Journal of Knot Theory and Its Ramifications

We introduce the bridge genus and the braid genus as two kinds of genera of a closed connected orientable 3-manifold, and compare them with the Heegaard genus. We show that for any closed connected orientable 3-manifold, the Heegaard genus is less than or equal to the bridge genus and the bridge genus is less than or equal to the braid genus. We consider four inequalities between these three genera and we construct a 3-manifold which satisfies each one of inequalities. Moreover, we show that there exist infinite many 3-manifolds which satisfy each one of inequalities.

By using Rasmussen's s-invariant derived from Khovanov homology, we give a combinatorial proof for the existence of countably many Casson handles.

This paper generalizes Van Cott's bounds on the Heegaard–Floer τ-invariant and Rasmussen's s-invariant of cabled knots to apply to all satellite knots.

We study Coxeter racks over ℤ_n and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.

A rack shadow or rack set is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.

Various theorems for classical piecewise-linear (PL) knots are shown to carry over to PL virtual knots. We then classify PL virtual knots up to edge index six.

In this paper, we study rational real algebraic knots in ℝP³. We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

The genus of knots is one of the fundamental invariants and can be seen as a complexity of knots. In this paper, we give a lower bound of the knot genus using the Dehornoy floor, which is a measure of complexity of braids in terms of the Dehornoy ordering.