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July, 2018 Alexander invariants of ribbon tangles and planar algebras
Celeste DAMIANI, Vincent FLORENS
J. Math. Soc. Japan 70(3): 1063-1084 (July, 2018). DOI: 10.2969/jmsj/75267526


Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball $B^4$, “bounding” 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathbf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathbf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau–Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathbf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [Jon99], and prove that the invariant $\mathbf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, through welded diagrams. We give a simple combinatorial description of $\mathbf{A}$ and of the algebra $\mathbf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [Arc10]. When restricted to diagrams without virtual crossings, $\mathbf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [BCF15].


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Celeste DAMIANI. Vincent FLORENS. "Alexander invariants of ribbon tangles and planar algebras." J. Math. Soc. Japan 70 (3) 1063 - 1084, July, 2018.


Received: 20 May 2016; Revised: 23 December 2016; Published: July, 2018
First available in Project Euclid: 18 June 2018

zbMATH: 06966974
MathSciNet: MR3830799
Digital Object Identifier: 10.2969/jmsj/75267526

Primary: 57M25 , 57M27 , 57Q45

Keywords: Alexander polynomials , planar algebras , tangles , welded knots

Rights: Copyright © 2018 Mathematical Society of Japan

Vol.70 • No. 3 • July, 2018
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