Journal of Knot Theory and Its Ramifications

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston–Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not.10 (2010) 1831–1859].

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

The Torelli group, , is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying : bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on , while SIP-maps have received relatively little attention until recently, due to an infinite presentation of introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP(S_g), and show it is an infinite index subgroup of .

For n = 1, 2 and 3, we construct a pair of transverse knots T₁ and , in the standard contact 3-sphere, satisfying the following properties:

(1) the topological knot type of T₁ is the same as that of ,

(2) the self-linking number of T₁ is equal to that of ,

(3) is obtained from a transverse knot T₂ by n stabilizations, and

(4) T₁ is not transversely isotopic to .

For N ≥ 2, we study a certain sequence of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of M(0,4), Math. Proc. Cambridge Philos Soc.141 (2006) 477–488] holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as p tends to infinity. Moreover, the limits describe the action of SL₂(ℤ) on the space of homogeneous polynomials of two variables of total degree N - 1.

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We determine the complete list (set) of pairs of integral surgeries along distinct torus knots whose resulting manifolds are orientation preserving/reversing homeomorphic lens spaces, and study the closed 4-manifolds constructed as above. The list consists of five sequences. All framed links and Kirby calculus are indexed by integers. As a bi-product, some sequences of embeddings of lens spaces into the standard 4-manifolds are constructed.

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n ≤ 10. In his thesis, Venzke mentions that these cannot be smallest volume for n ≥ 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n ≥ 60 we use a formula from work of Futer, Kalfagianni and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m + 1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n ≥ 60 or |m| ≥ 8, and we give computational data indicating this remains true for smaller n and |m|.

We prove a pattern theorem for 2-spheres in tubes in Z⁴ and use this to prove that all except exponentially few 2-spheres in a tube in Z⁴ are knotted. We sketch how the argument can be applied to prove the same result for p-spheres in a tube in Z^p+2, p > 2.

Let L = ℓ₁ ∪⋯∪ℓ_d+1 be an oriented link in 𝕊³, and let L(q) be the d-component link ℓ₁ ∪⋯∪ℓ_d regarded in the homology 3-sphere that results from performing 1/q-surgery on ℓ_d+1. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.

In this paper we propose a region choice problem for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.