings of the International Conference on Knot Theory and Related Topics. 10.1515/9783110875911.323 Joel Langer and David A. Singer. 1984. Knotted Elastic Curves in R3. Journal of the London Mathematical Society s2-30, 3 (1984), 512--520. 10.1112/jlms/s2-30.3.512 Nicolas Clauvelin, Basile Audoly, and Sébastien Neukirch. 2009. Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non- trivial contact topology. Journal of the Mechanics and Physics of Solids 57, 9 (2009), 1623--1656. 10.1016/j.jmps.2009.05.004 Basile Audoly, Nicolas Clauvelin, and Sébastien Neukirch. 2007. Elastic Knots. Physical review letters 99 (2007), 164301. 10.1103/PhysRevLett.99.164301 Bernard Coleman and David Swigon. 2000. Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids. Journal of Elasticity 60 (2000), 173--221. 10.1023/A:1010911113919 Shinji Fukuhara. 1988. Energy of a Knot. In A Fëte of Topology. Elsevier, 443--451. 10.1016/B978-0-12-480440-1.50025-3 Eric J. Rawdon and Jonathan K. Simon. 2006. Polygonal Approximation and Energy of Smooth Knots. Journal of Knot Theory and Its Ramifications 15, 04 (2006), 429--451. 10.1142/S0218216506004543 Oscar Gonzalez and John H. Maddocks. 1999. Global curvature, thickness, and the ideal shapes of knots. Proceedings of the National Academy of Sciences 96, 9 (1999), 4769--4773. 10.1073/pnas.96.9.4769 Andrzej Stasiak, Akos Dobay, Jacques Dubochet, and Giovanni Dietler. 1999. Knotted Fishing Line, Covalent Bonds, and Breaking Points. Science 286, 5437 (1999). 10.1126/science.286.5437.11a Chris Yu, Henrik Schumacher, and Keenan Crane. 2021. Repulsive Curves. ACM Trans. Graph. 40, 2 (2021), 1--21. 10.1145/3439429 Martin Gardner. 1983. Mathematical Games. Scientific American 249, 3 (1983), 18--E8. Jason Cantarella, Robert B. Kusner, and John M. Sullivan. 2002. On the minimum ropelength of knots and links. Inventiones mathematicae 150, 2 (2002), 257--286. 10.1007/s00222-002-0234-y Colin C. Adams. 2004. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Soc. Sören Bartels and Philipp Reiter. 2020. Numerical solution of a bending-torsion model for elastic rods. Numer. Math. 146, 4 (2020), 661--697. 10.1007/s00211-020-01156-6 Szymon Rusinkiewicz and Marc Levoy. 2001. Efficient Variants of the ICP Algorithm. In Proceedings of the Third Intl. Conf. on 3D Digital Imaging and Modeling. 145--152. 10.1109/IM.2001.924423 Jason Cantarella and Clayton Shonkwiler. 2016. The symplectic geometry of closed equilateral random walks in 3-space. e_1_2_2_4_1 e_1_2_2_24_1 e_1_2_2_49_1 e_1_2_2_6_1 e_1_2_2_22_1 e_1_2_2_20_1 e_1_2_2_2_1 e_1_2_2_41_1 e_1_2_2_62_1 e_1_2_2_43_1 e_1_2_2_64_1 e_1_2_2_85_1 e_1_2_2_8_1 e_1_2_2_28_1 e_1_2_2_45_1 e_1_2_2_66_1 e_1_2_2_26_1 e_1_2_2_47_1 e_1_2_2_68_1 e_1_2_2_83_1 e_1_2_2_60_1 e_1_2_2_81_1 e_1_2_2_13_1 e_1_2_2_38_1 e_1_2_2_59_1 e_1_2_2_11_1 e_1_2_2_30_1 e_1_2_2_51_1 e_1_2_2_76_1 e_1_2_2_19_1 e_1_2_2_32_1 e_1_2_2_53_1 e_1_2_2_74_1 e_1_2_2_17_1 e_1_2_2_34_1 e_1_2_2_55_1 e_1_2_2_15_1 e_1_2_2_36_1 e_1_2_2_57_1 e_1_2_2_78_1 e_1_2_2_70_1 e_1_2_2_25_1 e_1_2_2_48_1 e_1_2_2_5_1 e_1_2_2_23_1 e_1_2_2_7_1 e_1_2_2_21_1 e_1_2_2_1_1 Scharein Robert G. (e_1_2_2_72_1) e_1_2_2_3_1 e_1_2_2_40_1 e_1_2_2_63_1 e_1_2_2_42_1 e_1_2_2_84_1 e_1_2_2_9_1 e_1_2_2_29_1 e_1_2_2_44_1 e_1_2_2_67_1 e_1_2_2_27_1 e_1_2_2_46_1 e_1_2_2_69_1 e_1_2_2_82_1 e_1_2_2_61_1 e_1_2_2_80_1 e_1_2_2_14_1 e_1_2_2_37_1 e_1_2_2_12_1 e_1_2_2_39_1 e_1_2_2_10_1 John (e_1_2_2_65_1) 2006 e_1_2_2_52_1 e_1_2_2_75_1 e_1_2_2_31_1 e_1_2_2_54_1 e_1_2_2_73_1 e_1_2_2_18_1 e_1_2_2_33_1 e_1_2_2_56_1 e_1_2_2_16_1 e_1_2_2_35_1 e_1_2_2_58_1 e_1_2_2_77_1 Tait Peter Guthrie (e_1_2_2_79_1) 1884; 32 e_1_2_2_50_1 e_1_2_2_71_1 |
