Daniel Moskovich

Surgery untying of coloured knots

For p = 3 and for p = 5 we prove that there are exactly p equivalence classes of p-coloured knots modulo ± 1-framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n left-hand (p, 2)-torus knots with a given colouring when n = 1, 2,..., p. This gives a 3-colour and a 5-colour analogue of the surgery presentation of a knot.

Vanishing of 3-loop Jacobi diagrams of odd degree

We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop Vassiliev invariant can distinguish between a knot and its inverse.

Computing with Colored Tangles

We present a formalism to distribute an interactive proof between multiple interacting verifiers. Each verifier has a belief as to whether the claim to be proven is true or false, and verifiers convince one another based on input from a prover/oracle subject to a deformation parameter δ ∈ (0, 1) which introduces ‘noise’ into the system. Usual interactive proofs are recovered at the limit δ → 1. The utility of our theory is demonstrated by a network of nonadaptive 3-bit query PCP verifiers whose resulting soundness improves over the best known result for a single verifier of this class. There is a natural equivalence relation on such ‘tangled interactive proofs’ stemming from their analogy with low-dimensional topological objects. Two such proofs are considered equivalent if, roughly, one can be completely reconstructed from the other. This induces a notion of zero knowledge for our tangled interactive proofs.We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated colored tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams represent bisimilar computations. We prove that our model of computation is Turing complete and with bounded resources that it can decide any language in complexity class IP, sometimes with better performance parameters than corresponding classical protocols.

Surgery presentations of coloured knots and of their covering links

We consider knots equipped with a representation of their knot groups onto a dihedral group D2n (where n is odd). To each such knot there corresponds a closed 3‐ manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D2n ‐coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D2n ‐coloured knots are related by a sequence of surgeries along 1‐framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Zn ‐valued algebraic invariant of D2n ‐coloured knots).

Tsirelson’s Bound Prohibits Communication through a Disconnected Channel

Why does nature only allow nonlocal correlations up to Tsirelson’s bound and not beyond? We construct a channel whose input is statistically independent of its output, but through which communication is nevertheless possible if and only if Tsirelson’s bound is violated. This provides a statistical justification for Tsirelson’s bound on nonlocal correlations in a bipartite setting.