Torus knots $T(p,q)$ are defined as curves on the standard torus $S^1 \times S^1 \subset \mathbb{R}^3$ that wind $p$ times around the meridian and $q$ times around the longitude, where $\gcd(p,q) = 1$.

Classical examples:

• $T(2,3)$: the trefoil knot $3_1$
• $T(2,5)$: the $(2,5)$-torus knot $5_1$ with 5 crossings
• $T(3,4)$: the $(3,4)$-torus knot with 12 crossings

Torus knots are always alternating and prime. The crossing number satisfies $c(T(p,q)) = \min(p(q-1), q(p-1))$.

Pretzel knots $P(a_1, a_2, \ldots, a_n)$ are constructed from $n$ rational tangles arranged in a cyclic pattern, where $a_i$ denotes the number of half-twists (signed) in the $i$-th tangle.

Examples:

• $P(3,-3,-3)$: produces a 9-crossing non-alternating knot
• $P(-2,3,7)$: the pretzel knot with signature properties
• $P(2n+1, 2m+1, 2k+1)$: general odd pretzel knots

Pretzel knots generalize torus knots and provide examples with specific symmetry properties.

Satellite knots provide another construction. Given a knot $K$ (the companion) and a pattern knot $P$ in a solid torus, the satellite $S(K,P)$ is obtained by embedding the solid torus containing $P$ along $K$ in $S^3$.

Cable knots are special satellites where $P$ is a torus knot on the boundary of the solid torus. The $(p,q)$-cable of $K$ is denoted $C_{p,q}(K)$.

For instance, cabling the trefoil produces infinite families of increasingly complex knots, all inheriting properties from the original trefoil.