Finite type invariants for knotoids

Manousos Manouras, Sofia Lambropoulou, Louis H. Kauffman

Research output: Contribution to journalArticlepeer-review


We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.

Original languageEnglish
Article number103402
JournalEuropean Journal of Combinatorics
Publication statusPublished - Dec 2021




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