Encyclopedia of knot theory. (English) Zbl 1468.57001

Boca Raton, FL: CRC Press (ISBN 978-1-138-29784-5/hbk; 978-0-367-62304-3/pbk; 978-1-138-29821-7/ebook). xi, 941 p. (2021).

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This expensive and unwieldy 952 page book consists of 91 separately authored chapters of vastly uneven quality. The first, authored by one of its six distinguished editors, summarizes the early history of the subject and sets a low bar for accuracy by declaring that by the end of the 19th century “the enumeration of all knots up to 10 crossings was almost complete.” In fact, it was overcomplete, as evidenced by the celebrated duplication known for almost half a century as “the Perko pair” knot. Similarly, the chapter on hyperbolic knots completely ignores the work of Robert Riley, who first discovered the hyperbolic structure of certain knot complements, while the chapter on bridge number discusses the meridional rank conjecture without attributing it to Cappell and Shaneson or mentioning the recent work of Blair, Kjuchukova, Vilazquez and Villaneuva [R. Blair et al., Commun. Anal. Geom. 28, No. 2, 243–262 (2020; Zbl 1448.57005)]. A welcome exception is Bonahon’s excellent essay on arborescent knots, supplementing the as yet unpublished account of his joint work with Siebenmann more than 40 years ago. The book competently covers quantum, polynomial, homological, algebraic and combinatorial invariants, along with what the editors call “physical knot theory” (stick numbers and the like), and concludes with three chapters on the science of knots – DNA, proteins and synthetic molecules – no doubt the most important of its future applications. Most of its topics are brought correctly up to date, for which the book furnishes a valuable service, though perhaps not one fully commensurate with its price.
Finally, we note that the chapter on knot tabulation describes Conway’s table of non-alternating knots as containing “only a few errors” with a just passing reference to Caudron’s work that filled those gaps, while the chapter on alternating knots discusses the so-called “Tait Conjecture” on writhe without mention of Little’s false proof that it also holds for non-alternating knots (see page 36 of [A. Stoimenow, Diagram genus, generators, and applications. Boca Raton, FL: CRC Press (2016; Zbl 1336.57016)]).
Editorial remark: The articles of this volume will be announced individually.
Indexed articles:
Ludwig, Lewis D., Introduction to knots, 3-6 [Zbl 1465.57057]
Hoste, Jim, Link diagrams, 9-12 [Zbl 1465.57033]
Johnson, Inga, Gauss diagrams, 13-15 [Zbl 1465.57037]
Russell, Heather M., DT codes, 17-20 [Zbl 1465.57075]
Ludwig, Lewis D., Knot mosaics, 21-26 [Zbl 1465.57058]
Lee, Hwa Jeong, Arc presentations of knots and links, 27-34 [Zbl 1465.57053]
Lambropoulou, Sofia, Diagrammatic representations of knots and links as closed braids, 35-51 [Zbl 1465.57049]
Sullivan, Michael C., Knots in flows, 53-62 [Zbl 1465.57085]
Adams, Colin, Multi-crossing number of knots and links, 63-70 [Zbl 1465.57003]
Adams, Colin, Complementary regions of knot and link diagrams, 71-76 [Zbl 1465.57002]
Hoste, Jim, Knot tabulation, 77-82 [Zbl 1465.57034]
Lawrence, Emille Davie, What is a tangle?, 85-88 [Zbl 1465.57050]
