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In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. Since virtual knots and links are represented by embeddings of circles in thickened surfaces, we refer to embeddings of circles in the 3-sphere as {\it classical links}. Classical links are the same as virtual links that can be represented in a thickened 2-sphere and it is a fact that classical links, up to isotopy, embed in the collection of virtual links taken up to isotopy. We give a new invariant of classical links in the 3-sphere in the following sense: Consider a link $L$ in $S^3$ of two components. The complement of a tubular neighborhood of $L$ is a manifold whose boundary consists in two copies of a torus. Our invariants apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained. In this paper we calculate the topological quantum invariants of some examples explicitly. We conclude from our examples that our invariant is new and strong enough to distinguish some classical knots from one another. We also explain how different Our topological quantum invariants of 3-manifolds with boundary and the Reshetikhin-Turaev invariants are. (See the body for detail)., 53 pages, many figures. arXiv admin note: substantial text overlap with arXiv:2108.13547

For a knot diagram $K$, the classical knot group $\pi_1(K)$ is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this paper, we define a set of quotient groups associated to a knot diagram $K$. These quotient groups are invariant under the Reidemeister moves and the set includes the extended knot groups defined by Boden et al and Silver and Williams., Comment: 12 pages, 13 figures

In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots can not be unknotted by crossing change on only odd crossings., 15 pages, 8 figures

Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Ryo Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct and obstruction to cosmetic crossings in classical links., 7 pages, 4 figures

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links. Key Words: Jones polynomial, bracket polynomial, extended bracket polynomial, arrow polynomial, Miyazawa polynomial, Khovanov complex, Khovanov homology, Reidemeister moves, virtual knot theory, differential, partial differential, grading, dotted grading, vector grading., 34 pages; small stylistic changes