Journal of Knot Theory and Its Ramifications

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

Let be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom(K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism

We use the methods of Hedden, Juhász and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_n which will bound at least 2^2n-1 non-isotopic minimal spanning surfaces of genus n.

In this paper, an expression for the Homfly polynomial of a classical pretzel link with a fixed orientation was obtained. As an application, we proved that a (p, q, r)-pretzel knot is achiral if and only if {p, q, r} = {2, 1, 1}, {-2, -1, -1}, {2, -1, 3}, {-2, 1, -3}, {1, -1, k}, {1, k, -2 -k} or {-1, -k, 2 + k}, where k is an odd integer.

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11-crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

We elaborate the Chern–Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for U_q(𝔰𝔩_N), we present explicit evaluations of the HOMFLY invariants colored by symmetric representations for a variety of knots, two-component links and three-component links.