| References_xml | – reference: Chaim Even-Zohar. 2017. Models of random knots. Journal of Applied and Computational Topology 1, 2 (2017), 263--296. 10.1007/s41468-017-0007-8 – reference: Robert G. Scharein. 1998. Interactive Topological Drawing. Ph. D. Dissertation. Department of Computer Science, The University of British Columbia. – reference: J. W. Milnor. 1950. On the Total Curvature of Knots. The Annals of Mathematics 52, 2 (1950), 248. 10.2307/1969467 – reference: Paul Johanns, Paul Grandgeorge, Changyeob Baek, Tomohiko G. Sano, John H. Maddocks, and Pedro M. Reis. 2021. The shapes of physical trefoil knots. Extreme Mechanics Letters 43 (2021), 101172. 10.1016/j.eml.2021.101172 – reference: Adam Liwo, Jooyoung Lee, Daniel R Ripoll, Jaroslaw Pillardy, and Harold A Scheraga. 1999. Protein structure prediction by global optimization of a potential energy function. Proceedings of the National Academy of Sciences 96, 10 (1999), 5482--5485. – reference: B. Berg and D. Foerster. 1981. Random paths and random surfaces on a digital computer. Phys. Lett. B 106, 4 (1981), 323--326. 10.1016/0370-2693(81)90545-1 – reference: Sören Bartels and Philipp Reiter. 2020. Numerical solution of a bending-torsion model for elastic rods. Numer. Math. 146, 4 (2020), 661--697. 10.1007/s00211-020-01156-6 – reference: Stephen R. Quake. 1995. Fast Monte Carlo algorithms for knotted polymers. Physical Review E 52, 1 (1995), 1176--1180. 10.1103/PhysRevE.52.1176 – reference: Carlota Soler, Tobias Martin, and Olga Sorkine-Hornung. 2018. Cosserat Rods with Projective Dynamics. Computer Graphics Forum 37, 8 (2018), 137--147. 10.1111/cgf.13519 – reference: David Swigon, Bernard D. Coleman, and Irwin Tobias. 1998. The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophysical Journal 74, 5 (1998), 2515--2530. 10.1016/S0006-3495(98)77960-3 – reference: Chris Yu, Henrik Schumacher, and Keenan Crane. 2021. Repulsive Curves. ACM Trans. Graph. 40, 2 (2021), 1--21. 10.1145/3439429 – reference: Michalewicz Zbigniew. 1996. Genetic algorithms + data structures = evolution programs. In Computational Statistics. Springer-Verlag, 372--373. 10.1007/978-3-662-03315-9 – reference: Minchen Li, Zachary Ferguson, Teseo Schneider, Timothy Langlois, Denis Zorin, Daniele Panozzo, Chenfanfu Jiang, and Danny M. Kaufman. 2020. Incremental potential contact: intersection-and inversion-free, large-deformation dynamics. ACM Trans. Graph. 39, 4 (2020), 1--20. 10.1145/3386569.3392425 – reference: R. Gallotti and O. Pierre-Louis. 2007. Stiff knots. Physical Review E 75, 3 (2007), 031801. 10.1103/PhysRevE.75.031801 – reference: Zachary Ferguson et al. 2020. IPC Toolkit. https://ipc-sim.github.io/ipc-toolkit/ – reference: Helmut Alt and Michael Godau. 1995. Computing the Fréchet Distance between Two Polygonal Curves. Int. J. Comput. Geometry Appl. 5 (1995), 75--91. 