Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

Abstract

In this paper, we use theory of embedded graphs on oriented and compact [Formula: see text]-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter’s circles. For the case of signed Gauss words, we use a generating set of [Formula: see text], given in [G. Cairns and D. Elton, The Planarity problem for signed Gauss world, J. Knots Theor. Ramif. 2(4) (1993) 359–367], and the intersection pairing of immersed [Formula: see text]-normal curves to present a short solution of the signed Gauss word problem. We relate this solution with the one given by Cairns and Elton. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus [Formula: see text]-surface. We connect the characterization of signed Gauss paragraph with the recognition virtual links problem. Also we present a combinatorial algorithm to compute, in an easier way, skew-symmetric graded matrices [V. Turaev, Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] for virtual knots through the concept of triplets [M. Toro and J. Rodríguez, Triplets associated to virtual knot diagrams, Rev. Integración (2011)]. Therefore, we can prove that the Kishino’s knot is not classical, moreover, we prove that the virtual knots of the family [Formula: see text] given in [H. A. Dye, Virtual knots undetected by [Formula: see text] and [Formula: see text]-strand bracket polynomials, Topol. Appl. 153 (2005) 141–160] are not classical knots.