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Register a new userChord index for knots in thickened surfaces
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In this note, we construct a chord index homomorphism from a subgroup of $H_1(Sigma, mathbb{Z})$ to the group of chord indices of a knot $K$ in $Sigmatimes I$. Some knot invariants derived from this homomorphism are discussed.
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We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.
We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus$>1$ are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.
Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S times I$ to which these results apply are included.
In this note we make progress toward a conjecture of Durham--Fanoni--Vlamis, showing that every infinite-type surface with finite-invariance index 1 and no nondisplaceable compact subsurfaces fails to have a good curve graph, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and where the natural mapping class group action has infinite diameter orbits. Our arguments use tools developed by Mann--Rafi in their study of the coarse geometry of big mapping class groups.
In this paper, we prove that given two cubical links of dimension two in ${mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.
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