10.1142/S0218195995000064 – reference: Sören Bartels and Philipp Reiter. 2021. Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves. Math. Comp. 90, 330 (2021), 1499--1526. 10.1090/mcom/3633 – reference: Jason Cantarella, Robert B. Kusner, and John M. Sullivan. 2002. On the minimum ropelength of knots and links. Inventiones mathematicae 150, 2 (2002), 257--286. 10.1007/s00222-002-0234-y – reference: Neal Madras and Gordon Slade. 1996. The Self-Avoiding Walk. Birkhäuser Boston, Boston, MA. 10.1007/978-1-4612-4132-4 – reference: M. Khalid Jawed, Alyssa Novelia, and Oliver M. O'Reilly. 2018. A Primer on the Kinematics of Discrete Elastic Rods. Springer International Publishing, Cham. 10.1007/978-3-319-76965-3 – reference: Florence Bertails, Basile Audoly, Marie-Paule Cani, Bernard Querleux, Frédéric Leroy, and Jean-Luc Lévëque. 2006. Super-Helices for Predicting the Dynamics of Natural Hair. ACM Trans. Graph. 25, 3 (2006), 1180--1187. 10.1145/1141911.1142012 – reference: Henryk Gerlach, Philipp Reiter, and Heiko von der Mosel. 2017. The elastic trefoil is the twice covered circle. Archive for Rational Mechanics and Analysis 225, 1 (2017), 89--139. 10.1007/s00205-017-1100-9 – reference: Shinji Fukuhara. 1988. Energy of a Knot. In A Fëte of Topology. Elsevier, 443--451. 10.1016/B978-0-12-480440-1.50025-3 – reference: Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete Elastic Rods. ACM Trans. Graph. 27, 3 (2008), 1--12. 10.1145/1360612.1360662 – reference: C. Livingston and A. H. Moore. 2022. KnotInfo: Table of Knot Invariants. https://knotinfo.math.indiana.edu/index.php – reference: Peter Guthrie Tait. 1884. On Knots, Part II. Transactions of the Royal Society of Edinburgh 32 (1884), 318--334. – reference: Martin Gardner. 1983. Mathematical Games. Scientific American 249, 3 (1983), 18--E8. – reference: Oscar Gonzalez and John H. Maddocks. 1999. Global curvature, thickness, and the ideal shapes of knots. Proceedings of the National Academy of Sciences 96, 9 (1999), 4769--4773. 10.1073/pnas.96.9.4769 – reference: Tian Chen, Mark Pauly, and Pedro M. Reis. 2021. A reprogrammable mechanical metamaterial with stable memory. Nature 589, 7842 (2021), 386--390. 10.1038/s41586-020-03123-5 – reference: Nicolas Clauvelin, Basile Audoly, and Sébastien Neukirch. 2009. Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non- trivial contact topology. Journal of the Mechanics and Physics of Solids 57, 9 (2009), 1623--1656. 10.1016/j.jmps.2009.05.004 – reference: Szymon Rusinkiewicz and Marc Levoy. 2001. Efficient Variants of the ICP Algorithm. In Proceedings of the Third Intl. Conf. on 3D Digital Imaging and Modeling. 145--152. 10.1109/IM.2001.924423 – reference: Oliver M. O'Reilly. 2017. Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Springer International Publishing. 10.1007/978-3-319-50598-5 – reference: Patrick B. Furrer, Robert S. Manning, and John H. Maddocks. 2000. DNA Rings with Multiple Energy Minima. Biophysical Journal 79, 1 (2000), 116--136. 10.1016/S0006-3495(00)76277-1 – reference: Wolfgang Wenzel and Kay Hamacher. 1999. Stochastic tunneling approach for global minimization of complex potential energy landscapes. Physical Review Letters 82, 15 (1999), 3003. – reference: Bernard Coleman and David Swigon. 