Hex, Braids, the Crossing Rule, and XH-Search

An Extension of the H-Search Algorithm for Artificial Hex Players

Hex is a classic board game invented in the middle of the twentieth century by Piet Hein and rediscovered later by John Nash. The best Hex artificial players analyse the board positions by deducing complex virtual connections from elementary connections using the H-Search algorithm. In this paper, we extend the H-search with a new deduction rule. This new deduction rule is capable of discoverin...

Cosmetic Crossings of Twisted Knots

We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of inserting full twists in the strings of closed braids and the operation of forming certain satellites of winding number zero. As a consequences of the main results, we prove the nugatory crossing conjecture for twisted fibered braids, for closed 3-braids and for Whitehead doubles of prime no...

Closed 3-braids Are Nearly Fibred

We classify fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by adding a crossing. The proof uses Gabai’s method of disk decomposition. The topmost term in the knot Floer homology of closed 3-braids is also computed.

Crossing Matrices and Thurston's Canonical Form for Braids

Pure Virtual Braids Homotopic to the Identity Braid

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.