Journal of Knot Theory and Its Ramifications

The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic $\gamma$ from the root of the tree to the tip of a leaf an irreducible component $C_{\gamma }$ of the representation variety of the web, and a graded subalgebra $A_{\gamma }$ of $H^{\ast }(C_{\gamma };\mathbb{Q})$. The spider evaluation of geodesic $\gamma$ is the symmetrized Poincaré polynomial of $A_{\gamma }$. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

In this paper, we consider biquandle colorings for knotoids in ${ℝ}^2$ or $S^2$, and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski’s biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements.

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure ${\xi }_K$ is supported by the fibered knot $K\subset M$, we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure ${\xi }_{\overline{K}}$ supported by the mirror knot $\overline{K}\subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, nonplanar contact structures, and nonplanar Legendrian knots.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that nontrivial knots in $S^3$ do not admit purely cosmetic surgeries. In this paper, we confirm this conjecture for cable knots.