2000. Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids. Journal of Elasticity 60 (2000), 173--221. 10.1023/A:1010911113919 – reference: Gregory Buck and Jeremey Orloff. 1993. Computing canonical conformations for knots. Topology and its Applications 51, 3 (1993), 247--253. 10.1016/0166-8641(93)90079-S – reference: John G. Proakis and Dimitris K Manolakis. 2006. Digital Signal Processing (4 ed.). Prentice Hall. – reference: Ryan Blair, Alexandra Kjuchukova, Roman Velazquez, and Paul Villanueva. 2020. Wirtinger systems of generators of knot groups. Communications in Analysis and Geometry 28, 2 (2020), 243--262. 10.4310/CAG.2020.v28.n2.a2 – reference: Yuanan Diao, Claus Ernst, and Philipp Reiter. 2021. Knots with Equal Bridge Index and Braid Index. Journal of Knot Theory and Its Ramifications (2021). 10.1142/s0218216521500759 – reference: Dinesh K. Pai. 2002. STRANDS: Interactive Simulation of Thin Solids using Cosserat Models. Computer Graphics Forum 21, 3 (2002), 347--352. 10.1111/1467-8659.00594 – reference: Eric J. Rawdon and Jonathan K. Simon. 2006. Polygonal Approximation and Energy of Smooth Knots. Journal of Knot Theory and Its Ramifications 15, 04 (2006), 429--451. 10.1142/S0218216506004543 – reference: Damien Durville. 2012. Contact-friction modeling within elastic beam assemblies: an application to knot tightening. Computational Mechanics 49, 6 (2012), 687--707. 10.1007/s00466-012-0683-0 – reference: K. Kodama and M. Sakuma. 1992. Symmetry groups of prime knots up to 10 crossings. In Knot 90, Proceedings of the International Conference on Knot Theory and Related Topics. 10.1515/9783110875911.323 – reference: Joel Hass. 1998. Algorithms for recognizing knots and 3-manifolds. Chaos, Solitons & Fractals 9, 4--5 (1998), 569--581. 10.1016/S0960-0779(97)00109-4 – reference: G. M. Guisewite. 1995. Network Problems. Springer US, Boston, MA, 609--648. 10.1007/978-1-4615-2025-2_11 – reference: Jonas Spillmann and Matthias Teschner. 2007. CORDE: Cosserat Rod Elements for the Dynamic Simulation of One-Dimensional Elastic Objects. In Eurographics/SIGGRAPH Symposium on Computer Animation. 10.2312/SCA/SCA07/063-072 – reference: Ran Zhang, Thomas Auzinger, and Bernd Bickel. 2021. Computational Design of Planar Multistable Compliant Structures. ACM Trans. Graph. 40, 5 (2021), 1--16. 10.1145/3453477 – reference: R Scharein, K Ishihara, J Arsuaga, Y Diao, K Shimokawa, and M Vazquez. 2009. Bounds for the minimum step number of knots in the simple cubic lattice. Journal of Physics A: Mathematical and Theoretical 42, 47 (2009). 10.1088/1751-8113/42/47/475006 – reference: Jun O'Hara. 2003. Energy of knots and conformal geometry. Series on knots and everything, Vol. 33. World Scientific Pub. 10.1142/5229 – reference: Gregory Buck and Eric J. Rawdon. 2004. Role of flexibility in entanglement. Physical Review E 70, 1 (2004), 011803. 10.1103/PhysRevE.70.011803 – reference: Daniel Carroll, Eleanor Hankins, Emek Kose, and Ivan Sterling. 2014. A Survey of the Differential Geometry of Discrete Curves. The Mathematical Intelligencer 36, 4 (2014), 28--35. 