In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [19] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [1]. The relation between this new polynomial invariant and the affine index polynomial [14,3] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

The last decade has seen the development of a wide set of tools, such as wavefront shaping, computational or fundamental methods, that allow to understand and control light propagation in a complex medium, such as biological tissues or multimode fibers. A vibrant and diverse community is now working on this field, that has revolutionized the prospect of diffraction-limited imaging at depth in tissues. This roadmap highlights several key aspects of this fast developing field, and some of the challenges and opportunities ahead.

The aim of this paper is to introduce a polynomial invariant f K (t) for virtual knots. We show that f K (t) can be used to distinguish some virtual knot from its inverse and mirror image. The behavior of f K (t) under a connected sum is also given. Finally, we discuss which kinds of polynomials can be realized as f K (t) for some virtual knot K.

In a recent work of Ayaka Shimizu [5] , she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams.In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals to c − n + 1, here c denotes the size of the graph.

One of the challenges in surface defects evaluation of large fine optics is to detect defects of microns on surfaces of tens or hundreds of millimeters. Sub-aperture scanning and stitching is considered to be a practical and efficient method. But since there are usually few defects on the large aperture fine optics, resulting in no defects or only one run-through line feature in many sub-aperture images, traditional stitching methods encounter with mismatch problem. In this paper, a feature-based multi-cycle image stitching algorithm is proposed to solve the problem. The overlapping areas of sub-apertures are categorized based on the features they contain. Different types of overlapping areas are then stitched in different cycles with different methods. The stitching trace is changed to follow the one that determined by the features. The whole stitching procedure is a region-growing like process. Sub-aperture blocks grow bigger after each cycle and finally the full aperture image is obtained. Comparison experiment shows that the proposed method is very suitable to stitch sub-apertures that very few feature information exists in the overlapping areas and can stitch the dark-field microscopic sub-aperture images very well.