arXiv research

The paper proves link homotopy invariants from tree invariants and explains indeterminacy.

Width trees link link invariants and bridge number.

New proof and formula linking fusion trees to quantum knot invariants.

New invariant for rooted trees with connections to knot theory.

Study configuration space integrals for Milnor invariants of string links and trivalent trees.

New spanning tree model connects knot homology, s-invariant, and exotic discs.

New equivalence found between knot invariants.

Trees represent critical points, linking function topology.

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depe…

The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…

Graph homomorphism numbers embed graphs for classification.

Proves a conjecture about 3-manifold invariants using trees and asymptotic formulas.

Improved Shapley Values for tree-based models, more accurate than existing methods.

It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of…

Develops a relative Rips machine for studying group actions on R-trees.

It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtrat…

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term rela…

New tree structure for pseudo-Anosovs from interval maps.

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.

We relate the tree part of the Aarhus integral to Milnor's mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum.

Study on embedding tree products into groups, distinguishing them.

Invariant detects triple points in sphere immersions.

We extend Forester's rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups).

TD-GEN generates graphs using tree decomposition, improving efficiency and performance.

Graphs with specific spanning trees yield RAAGs, with applications to BBGs.

Theory of structure trees applied to network max-flow min-cut theorem and group theory.

Link concordance and Whitney towers linked to Milnor invariants.

Let G be a finitely presented group. Scott and Swarup have constructed a canonical splitting of G which encloses all almost invariant sets over virtually polycyclic subgroups of a given length. We give an alternative construction of this regular neighbourhood, by showing that it is the tree of cylinders of a JSJ splitt…

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants…

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 Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to dias…

A new knot invariant using tangle-valued 1-cocycles.

Proposes a new BSP-Tree process for flexible space partition modeling.

Determinants of theta curves and symmetric graphs are studied.

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin o…

This paper and its companion arXiv:0911.3173 have been replaced by arXiv:1602.05139. We define the compatibility JSJ tree of a group G over a class of subgroups. It exists whenever G is finitely presented and leads to a canonical tree (not a deformation space) which is invariant under automorphisms. Under acylindricity…

Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links. We present a different and conjecturally complete construction, that uses element…

Treant creates evasion-resistant decision trees.

The paper proposes a method for interpretable mixture density estimation using a tree structure.

New method finds knots without low treewidth diagrams.

We define a variation of Khovanov homology with an explicit description in terms of the spanning trees of a link projection. We prove that this new theory is a link invariant and describe some of its properties. Finally, we provide some the results of some computer computations of the invariant.

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree …

Energy trees handle complex data structures with multiple variable types.

Extended Jarrow-Rudd model with skewness and kurtosis for option pricing.

New connection found between complex polynomials and surface homeomorphisms.

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $Λ_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $Λ_K$, the homology of a di…

Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.