Journal of Knot Theory and Its Ramifications

We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.

In this paper we present an explicit isomorphism between Khovanov–Rozansky sl₂-homology and ordinary Khovanov homology. This result was originally claimed in Khovanov and Rozansky's paper [Matrix factorizations and link homology, Fund. Math.199(1) (2008) 1–91, MR 2391017 (2010a:57011)], though the proof was never presented. The main missing detail is providing a coherent choice of signs when identifying variables in the sl₂-homology. Along with the behavior of the signs and local orientations in the sl₂-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the sl₂-homology may instead correspond to a different variant of Khovanov homology. In this paper we describe both theories and prove that they are in fact isomorphic by showing that a coherent choice of signs can be made. In doing so we emphasize the interpretation of the sl₂-complex as a cube of resolutions.

For a 1- or 2-dimensional knot, we give a lower bound log₂ n + 1 of the minimum number of distinct colors for all effective n-colorings. In particular, we prove that any effectively 9-colorable 1- or ribbon 2-knot is presented by a diagram where exactly five colors of nine are assigned to the arcs or sheets.

Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot colorability to this broader class of knot-like objects. We use these definitions to define the determinant of a pseudoknot (i.e. the pseudodeterminant) that agrees with the classical determinant for classical knots. Moreover, we extend Conway notation to pseudoknots to facilitate the investigation of families of pseudoknots and links. The general formulae for pseudodeterminants of pseudoknot families may then be used as a criterion for p-colorability of pseudoknots.

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

In this paper, we prove the validity of a formula for computing the Alexander invariant, which was originally conjectured by Bar-Natan and Dancso in [2].

Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of the Q-polynomial is less than its determinant. We give a refinement of their evaluation.