10.1007/s00283-014-9472-2 – reference: Brian S. Everitt, Sabine Landau, Morven Leese, and Daniel Stahl. 2011. Cluster Analysis (1 ed.). Wiley. 10.1002/9780470977811 – reference: Bolun Wang, Zachary Ferguson, Teseo Schneider, Xin Jiang, Marco Attene, and Daniele Panozzo. 2021. A Large-scale Benchmark and an Inclusion-based Algorithm for Continuous Collision Detection. ACM Trans. Graph. 40, 5 (2021), 1--16. 10.1145/3460775 – reference: Chengzi Liang and Kurt Mislow. 1994. On amphicheiral knots. Journal of Mathematical Chemistry 15, 1 (1994), 1--34. 10.1007/BF01277546 – reference: Alain C. Vaucher and Markus Reiher. 2018. Minimum Energy Paths and Transition States by Curve Optimization. Journal of Chemical Theory and Computation 14, 6 (2018), 3091--3099. 10.1021/acs.jctc.8b00169 – reference: Thomas A. Gittings. 2004. Minimum Braids: A Complete Invariant of Knots and Links. (2004). 10.48550/arXiv.math/0401051 – reference: Julian Panetta, Mina Konaković-Luković, Florin Isvoranu, Etienne Bouleau, and Mark Pauly. 2019. X-Shells: a new class of deployable beam structures. ACM Trans. Graph. 38, 4 (2019), 1--15. 10.1145/3306346.3323040 – reference: Bernard Coleman and David Swigon. 2004. Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 362 (2004), 1281--99. 10.1098/rsta.2004.1393 – reference: Benjamin A. Burton. 2020. The Next 350 Million Knots. In 36th International Symposium on Computational Geometry (SoCG 2020), Vol. 164. 25:1--25:17. 10.4230/LIPIcs.SoCG.2020.25 – reference: Sotero Alvarado, Jorge Alberto Calvo, and Kenneth C. Millett. 2011. The Generation of Random Equilateral Polygons. Journal of Statistical Physics 143, 1 (2011), 102--138. 10.1007/s10955-011-0164-4 – reference: Miklós Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2010. Discrete Viscous Threads. ACM Trans. Graph. 29, 4 (2010), 1--10. 10.1145/1778765.1778853 – reference: János D. Pintér. 2002. Global Optimization: Software, Test Problems, and Applications. Springer US, Boston, MA, 515--569. 10.1007/978-1-4757-5362-2_15 – reference: Christoph Meier, Alexander Popp, and Wolfgang A. Wall. 2014. An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Computer Methods in Applied Mechanics and Engineering 278 (2014), 445--478. 10.1016/j.cma.2014.05.017 – reference: Basile Audoly, Nicolas Clauvelin, and Sébastien Neukirch. 2007. Elastic Knots. Physical review letters 99 (2007), 164301. 10.1103/PhysRevLett.99.164301 – reference: Minchen Li, Danny M. Kaufman, and Chenfanfu Jiang. 2021. Codimensional incremental potential contact. ACM Trans. Graph. 40, 4 (2021), 1--24. 10.1145/3450626.3459767 – reference: Jason Cantarella and Clayton Shonkwiler. 2016. The symplectic geometry of closed equilateral random walks in 3-space. The Annals of Applied Probability 26, 1 (2016), 549 -- 596. 10.1214/15-AAP1100 – reference: Gi-Woo Kim and Jaehwan Kim. 2012. Compliant bistable mechanism for low frequency vibration energy harvester inspired by auditory hair bundle structures. Smart Materials and Structures 22, 1 (2012). 10.1088/0964-1726/22/1/014005 – reference: Andrew Choi, Dezhong Tong, Mohammad K. Jawed, and Jungseock Joo. 2021. Implicit Contact Model for Discrete Elastic Rods in Knot Tying. Journal of Applied Mechanics 88, 5 (2021), 051010. 