Lawrence, Emille Davie, Rational and non-rational tangles, 89-95 [Zbl 1465.57051]
Silver, Daniel S.; Williams, Susan G., Persistent invariants of tangles, 97-101 [Zbl 1465.57081]
Callahan, Jason, Torus knots, 105-109 [Zbl 1465.57014]
Wilson, Robin T., Rational knots and their generalizations, 111-119 [Zbl 1465.57090]
Bonahon, Francis, Arborescent knots and links, 121-146 [Zbl 1465.57010]
Schultens, Jennifer, Satellite knots, 147-157 [Zbl 1465.57078]
Adams, Colin, Hyperbolic knots and links, 159-166 [Zbl 1465.57004]
Menasco, William W., Alternating knots, 167-178 [Zbl 1465.57063]
Naik, Swatee, Periodic knots, 179-184 [Zbl 1465.57066]
Brittenham, Mark, Seifert surfaces and genus, 187-195 [Zbl 1465.57012]
Kindred, Thomas, Non-orientable spanning surfaces for knots, 197-204 [Zbl 1465.57047]
Kalfagianni, Efstratia, State surfaces of links, 205-212 [Zbl 1465.57040]
Kim, Seungwon; Hofman, Ilya, Turaev surfaces, 213-220 [Zbl 1465.57046]
Zupan, Alexander, Crossing numbers, 223-228 [Zbl 1465.57094]
Schultens, Jennifer, The bridge number of a knot, 229-242 [Zbl 1465.57079]
Lowrance, Adam, Alternating distances of knots, 243-250 [Zbl 1465.57056]
Adams, Colin, Superinvariants of knots and links, 251-257 [Zbl 1465.57005]
Kauffman, Louis H., Virtual knot theory, 261-310 [Zbl 1465.57044]
Chrisman, Micah, Virtual knots and surfaces, 311-318 [Zbl 1465.57022]
Dye, Heather A.; Kaestner, Aaron, Virtual knots and parity, 319-325 [Zbl 1465.57027]
Nelson, Sam, Forbidden moves, welded knots and virtual unknotting, 327-334 [Zbl 1465.57069]
Petit, Nicolas, Virtual strings and free knots, 335-342 [Zbl 1465.57073]
Kamada, Naoko, Abstract and twisted links, 343-349 [Zbl 1465.57041]
Chapman, Harrison, What is a knotoid?, 351-358 [Zbl 1465.57019]
Gügümcü, Neslihan, What is a braidoid?, 359-370 [Zbl 1465.57032]
Dancso, Zsuzsanna, What is a singular knot?, 371-378 [Zbl 1465.57023]
Johnson, Inga, Pseudoknots and singular knots, 379-384 [Zbl 1465.57038]
Traynor, Lisa, An introduction to the world of Legendrian and transverse knots, 385-392 [Zbl 1465.57087]
Cahn, Patricia, Classical invariants of Legendrian and transverse knots, 393-401 [Zbl 1465.57013]
Sabloff, Joshua M., Ruling and augmentation invariants of Legendrian knots, 403-410 [Zbl 1465.57076]
Carter, J. Scott; Saito, Masahico, Broken surface diagrams and Roseman moves, 413-418 [Zbl 1465.57017]
Carter, J. Scott; Saito, Masahico, Movies and movie moves, 419-424 [Zbl 1465.57018]
Kamada, Seiichi, Surface braids and braid charts, 425-431 [Zbl 1465.57042]
Lee, Sang Youl, Marked graph diagrams and Yoshikawa moves, 433-441 [Zbl 1465.57054]
Zupan, Alexander, Knot groups, 443-449 [Zbl 1465.57095]
Kearney, Kate, Concordance groups, 451-458 [Zbl 1465.57045]
Friedl, Stefan; Herrmann, Gerrit, Spatial graphs, 461-466 [Zbl 1465.57030]
Naimi, Ramin, A brief survey on intrinsically knotted and linked graphs, 467-475 [Zbl 1465.57067]
Howards, Hugh, Chirality in graphs, 477-483 [Zbl 1465.57036]
Flapan, Erica, Symmetries of graphs embedded in \(S^3\) and other 3-manifolds, 485-490 [Zbl 1465.57028]
Mellor, Blake, Invariants of spatial graphs, 491-501 [Zbl 1465.57061]
O’Donnol, Danielle, Legendrian spatial graphs, 503-512 [Zbl 1465.57070]
Pavelescu, Elena, Linear embeddings of spatial graphs, 513-519 [Zbl 1465.57072]
