Recent zbMATH articles in MSC 57-02https://zbmath.org/atom/cc/57-022022-07-25T18:03:43.254055ZWerkzeugPolynomial one-cocycles for knots and closed braidshttps://zbmath.org/1487.570012022-07-25T18:03:43.254055Z"Fiedler, Thomas"https://zbmath.org/authors/?q=ai:fiedler.thomasIn this research monograph, the author expounds the current status and results of an ongoing long-term project, where the goal is to find one-cocycles in spaces of knots (that is, homotopy invariants of one-parameter families of knots), and the hope is that they will capture geometric information, be stronger than classical invariants, or shed a new light on classical invariants.
All one-cocycles found in the book are of combinatorial nature in that they can be computed by hand given a sequence of knot diagrams that differ each from the next by one Reidemeister move. In other words they can be seen as intersection forms with ($1$-codimensional) Reidemeister strata with values in either the ring of integers or a ring of polynomials. The cocyclicity condition amounts to the vanishing on small loops around $2$-codimensional strata from a higher Reidemeister theorem (e.g., the integer or polynomial associated to, first, creating two crossings in a diagram by two successive Reidemeister I moves, and then killing them off in one Reidemeister II move, should be $0$). Finding one-cocycles from this point of view requires to solve large systems of equations coming from this higher Reidemeister theorem, including what are called the tetrahedron equation and the cube equation.
Experience shows that non-trivial solutions are sparse and hard to find. The author comes up with several ideas that will result in making the system of equations bigger, making solutions more numerous, and making solutions perhaps more likely to capture interesting information about the knots under consideration. These ideas come in two flavours:
\begin{itemize}
\item[1.] Considering different kinds of spaces of knots, subject to specific constraints.
Oftentimes in the book these constraints are diagram-dependent, hence the phrase ``space of knot diagrams'' used by the author. The first historical step towards such considerations was the partial shift of interest from classical compact knots to long knots.
Here are the three spaces under consideration in the book. It takes a while to get used to the fact that $M$ depends on the integer $n$ just as $M_n$ and $M_n^+$ do.
\begin{itemize}
\item[a.] $M$ stands for the space of all knots that are embedded in a standard solid torus $V$ ($=S^1\times D^2$, $S^1$ being oriented) and which represent the homology class $n$ in $H_1(V)$.
\item[b.] $M_n$ stands for the space of all knots in $M$ whose diagram (as a $4$-valent graph embedded in $S^1\times[0,1]$) contains no oriented loop with a negative homology class. It is worth noting that $M_1$ is exactly the space of long knots.
\item[c.] $M_n^+$ is the space of all knots in $M_n$ whose diagram contains no oriented loop with a non-positive homology class; in other words $M_n^+$ is the space of closed braids with $n$ strands which are knots (i.e. have only one component as a link).
\end{itemize}
\item[2.] Using topological data attached to each crossing (point of having the knots embedded in $V$).
The most ubiquitous of these data is a homological marking. It is the element of $H_1(V)\cong H_1(S^1)$ defined by the half of the diagram that runs from the top branch of a crossing to the bottom one.
The second kind of topological data is the trace circle associated with each crossing: keep track of the crossing as a loop unfolds in the space of knots; if it ever gets cancelled out in a Reidemeister II move, then reverse time and follow the twin crossing. When both the crossing and the knot itself come back to their exact original position, the crossing has described a trace circle. The properties of such circles allow for a refinement of the one-cocycles in even chapters (Sections 2.2 and 4.9-4.10)
\end{itemize}
Each of the Chapters 2 to 4 corresponds to one particular space of knots being considered.
Chapter 2 deals with one-cocycles in $M_n$ where $n>0$. This covers two important families of knot[ diagram]s: closed braids (from $M_n^+$), and long knots (where $n=1$) for which the author finds a formula that generalizes the well-known Teiblum-Turchin cocycle. The cocycles found depend on $n$ as well as up to 2 additional parameters.
The cocycles from Section 2.1 are integer-valued, based on the homological markings of the crossings, while the one from Section 2.2 is polynomial-valued. This last one is not really a one-cocycle though, for two reasons. First, it is defined as a $0$-cocycle on a loop space; in other words you will not be able to evaluate it on an arbitrary path in the space of knots, only on a loop. Second, its evaluation will not always be the same on two homotopic loops, when one of them kills off two crossings in a Reidemeister II move to bring them back later in the opposite move, while the other keeps the crossings there all along. The three-line convention between Definition 2.9 and Proposition 2.1 is dubious and probably not what the author intended.
There are 10 pages of notations/definitions of the cocycles, 10 pages of examples being worked out, and 70 pages of proofs of cocyclicity that can be skipped upon first reading.
In Chapter 3 the only one-cocycle in $M$ in the whole book is shown. The only requirement is that $n$ must not be $1$, meaning that long knots are off the table. It is also the only place in the book where $n$ is allowed to be $0$. Homological markings are regarded modulo $n$ for the particular case of this cocycle only.
Chapter 4 opens with a long digression, from Section 4.1 to 4.6 included, that can be ignored upon first reading. The purpose of these sections is first to lay theoretical ground for the higher Reidemeister theorem, and second to prove Theorem 4.1: two links in $V$ are isotopic if and only if their trace graphs are equivalent [under some set of moves], a result that is not used in the book.
Section 4.7 is an independent rewriting of Sections 4.1 to 4.6 in the particular case of closed braids (from $M_n^+$) instead of arbitrary links. Theorem 4.4 is the closed braid version of Theorem 4.1. Results from this section are not used until the very last pages of the book.
Finally, Section 4.8 introduces one-cocycles in $M_n^+$, which are refined in the subsequent sections using trace circles. Unlike Section 2.2 where the trace circles are derived by unfolding the very loop at which the cocycle is being evaluated, the trace circles used in Chapter 4 (including Sections 4.1 through 4.7) are always based on performing a full rotation of the solid torus around its core. An interesting property of these circles in the case of closed braids (in $M_n^+$) is Lemma 4.20: two crossings lie on the same trace circle if and only if they have the same homological marking.
Editorial remark: The book would have benefitted from careful
proofreading by the author and the copy editor that could have
eliminated a plethora of typos.
Reviewer: Arnaud Mortier (Caen)