Parity in knot theory and graph-links. (English. Russian original) Zbl 1320.57001
J. Math. Sci., New York 193, No. 6, 809-965 (2013); translation from Sovrem. Mat., Fundam. Napravl. 41, 3-163 (2011).
This monograph contains detailed accounts of two important ideas associated with virtual knots, extracted (in great part) from the broader discussion of virtual knot theory in a recent book by the first two authors [V. O. Manturov and D. P. Ilyutko, Virtual knots. The state of the art. Hackensack, NJ: World Scientific (2013; Zbl 1270.57003)].
The first of these important ideas is the notion of parity, which was introduced by V. O. Manturov [Sb. Math. 201, No. 5, 693–733 (2010; Zbl 1210.57010)]. The origin of this notion lies in an observation of Gauss: if one walks along a classical knot diagram starting at a crossing, then one will encounter an even number of crossings before returning to the starting point. The definition of parity reflects the insight that although Gauss’ observation does not hold for virtual knots (as virtual crossings are not counted), the crossings for which Gauss’ observation fails are of a special type, which does not appear in classical knot theory.
The second of the important ideas is the notion of graph-links, different forms of which were introduced independently by D. P. Ilyutko and V. O. Manturov [J. Knot Theory Ramifications 18, No. 6, 791–823 (2009; Zbl 1189.57006)] and L. Zulli and the reviewer [J. Knot Theory Ramifications 18, No. 12, 1681–1709 (2009; Zbl 1204.57009)]. The interlacement graph of an Euler circuit of a virtual link diagram \(D\) is the graph \(I(D)\) whose vertices are the crossings of \(D\), with two vertices \(v\) and \(w\) connected by an edge in \(I(D)\) if and only if when the Euler circuit is traversed beginning at \(v\), the crossing \(w\) is encountered precisely once before returning to \(v\). Many knot-theoretic ideas regarding virtual link diagrams yield graph-theoretic ideas about interlacement graphs in a natural way; for instance, Reidemeister moves on virtual link diagrams induce corresponding moves on interlacement graphs. The theory of graph-links is the result of extending these images of knot-theoretic ideas from interlacement graphs to general graphs. The monograph closes with two especially interesting extensions of this sort, which provide versions of the odd Khovanov homology for graph-links.
The first of these important ideas is the notion of parity, which was introduced by V. O. Manturov [Sb. Math. 201, No. 5, 693–733 (2010; Zbl 1210.57010)]. The origin of this notion lies in an observation of Gauss: if one walks along a classical knot diagram starting at a crossing, then one will encounter an even number of crossings before returning to the starting point. The definition of parity reflects the insight that although Gauss’ observation does not hold for virtual knots (as virtual crossings are not counted), the crossings for which Gauss’ observation fails are of a special type, which does not appear in classical knot theory.
The second of the important ideas is the notion of graph-links, different forms of which were introduced independently by D. P. Ilyutko and V. O. Manturov [J. Knot Theory Ramifications 18, No. 6, 791–823 (2009; Zbl 1189.57006)] and L. Zulli and the reviewer [J. Knot Theory Ramifications 18, No. 12, 1681–1709 (2009; Zbl 1204.57009)]. The interlacement graph of an Euler circuit of a virtual link diagram \(D\) is the graph \(I(D)\) whose vertices are the crossings of \(D\), with two vertices \(v\) and \(w\) connected by an edge in \(I(D)\) if and only if when the Euler circuit is traversed beginning at \(v\), the crossing \(w\) is encountered precisely once before returning to \(v\). Many knot-theoretic ideas regarding virtual link diagrams yield graph-theoretic ideas about interlacement graphs in a natural way; for instance, Reidemeister moves on virtual link diagrams induce corresponding moves on interlacement graphs. The theory of graph-links is the result of extending these images of knot-theoretic ideas from interlacement graphs to general graphs. The monograph closes with two especially interesting extensions of this sort, which provide versions of the odd Khovanov homology for graph-links.
Reviewer: Lorenzo Traldi (Easton)
MSC:
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
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