Taylor, Scott A., Abstractly planar spatial graphs, 521-528 [Zbl 1465.57086]
Yetter, D. N., Quantum link invariants, 531-545 [Zbl 1465.57092]
Morton, H. R., Satellite and quantum invariants, 547-558 [Zbl 1465.57065]
Lawrence, Ruth, Quantum link invariants: from QYBE and braided tensor categories, 559-579 [Zbl 1465.57052]
Kauffman, Louis H., Knot theory and statistical mechanics, 581-600 [Zbl 1465.57043]
Frohman, Charles, What is the Kauffman bracket?, 603-609 [Zbl 1465.57031]
Stoltzfus, Neal, Span of the Kauffman bracket and the Tait conjectures, 611-616 [Zbl 1465.57084]
Bakshi, Rhea Palak; Przytycki, Józef H.; Wong, Helen, Skein modules of 3-manifolds, 617-623 [Zbl 1465.57007]
Chmutov, Sergei, The Conway polynomial, 625-630 [Zbl 1465.57020]
Vidussi, Stefano, Twisted Alexander polynomials, 631-636 [Zbl 1465.57089]
Hoste, Jim, The HOMFLYPT polynomial, 637-641 [Zbl 1465.57035]
Zhong, Jianyuan k., The Kauffman polynomials, 643-648 [Zbl 1465.57093]
Caprau, Carmen, Kauffman polynomial on graphs, 649-655 [Zbl 1465.57015]
Bakshi, Rhea Palak; Przytycki, Józef; Wong, Helen, Kauffman bracket skein modules of 3-manifolds, 657-666 [Zbl 1465.57008]
Sazdanovic, Radmila, Khovanov link homology, 669-680 [Zbl 1465.57077]
Stipsicz, András I., A short survey on knot Floer homology, 681-700 [Zbl 1465.57083]
Stipsicz, András, An introduction to grid homology, 701-722 [Zbl 1465.57082]
Mazorchuk, Volodymyr, Categorification, 723-731 [Zbl 1465.57059]
Shumakovitch, Alexander N., Khovanov homology and the Jones polynomial, 733-749 [Zbl 1465.57080]
Rushworth, William, Virtual Khovanov homology, 751-762 [Zbl 1465.57074]
Lopes, Pedro, Knot colorings, 765-772 [Zbl 1465.57055]
Carter, J. Scott, Quandle cocycle invariants, 773-779 [Zbl 1465.57016]
Oshiro, Kanako, Kei and symmetric quandles, 781-787 [Zbl 1465.57071]
Nelson, Sam, Racks, biquandles and biracks, 789-794 [Zbl 1465.57068]
Vendramin, Leandro, Quantum invariants via Hopf algebras and solutions to the Yang-Baxter equation, 795-800 [Zbl 1465.57088]
Bigelow, Stephen, The Temperley-Lieb algebra and planar algebras, 801-805 [Zbl 1465.57009]
Chmutov, Sergei; Stoimenow, Alexander, Vassiliev/finite-type invariants, 807-815 [Zbl 1465.57021]
Meilhan, Jean-Baptiste, Linking number and Milnor invariants, 817-829 [Zbl 1465.57060]
Adams, Colin, Stick number for knots and links, 833-840 [Zbl 1465.57006]
Millett, Kenneth, Random knots, 841-857 [Zbl 1465.57064]
Dorier, Julien; Goundaroulis, Dimos; Rawdon, Eric J.; Stasiak, Andrzej, Open knots, 859-875 [Zbl 1465.57026]
Kozai, Kenji, Random and polygonal spatial graphs, 877-883 [Zbl 1465.57048]
Denne, Elizabeth, Folded ribbon knots in the plane, 885-897 [Zbl 1465.57025]
Darcy, Isabel K., DNA knots and links, 901-910 [Zbl 1465.57024]
Wong, Helen, Protein knots, links and non-planar graphs, 911-917 [Zbl 1465.57091]
Flapan, Erica, Synthetic molecular knots and links, 919-925 [Zbl 1465.57029]

Reviewer: Kenneth A. Perko Jr. (Scarsdale)

MSC:

57-00	General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to manifolds and cell complexes
57-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57K10	Knot theory
00B15	Collections of articles of miscellaneous specific interest

Keywords:

knots

Citations:

Zbl 1448.57005; Zbl 1336.57016