Ideal coset invariants for surface-links in \(\mathbb R^{4}\) (Q2850721)

A surface-link is a closed 2-manifold smoothly embedded in the Euclidean 4-space. A surface-link is presented by a diagram of a 4-regular spatial graph with a marker at each 4-valent vertex, called a marked graph diagram or a ch-diagram, and for two marked graph diagrams of surface-links, the surface-links are equivalent if and only if the marked graph diagrams are related by a finite sequence of local moves called Yoshikawa moves; see \textit{K. Yoshikawa} [Osaka J. Math. 31, No. 3, 497--522 (1994; Zbl 0861.57033)], \textit{F. J. Swenton} [J. Knot Theory Ramifications 10, No. 8, 1133--1141 (2001; Zbl 1001.57044)], and \textit{C. Kearton} and \textit{V. Kurlin} [Algebr. Geom. Topol. 8, No. 3, 1223--1247 (2008; Zbl 1151.57027)]. For a marked graph diagram \(D\) of a surface-link, a polynomial \([[D]]\) was introduced by the third author of this paper, by using a state-sum model involving a given invariant \([\;]\) for classical links, and several invariants for surface-links were constructed from \([[D]]\) by taking \([\;]\) to be the Kauffman bracket polynomial; see \textit{S. Y. Lee} [Trans. Am. Math. Soc. 361, No. 1, 237--265 (2009; Zbl 1160.57021)]. In this paper, the authors introduce an invariant for surface-links by investigating obstructions to obtain from \([[D]]\) an invariant under Yoshikawa moves.NEWLINENEWLINEThe main result is that for a marked graph diagram \(D\) of a surface-link, an ideal coset \(\langle\langle D \rangle\rangle_{\phi}+I_{\phi}\) in a quotient ring \(S/I_{\phi}\) is an invariant of surface-link types, where, for a normalized polynomial \(\langle\langle D \rangle\rangle\) of \([[D]]\) called the L-polynomial of \(D\) associated to a classical link invariant \([\;]\), \(\langle \langle D \rangle \rangle_{\phi}\) is the image of \(\langle\langle D \rangle\rangle\) by a homomorphism \(\phi\) valued in a certain ring \(S\) of polynomials, and \(I_{\phi}\) is an ideal of \(S\) called the obstruction ideal determined from \(\phi\) and a finite number of certain polynomials. According to this method, by taking the classical link invariant \([\;]\) to be the number of components, the authors introduce an ideal coset invariant for surface-links. Further, they show how to find a unique representative of the ideal coset invariant by using a Gröbner basis for the obstruction ideal, and they compute examples for surface-links represented by marked graph diagrams with ch-index up to 10 in Yoshikawa's table with the help of the computer.