Narrow Results

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 .

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.

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

A point in the $(N,q)$-torus knot in ${ℝ}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto (sin(qt+\varphi ),cos(pt+\psi )))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $\mathrm{\text{gcd}}(q,p)=1$, $K(N,q,p)$ is ribbon; if $\mathrm{\text{gcd}}(q,p)=d>1$, $B_{N,q,p}$ is the $d$th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$’s which are trivial knots.

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let $T(p,q)$ be any torus knot with $p$ even and $q$ odd. The difference between these two invariants on $T(p,q)$ is at least $\frac{k}{2}$, where $p=qk+a$ and $0<a<q$ and $k\ge 0$. Hence, the difference between the two invariants on torus knots $T(p,q)$ grows arbitrarily large for any fixed odd $q$, as $p$ ranges over values of a fixed congruence class modulo $q$. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot $T(p,q)$ is $\frac{1}{2}(p-1)(q-1)$, and Kronheimer and Mrowka later proved that the orientable four genus of $T(p,q)$ is also this same value.

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.

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.

A petal projection of a knot $K$ is a projection of a knot which consists of single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows:

Let $r$ be an odd integer, $r\ge 3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.

The minimal number of straight line segments required to form a given knot or link in ℝ³ is determined for a family of torus knots and links.

The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.