The Combinatorial Revolution in Knot Theory Sam Nelson not theory is usually understood to be the study of embeddings of topologi- cal spaces in other topological spaces. Classical knot theory, in particular, is concerned with the ways in which a cir- Kcle or a disjoint union of circles can be embedded in R3. Knots are usually described via knot dia- grams, projections of the knot onto a plane with Figure 1. Knot diagrams. breaks at crossing points to indicate which strand passes over and which passes under, as in Fig- ure 1. However, much as the concept of “numbers” seriously, however, has led to the discovery of has evolved over time from its original meaning new types of generalized knots and links that do of cardinalities of finite sets to include ratios, not correspond to simple closed curves in R3. equivalence classes of rational Cauchy sequences, Like the complex numbers arising from missing roots of polynomials, and more, the classical con- roots of real polynomials, the new generalized cept of “knots” has recently undergone its own knot types appear as abstract solutions in knot evolutionary generalization. Instead of thinking equations that have no solutions among the classi- of knots topologically as ambient isotopy classes cal geometric knots. Although they seem esoteric of embedded circles or geometrically as simple at first, these generalized knots turn out to have R3 closed curves in , a new approach defines knots interpretations such as knotted circles or graphs combinatorially as equivalence classes of knot di- in three-manifolds other than R3, circuit diagrams, agrams under an equivalence relation determined and operators in exotic algebras. Moreover, clas- by certain diagrammatic moves. No longer merely sical knot theory emerges as a special case of the symbols standing in for topological or geomet- new generalized knot theory. ric objects, the knot diagrams themselves have This diagram-based combinatorial approach to become mathematical objects of interest. knot theory has revived interest in a related Doing knot theory in terms of knot diagrams, approach to algebraic knot invariants, applying of course, is nothing new; the Reidemeister moves techniques from universal algebra to turn the date back to the 1920s [21], and identifying knot combinatorial structures into algebraic ones. The invariants (functions used to distinguish differ- resulting algebraic objects, with names such as kei, ent knot types) by checking invariance under the quandles, racks, and biquandles, yield new invari- moves has been common ever since. A recent shift ants of both classical and generalized knots and toward taking the combinatorial approach more provide new insights into old invariants. Much like Sam Nelson is assistant professor of mathematics at Clare- groups arising from symmetries of geometric ob- mont McKenna College, a member of the Claremont Uni- jects, these knot-inspired algebraic structures have versity Consortium in Claremont, CA. His email address connections to vector spaces, groups, Lie groups, is [email protected]. Hopf algebras, and other mathematical structures. December 2011 Notices of the AMS 1553 Consequentially, they have potential applications in disciplines from statistical mechanics to bio- chemistry to other areas of mathematics, with many promising open questions. Virtual Knots A geometric knot is a simple closed curve in R3; a geometric link is a union of knots that may be linked together. A diagram D of a knot or link K is a projection of K onto a plane such that no point in D comes from more than two points in K. Every point in the projection with two preimages is then a crossing point; if the knot were a physical rope laid on the plane of the paper, the crossing points would be the places where the rope touches Figure 3. Reidemeister moves. itself. We indicate which strand of the knot goes over and which goes under at a crossing point by drawing the undercrossing strand with gaps. graphs could not be embedded in the plane still Combinatorially, then, a knot or link diagram behaved like knots in certain ways—for instance, is a four-valent graph embedded in a plane with knot invariants defined via combinatorial pairings vertices decorated to indicate crossing informa- of Gauss codes still gave valid invariants when tion. We can describe such a diagram by giving ordinary knots were paired with nonplanar Gauss a list of crossings and specifying how the ends codes. A planar Gauss code always describes a are to be connected; for example, a Gauss code simple closed curve in three-space; what kind of is a cyclically ordered list of over- and under- thing could a nonplanar Gauss code be describing? crossing points with connections determined by Resolving the vertices in a planar four-valent the ordering. See Figure 2. graph as crossings yields a simple closed curve in three-dimensional space.1 To draw nonplanar graphs, we normally turn edge-intersections into crossings, but here all the crossings are already supposed to be present as vertices in the graph. O1U2O3U1O2U3 Thus we need a new kind of crossing, a virtual