62 citations · 152 across the 18 of their papers we have counts for
18 papers
Encoding knots by clasp diagrams
Jacob Mostovoy, Michael Polyak
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for buildi…
Pseudotropical curves
Sergei Lanzat, Michael Polyak
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves a…
Integrating curvature: from Umlaufsatz to J^+ invariant
Sergei Lanzat, Michael Polyak
Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral…
Diassociative algebras and Milnor's invariants for tangles
Olga Kravchenko, Michael Polyak
We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in…
Alexander-Conway invariants of tangles
Michael Polyak
We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The…
Counting real curves with passage/tangency conditions
Sergei Lanzat, Michael Polyak
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree…