Narrow Results

In this paper, we give a combinatorial description of the concordance invariant $𝜀$ defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of $𝜀$ for $(p,q)$ torus knots and prove that $𝜀({𝔾}_{+})=1$ if ${𝔾}_{+}$ is a grid diagram for a positive braid. Furthermore, we show how $𝜀$ behaves under $(p,q)$-cabling of negative torus knots.

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant $\mathrm{{\rm Y}}$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that $\mathrm{{\rm Y}}$, as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that $\mathrm{{\rm Y}}$ provides a lower bound on the unknotting number.

The $\mathrm{{\rm Y}}$ invariant is a concordance invariant using knot Floer homology. Földvári [The knot invariant $\mathrm{{\rm Y}}$ using grid homologies, J. Knot Theory Ramifications30(7) (2021) 2150051] gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\mathrm{{\rm Y}}$ invariant for balanced spatial graphs. Regarding links as spatial graphs, we give an upper and lower bound for the $\mathrm{{\rm Y}}$ invariant when two links are connected by a cobordism. Also, we show that the combinatorial $\mathrm{{\rm Y}}$ invariant is a concordance invariant for knots.

Grid homology, a combinatorial version of link Floer homology introduced by Manolescu, Ozsváth, Szabó, Thurston [On combinatorial link Floer homology, Geom. Topol. 11 (2007) 2339–2412, MR 2372850], has been extended to transverse spatial graphs by Harvey and O’Donnol [Heegaard Floer homology of spatial graphs, Algebr. Geom. Topol. 17(3) (2017) 1445–1525, MR 3677933]. In this paper, we define grid homology for MOY graphs, a specific class of transverse spatial graphs, and show properties such as the oriented skein relation, the effect of contracting an edge, and the effect of uniting parallel edges with the same orientation.