crossing which isn’t really there, to resolve the nonvertex edge-intersections in a nonplanar Gauss code graph. To keep the crossing types distinct, we drawa virtual crossing as a circled self-intersection Figure 2. A knot and its Gauss code. with no over-or-under information. See Figure 4. Intuitively, moving a knot or link around in space without cutting or retying it shouldn’t change the kind of knot or link we have. Thus we really want topological knots and links, where two geometric knots are topologically equivalent if one can be continuously deformed into the other. Formally, topological knots are ambient isotopy classes of geometric knots and links. In the 1920s Kurt Reidemeister showed that ambient isotopy of simple closed curves in R3 corresponds to equiv- alence of knot diagrams under sequences of the moves in Figure 3 [21]. In these moves, the portion of the diagram outside the pictured neighborhood remains fixed. The proof that a function defined on knot diagrams is a topological knot invariant Figure 4. Virtual moves. is then reduced to a check that the function value is unchanged by the three moves. It turned out that Reidemeister equivalence In the mid-1990s various knot theorists (e.g., of these nonplanar “knot diagrams” was impor- [11, 14, 17]), studying combinatorial methods of tant in defining and calculating the invariants. computing knot invariants using Gauss codes and similar schemes for encoding knot diagrams, no- 1Actually, we only need R2 × (−ǫ, ǫ)—knots are almost ticed that even the Gauss codes whose associated two-dimensional! 1554 Notices of the AMS Volume 58, Number 11 Since Reidemeister equivalence classes of planar knot diagrams coincide with classical knots, it seemed natural to refer to the generalized equiv- alence classes as “knots” despite their inclusion of nonplanar diagrams. The computations were suggesting that familiar topological knots are a special case of a more general kind of thing, namely Reidemeister equivalence classes of “knot diagrams”which mayor may not be planar. The re- sulting “combinatorial revolution” was a shift from thinking of the diagrams as symbols represent- ing topological objects (ambient isotopy classes of simple closed curves in R3) to thinking of the objects themselves as equivalence classes of dia- Figure 5. A nonclassical virtual knot. grams. To reject these virtual knots on the grounds that they do not represent simple closed curves in R3 would mean throwing away infinite classes of knot invariants, akin to ignoring complex roots Generalized Knots of polynomials on the grounds that they are not Once we have one new type of crossing, it is natural “real” numbers. to consider others, spawning a zoo of new species Virtual crossings and their rules of interaction of generalized knot types. Introducing a new kind were introduced in 1996 by Louis Kauffman [17]. of crossing requires new Reidemeister-style rules of interaction. These rules or moves are deter- Since the virtual crossings are not really there, mined by the desired combinatorial, topological, any strand with only virtual crossings should or geometric interpretation. For example, Figure 6 be replaceable with any other strand with the illustrates how the geometric understanding of same endpoints and only virtual crossings. This virtual knots as knot diagrams drawn on surfaces is known as the detour move; it breaks down into with genus permits some moves while forbidding the four virtual moves in Figure 4. others. Despite their abstract origin, virtual knots do have a concrete geometric interpretation. Virtual crossings can be avoided by drawing nonpla- nar knot diagrams on compact surfaces which 2 may have nonzero genus, providing “bridges”Σ ∼ or “wormholes” that can be used to avoid edge- crossings. A virtual knot is then a simple closed curve in an ambient space of the form × (−ǫ, ǫ) (called a thickened surface or trivial I-bundleΣ ). Vir- tual crossings are not crossings in the classical sense of two strands close together—the strands meeting in a virtual crossing are on opposite sides ∼ of the ambientspaceor “universein which the knot lives”—but are instead artifacts of forcing a knot in a nonplanar ambient space into the plane. This diagrams-on-surfaces approach was introduced Figure 6. Geometric motivation for moves. by Naoko and Seiichi Kamada, who dubbed the results abstract knots in [14]. Other ideas related to virtual knots conceived independently include We’ve seen that virtual crossings represent Vladimir Turaev’s virtual strings [24] and Roger genus in the surface on which the knot diagram is Fenn, Richárd Rimányi, and Colin Rourke’s welded drawn. Flat crossings are classical crossings where braids [8]. we forget which strand goes over and which goes Figure 5 shows the smallest nonclassical virtual under; these are convenient for studying the virtual knot interpreted as a list of labeled crossings structure separately from the classical structure which cannot be realized in the plane, a virtual and can be understood as “shadows” of classical knot diagram and a knot diagram drawn on a crossings.