Journal of the Mathematical Society of Japan

Alexander invariants of ribbon tangles and planar algebras

Celeste DAMIANI and Vincent FLORENS

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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].

Article information

J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1063-1084.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

Alexander polynomials tangles welded knots planar algebras


DAMIANI, Celeste; FLORENS, Vincent. Alexander invariants of ribbon tangles and planar algebras. J. Math. Soc. Japan 70 (2018), no. 3, 1063--1084. doi:10.2969/jmsj/75267526.

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