10.1115/1.4050238 – reference: Richard A. Hord. 1972. Torsion at an inflection point of a space curve. The American Mathematical Monthly 79, 4 (1972), 371--374. – reference: Colin C. Adams. 2004. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Soc. – reference: Alexandra Gilsbach, Philipp Reiter, and Heiko von der Mosel. 2021. Symmetric elastic knots. Math. Ann. 385 (2021), 811--844. 10.1007/s00208-021-02346-9 – reference: Sergey Avvakumov and Alexey Sossinsky. 2014. On the Normal Form of Knots. Russian Journal of Mathematical Physics 21 (2014), 421--429. 10.1134/S1061920814040013 – reference: Joel Langer and David A. Singer. 1984. Knotted Elastic Curves in R3. Journal of the London Mathematical Society s2-30, 3 (1984), 512--520. 10.1112/jlms/s2-30.3.512 – reference: Dmitri Kozlov. 2013. Structures of Periodical Knots and Links as Geometric Models of Complex Surfaces for Designing. Nexus Network Journal 15, 2 (2013), 241--255. 10.1007/s00004-013-0154-8 – reference: Adam Roetter, Craig Lusk, and Rajiv Dubey. 2009. Bistable Compliant Extension Aid for a Polycentric Prosthetic Knee (International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 7: 33rd Mechanisms and Robotics Conference). 241--248. 10.1115/DETC2009-86990 – reference: Paweł Strzelecki and Heiko von der Mosel. 2017. Geometric curvature energies: facts, trends, and open problems. In New Directions in Geometric and Applied Knot Theory. 8--35. 10.1515/9783110571493-002 – reference: G. Călugăreanu. 1961. Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Mathematical Journal 11 (1961), 588--625. 10.21136/CMJ.1961.100486 – reference: David J. Earl and Michael W. Deem. 2005. Parallel tempering: Theory, applications, and new perspectives. Physical Chemistry Chemical Physics 7, 23 (2005), 3910--3916. – reference: Konstantin Klenin and Jörg Langowski. 2000. Computation of writhe in modeling of supercoiled DNA. Biopolymers 54, 5 (2000), 307--317. – reference: James W. Alexander. 1923. A Lemma on Systems of Knotted Curves. Proceedings of the National Academy of Sciences of the United States of America 9, 3 (1923), 93--95. – reference: E. J. Janse van Rensburg and S. G. Whittington. 1991. The BFACF algorithm and knotted polygons. Journal of Physics A: Mathematical and General 24, 23 (1991), 5553--5567. 10.1088/0305-4470/24/23/021 – reference: Andrzej Stasiak, Akos Dobay, Jacques Dubochet, and Giovanni Dietler. 1999. Knotted Fishing Line, Covalent Bonds, and Breaking Points. Science 286, 5437 (1999). 10.1126/science.286.5437.11a – reference: Joel Langer and David A. Singer. 1985. Curve straightening and a minimax argument for closed elastic curves. Topology 24, 1 (1985), 75--88. 10.1016/0040-9383(85)90046-1 – reference: Jonas Spillmann and Matthias Teschner. 2008. An Adaptive Contact Model for the Robust Simulation of Knots. Computer Graphics Forum 27, 2 (2008), 497--506. 10.1111/j.1467-8659.2008.01147.x – reference: Derek E. Moulton, Paul Grandgeorge, and Sébastien Neukirch. 2018. Stable elastic knots with no self-contact. Journal of the Mechanics and Physics of Solids 116 (2018), 33--53. 