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Register a new userAn independence system as knot invariant
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In this article, we define an independence system for a classical knot diagram and prove that the independence system is a knot invariant for alternating knots. We also discuss the exchange property for minimal unknotting sets. Finally, we show that there are knot diagrams where the independence system is a matroid and there are knot diagrams where it is not.
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We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnolds curve invariant St, as well as the relation with Hass and Nowiks invariant, which generalizes cowrithe. In particular, the change of SCI under {Omega}3 moves depends only on the forward/backward character of the move, similar to how the change of St or cowrithe depends only on the positive/negative quality of the move.
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.
For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof involves the study of a relation between the genus of a virtual knot diagram and the genus of a knotoid diagram, the former of which has been introduced by Stoimenow, Tchernov and Vdovina, and the latter by Turaev recently. Our operation has a simple interpretation in terms of Gauss codes and hence can easily be computer-implemented.
In this paper we introduce a chain complex $C_{1 pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to delta-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra A_n. We show that A_n is isomorphic to B(2n+1, n), the algebra used for the DA-bimodule constructed by Ozsvath and Szabo in their algebraic construction of knot Floer homology.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of knot Floer homology.
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