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Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem


This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their “usual” counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar “virtual” knot diagrams, hence enlarging the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the “overcrossings commute” relation, making w-knotted objects a bit weaker once again. Satoh (J. Knot Theory Ramif. 9-4:531–542, 2000) studied several classes of w-knotted objects (under the name “weakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in \({\mathbb R}^4\). In this article we study finite type invariants of w-tangles and w-trivalent graphs (also referred to as w-tangled foams). Much as the spaces \({\mathcal A}\) of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces \({\mathcal A}^w\) of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara and Vergne (Invent. Math. 47:249–272, 1978) conjecture and much of the Alekseev and Torossian (Ann. Math. 175:415–463, 2012) work on Drinfel’d associators and Kashiwara–Vergne can be re-interpreted as a study of w-foams.

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  1. q-tangles” in [31], “non-associative tangles” in [5].

  2. Alternatively define “algebraic structures” using the theory of “multicategories” [32]. Using this language, an algebraic structure is simply a functor from some “structure” multicategory \({\mathcal C}\) into the multicategory Set (or into Vect, if all \({\mathcal O}_i\) are vector spaces and all operations are multi-linear). A “morphism” between two algebraic structures over the same multicategory \({\mathcal C}\) is a natural transformation between the two functors representing those structures.

  3. Indeed, if \({\mathcal O}\) is finitely presented then finding such a morphism \(Z:{\mathcal O}\rightarrow {{\text {grad}}\,}{\mathcal O}\) amounts to finding its values on the generators of \({\mathcal O}\), subject to the relations of \({\mathcal O}\). Thus it is equivalent to solving a system of equations written in some graded spaces.

  4. The Drinfel’d graded Grothendieck–Teichmuller group \( GRT \) is an example of such an automorphism group. See [6, 23].

  5. A Leibniz algebra is a Lie algebra without anti-commutativity, as defined by Loday in [34].

  6. Or have they, and we have been looking the wrong way?

  7. We mean “pairing” in the sense of combinatorics, not in the sense of linear algebra. That is, an involution without fixed point.

  8. By convention we label the boundary points of such circuits \(1,\ldots ,p+q\), with the first p labels reserved for the incoming wires and the last q for the outgoing. The inputs of wiring diagrams must be labeled in the opposite way for the numberings to match.

  9. We usually short this to “w-Jacobi diagram”, or sometimes “arrow diagram” or just “diagram”.

  10. Oriented graphs with vertex degrees either 1 or 3, where trivalent vertices must have two edges incoming and one edge outgoing and are cyclically oriented.

  11. The core of Lord Voldemort’s wand was made of a phoenix feather.

  12. In practice this simply means that the value of the crossing is an exponential.

  13. The formal definition of the group-like property is along the lines of [10, Section]. In practice, it means that the Z-values of the vertices, crossings, and cap (denoted VR and C below) are exponentials of linear combinations of connected diagrams.

  14. For a detailed explanation of this minor point see the third paragraph of the proof.

  15. We apologize for the annoying \(2\leftrightarrow 1\) transposition in this equation, which makes some later equations, especially (22), uglier than they could have been. There is no depth here, just mis-matching conventions between us and Alekseev–Torossian.

  16. We need not specify how to unzip an edge e that carries a wen. To unzip such e, first use the TV relation to slide the wen off e.

  17. It will become apparent that in the proof we only use slightly weaker but less aesthetic conditions on \(Z^u\).

  18. An even nicer theorem would be a classification of homomorphic expansions for the combined algebraic structure \(\left( { s \!K\!T\!G}\overset{a}{\longrightarrow }{ w \!T\!F}\right) \) in terms of solutions of the KV problem. The two obstacles to this are clarifying whether there is a free choice of n for \(Z^u\), and — probably much harder—how much of the horizontal chord condition is necessary for a compatible \(Z^w\) to exist.

  19. Note that in [2] “\(\Phi '\) is an associator” means that \(\Phi '\) satisfies the pentagon equation, mirror skew-symmetry, and positive and negative hexagon equations in the space \({\text {SAut}}_3\). These equations are stated in [2] as equations (25), (29), (30), and (31), and the hexagon equations are stated with strands 1 and 2 re-named to 2 and 1 as compared to [8, 22]. This is consistent with \(F=e^{D^{21}}\).

  20. The condition \(j(\phi ')=0\) is equivalent to the condition \(\Phi \in KRV^0_3\) in [2]. The relevant definitions in [2] can be found in Remark 4.2 and at the bottom of page 434 (before Section 5.2).


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We wish to thank Anton Alekseev, Jana Archibald, Scott Carter, Karene Chu, Iva Halacheva, Joel Kamnitzer, Lou Kauffman, Peter Lee, Louis Leung, Jean-Baptiste Meilhan, Dylan Thurston, Lucy Zhang and the anonymous referees for comments and suggestions.

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Correspondence to Zsuzsanna Dancso.

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This work was partially supported by NSERC Grant RGPIN 262178. This paper is part 2 of a 4-part series whose first two parts originally appeared as a combined preprint [9].

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Bar-Natan, D., Dancso, Z. Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem. Math. Ann. 367, 1517–1586 (2017).

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