10.1016/j.jmps.2018.03.019 – ident: e_1_2_2_7_1 doi: 10.1007/s00211-020-01156-6 – ident: e_1_2_2_26_1 doi: 10.1142/s0218216521500759 – ident: e_1_2_2_12_1 doi: 10.1145/1141911.1142012 – ident: e_1_2_2_36_1 doi: 10.1007/s00205-017-1100-9 – ident: e_1_2_2_81_1 doi: 10.1145/3460775 – ident: e_1_2_2_3_1 doi: 10.1142/S0218195995000064 – ident: e_1_2_2_54_1 – ident: e_1_2_2_30_1 doi: 10.1002/9780470977811 – ident: e_1_2_2_66_1 doi: 10.1103/PhysRevE.52.1176 – ident: e_1_2_2_5_1 doi: 10.1103/PhysRevLett.99.164301 – ident: e_1_2_2_19_1 doi: 10.1007/s00283-014-9472-2 – ident: e_1_2_2_46_1 doi: 10.1002/1097-0282(20001015)54:5<307::AID-BIP20>3.0.CO;2-Y – ident: e_1_2_2_15_1 doi: 10.1103/PhysRevE.70.011803 – ident: e_1_2_2_70_1 doi: 10.1109/IM.2001.924423 – ident: e_1_2_2_4_1 doi: 10.1007/s10955-011-0164-4 – ident: e_1_2_2_13_1 doi: 10.4310/CAG.2020.v28.n2.a2 – ident: e_1_2_2_14_1 doi: 10.1016/0166-8641(93)90079-S – ident: e_1_2_2_47_1 doi: 10.1515/9783110875911.323 – ident: e_1_2_2_31_1 – ident: e_1_2_2_49_1 doi: 10.1112/jlms/s2-30.3.512 – ident: e_1_2_2_53_1 doi: 10.1007/BF01277546 – ident: e_1_2_2_62_1 doi: 10.1111/1467-8659.00594 – ident: e_1_2_2_68_1 doi: 10.1088/0305-4470/24/23/021 – ident: e_1_2_2_25_1 doi: 10.21136/CMJ.1961.100486 – ident: e_1_2_2_73_1 doi: 10.1111/cgf.13519 – ident: e_1_2_2_16_1 doi: 10.4230/LIPIcs.SoCG.2020.25 – ident: e_1_2_2_38_1 doi: 10.48550/arXiv.math/0401051 – ident: e_1_2_2_9_1 doi: 10.1016/0370-2693(81)90545-1 – ident: e_1_2_2_39_1 doi: 10.1073/pnas.96.9.4769 – volume-title: Proakis and Dimitris K Manolakis year: 2006 ident: e_1_2_2_65_1 – ident: e_1_2_2_50_1 doi: 10.1016/0040-9383(85)90046-1 – ident: e_1_2_2_52_1 doi: 10.1145/3450626.3459767 – ident: e_1_2_2_32_1 doi: 10.1016/B978-0-12-480440-1.50025-3 – ident: e_1_2_2_11_1 doi: 10.1145/1360612.1360662 – ident: e_1_2_2_75_1 doi: 10.1111/j.1467-8659.2008.01147.x – ident: e_1_2_2_41_1 doi: 10.1016/S0960-0779(97)00109-4 – ident: e_1_2_2_37_1 doi: 10.1007/s00208-021-02346-9 – ident: e_1_2_2_59_1 doi: 10.1016/j.jmps.2018.03.019 – volume: 32 start-page: 318 year: 1884 ident: e_1_2_2_79_1 article-title: On Knots publication-title: Part II. Transactions of the Royal Society of Edinburgh – ident: e_1_2_2_44_1 doi: 10.1016/j.eml.2021.101172 – ident: e_1_2_2_76_1 doi: 10.1126/science.286.5437.11a – ident: e_1_2_2_64_1 doi: 10.1007/978-1-4757-5362-2_15 – ident: e_1_2_2_18_1 doi: 10.1214/15-AAP1100 – ident: e_1_2_2_6_1 doi: 10.1134/S1061920814040013 – ident: e_1_2_2_57_1 doi: 10.1016/j.cma.2014.05.017 – ident: e_1_2_2_23_1 doi: 10.1023/A:1010911113919 – ident: e_1_2_2_29_1 doi: 10.1007/s41468-017-0007-8 – ident: e_1_2_2_35_1 doi: 10.1038/scientificamerican0983-18 – ident: e_1_2_2_2_1 doi: 10.1073/pnas.9.3.93 – ident: e_1_2_2_55_1 doi: 10.1073/pnas.96.10.5482 – ident: e_1_2_2_33_1 doi: 10.1016/S0006-3495(00)76277-1 – ident: e_1_2_2_45_1 doi: 10.1088/0964-1726/22/1/014005 – ident: e_1_2_2_60_1 doi: 10.1142/5229 – ident: e_1_2_2_22_1 doi: 10.1016/j.jmps.2009.05.004 – ident: e_1_2_2_1_1 – ident: e_1_2_2_77_1 doi: 10.1515/9783110571493-002 – ident: e_1_2_2_56_1 doi: 10.1007/978-1-4612-4132-4 – ident: e_1_2_2_28_1 doi: 10.1039/b509983h – ident: e_1_2_2_67_1 doi: 10.1142/S0218216506004543 – ident: e_1_2_2_42_1 doi: 10.1080/00029890.1972.11993052 – ident: e_1_2_2_71_1 doi: 10.1088/1751-8113/42/47/475006 – ident: e_1_2_2_82_1 doi: 10.1103/PhysRevLett.82.3003 – ident: e_1_2_2_17_1 doi: 10.1007/s00222-002-0234-y – ident: e_1_2_2_43_1 doi: 10.1007/978-3-319-76965-3 – ident: e_1_2_2_48_1 doi: 10.1007/s00004-013-0154-8 – ident: e_1_2_2_8_1 doi: 10.1090/mcom/3633 – ident: e_1_2_2_58_1 doi: 10.2307/1969467 – ident: e_1_2_2_78_1 doi: 10.1016/S0006-3495(98)77960-3 – ident: e_1_2_2_40_1 doi: 10.1007/978-1-4615-2025-2_11 – ident: e_1_2_2_63_1 doi: 10.1145/3306346.3323040 – ident: e_1_2_2_69_1 doi: 10.1115/DETC2009-86990 – ident: e_1_2_2_10_1 doi: 10.1145/1778765.1778853 – ident: e_1_2_2_27_1 doi: 10.1007/s00466-012-0683-0 – ident: e_1_2_2_74_1 doi: 10.2312/SCA/SCA07/063-072 – ident: e_1_2_2_85_1 doi: 10.1145/3453477 – ident: e_1_2_2_84_1 doi: 10.1007/978-3-662-03315-9 – ident: e_1_2_2_24_1 doi: 10.1098/rsta.2004.1393 – volume-title: Interactive Topological Drawing. Ph. D. Dissertation. Department of Computer Science ident: e_1_2_2_72_1 – ident: e_1_2_2_61_1 doi: 10.1007/978-3-319-50598-5 – ident: e_1_2_2_83_1 doi: 10.1145/3439429 – ident: e_1_2_2_34_1 doi: 10.1103/PhysRevE.75.031801 – ident: e_1_2_2_21_1 doi: 10.1115/1.4050238 – ident: e_1_2_2_51_1 doi: 10.1145/3386569.3392425 – ident: e_1_2_2_20_1 doi: 10.1038/s41586-020-03123-5 – ident: e_1_2_2_80_1 doi: 10.1021/acs.jctc.8b00169 |
| SSID | ssj0006446 |
| Score | 2.4435933 |
| Snippet | We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded... |
| SourceID | crossref acm |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 1 |
| SubjectTerms | Computer graphics Computing methodologies Modeling and simulation Shape modeling |
| SubjectTermsDisplay | Computing methodologies -- Computer graphics -- Shape modeling Computing methodologies -- Modeling and simulation |
| Title | Computational Exploration of Multistable Elastic Knots |
| URI | https://dl.acm.org/doi/10.1145/3592399 |
| Volume | 42 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV05T8MwFLagLDAgTnErA2ILNPGRZKw4BZSBSzBVju1IlSBFJR3g1_Mc24k5BmCJosR1m3yv733PfgdCu3EikgJoalgU4JsQoWSY4kiEgkqeEZ4mXNZRvlfs7I6cP9CHtn9nnV1S5fvi_ce8kv-gCtcAV50l-wdkm0nhApwDvnAEhOH4K4xNSwa3nGfC6RoKWKfWAvfTqVHHwJF1ZdaLcmQqNzWFZw_7ukuEaxle7x3UNay9IPj7oXRd003gaCML10ZnPYL5e3MmsN6OKlVlWKlOv27l73Q8LF9fJmbJZ1jZG3bNIcZNxJuTkr6npkBHhNqxMhbFqlGahAk2DXOcniWxJ0_EU5qRZ31N4uV3vU50CQxMM52K25quJqDQ3plGMzG4C6DvZnpH_cubxiYD66t3rd1vNenTetoD-1HNTsSzx048mnG7gOatfxD0DNiLaEqVS2jOqxq5jNgn2AMP9mBUBB7sgYU9qGFfQXcnx7eHZ6FtfxFyTHAVRpiqnMF_hnUFBjc-ZVmsZKa6iVKUJ0UE2lOmlGVcdHOc5jwBqkiw7iIEnIN38SrqlKNSraEglaCYGTB9xiXJucgimROgipkSkhYxW0dL8PCDF1PgZGBfyTracy9jIGzFeN245GlgstlpOzBoBro5vgzZ-PEbNtFsK15bqFONJ2obSF6V71gEPwBG2VHC |
| linkProvider | EBSCOhost |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Computational+Exploration+of+Multistable+Elastic+Knots&rft.jtitle=ACM+transactions+on+graphics&rft.au=Vidulis%2C+Michele&rft.au=Ren%2C+Yingying&rft.au=Panetta%2C+Julian&rft.au=Grinspun%2C+Eitan&rft.date=2023-08-01&rft.pub=ACM&rft.issn=0730-0301&rft.eissn=1557-7368&rft.volume=42&rft.issue=4&rft.spage=1&rft.epage=16&rft_id=info:doi/10.1145%2F3592399&rft.externalDocID=3592399 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0730-0301&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0730-0301&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0730-0301&client=summon |