Non-peripheral ideal decompositions of alternating knots

Abstract

An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is nonperipheral if no edge of T is homotopic to a curve in the boundary torus of M . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M . A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are ideal triangulations that use 4 (resp., 5) ideal tetrahedra per crossing. Our main result is that these ideal triangulations are non-peripheral for all planar, reduced, alternating projections of hyperbolic knots. Our proof uses the small cancellation properties of the Dehn presentation of alternating knot groups, and an explicit solution to their word and conjugacy problems. In particular, we describe a planar complex that encodes all geodesic words that represent elements of the peripheral subgroup of an alternating knot group. This gives a polynomial time algorithm for checking if an element in an alternating knot group is peripheral. Our motivation for this work comes from the Volume Conjecture for knots.

An application of non-positively curved cubings of alternating links

“…The authors realized that the results of this paper can be derived from the existence of non-positively curved cubings of alternating hyperbolic link exteriors in 2015, and it was announced by the second author at a conference at Waseda University in honour of the 20th anniversary of the Volume Conjecture. During the conference, we learned from Stavros Garoufalidis that essentially the same results had been already obtained in 2002 by the joint work of S. Garoufalidis, I. Moffatt and D. Thurston in [6], which was completed in 2007, but has not been published. Their method is based on the small cancellation property of the Dehn presentation.…”

“…To be precise, for a link L in S 3 , the arc in its exterior E(L) := S 3 \ N (L) obtained from the ideal edge of S is homotopic, relative to its endpoints, to an arc in ∂E(L). (Here, N (L) is an open regular neighborhood of L.) In [6], such an arc is called peripheral. An inessentail ideal edge has no geodesic representative in the hyperbolic manifold M , and conversely, if all ideal edges of S are essential (i.e., not inessential), then the edges have unique geodesic representatives, which gives a geometric solution to the hyperbolicity equations (though some of the tetrahedra may be flat or negatively oriented).…”

On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds

“…We assume the planar projection of an alternating knot is always reduced. For the topological ideal triangulation of S 3 \K, we take the triangulation of S 3 \{K ∪ two points} described by [3,9], the octahedral 4-term triangulation, and to give the geometric structure to M we send two points to ∂H 3 equivariantly, making sure that the resulting ideal simplices are non-degenerate (all four vertices distinct) as in [30, p.465]. With this ideal triangulation, a solution to the gluing and completeness equations can always be obtained to recover the hyperbolic structure of S 3 \K [9,35].…”

“…Recall that we defined the sequences (P j ) j∈Z , (Q j ) j∈Z , and (B j ) j∈Z by recursion relation (9) with the initial conditions…”