Narrow Results

We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.

We prove that the N-colored Jones polynomial for the torus knot satisfies the second order difference equation, which reduces to the first order difference equation for a case of . We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for .

A double-torus knot is a knot embedded in a genus two Heegaard surface in S³. We consider double-torus knots L such that is connected, and consider fibred knots in various classes.

We prove that the 3-sphere is a covering of itself branched over any volume zero knot.

We consider a family of words in a free group of rank n which determine 3-manifolds ℳ_n(p,q). We prove that the fundamental groups of ℳ_n(p,q) are cyclically presented, and that ℳ_n(p,q) is the n-fold cyclic covering of the 3-sphere branched over the torus knots T(p,q) if p is odd and q≡±2(mod p). We also obtain an explicit Dunwoody parameters for the torus knots T(p,q) for odd p and q≡±2(mod p).

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids , where δ_p = σ_p-1σ_p-2 · σ₂σ₁, or equally of the q-string braids . As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K) : c(K) ≤ ⌊(g(K)+9)/6⌋ and c(K) ≤ ⌊(n(K) + 16)/12⌋. The (6n - 2,3) torus knots show that these bounds are sharp.

Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng.

In particular, the torus knot T_p,q can be parameterized as

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

We give a direct computation of the Khovanov knot homology of the (3, m) torus knots/links. Our computation yields complete results with ℤ[½] coefficients, though we leave a slight ambiguity concerning 2-torsion when integer coefficients are used. Our computation uses only the basic long exact sequence in knot homology and Rasmussen's result on the triviality of the embedded surface invariant.

The 2-bridge number of knots was introduced by Hass, Rubinstein and Thompson [Knots and k-width, Geom. Dedicata143 (2009) 7–18] as a natural generalization of the bridge number introduced by Schubert [Über eine numerisch knoteninvariante, Math. Z.61 (1954) 245–288]. We show that the 2-bridge number of a torus knot of type (p, q) is (p - 1)q + 2 if 1 < p < q.

The notion of Gem–Matveev complexity (GM-complexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GM-complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base 𝔻² and two exceptional fibers and, therefore, for all torus knot complements.

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

We classify Legendrian torus knots in S¹ × S² with its standard tight contact structure up to Legendrian isotopy.

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot $K_{(r,s)}$. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on ${𝕋}_{p/q}$, the result of $p/q$ surgery in the left-handed trefoil knot $𝕋$, J. Knot Theory Ramifications24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Let $t_{\alpha ,\beta }\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $\alpha$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{\alpha ,\beta }$ if $\alpha \ge 3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{\alpha ,\beta }$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.

For an effectively $n$-colorable knot $K$, the palette number ${\mathrm{\text{C}}}_n^{\ast }(K)$ is the minimum number of distinct colors for all effective $n$-colorings of $K$. It is known that ${\mathrm{\text{C}}}_n^{\ast }(K)\ge 2+\lfloor {log}_{2}n\rfloor$ for any effectively $n$-colorable knot $K$. In this paper, we show that for any odd $n\ge 3$ and effectively $n$-colorable torus knot $K$ it holds that ${\mathrm{\text{C}}}_n^{\ast }(K)=2+\lfloor {log}_{2}n\rfloor$ namely, any effectively $n$-colorable torus knot has an effectively $n$-colored diagram with $2+\lfloor {log}_{2}n\rfloor$ colors.

Dean introduced twisted torus knots, which are obtained from a torus knot and a torus link by splicing them together along a number of adjacent strands of each of them. We study the knot types of these knots.