2021年7月23日 · A long knot is a smooth embedding $f\colon \mathbb{R} \to \mathbb{R}^3$ such that $f(t)=(t, 0, 0)$ for all $|t| > 1$. Given a smooth oriented knot $g\colon S^1 \to \mathbb{R}^3$, we can cut it at a point and extend it to a long knot, and given any long knot, we can add a closure strand to recover the knot. Connected sums for knots correspond to ...

The knot (/ nɒt /) is a unit of speed equal to one nautical mile per hour, exactly 1.852 km/h (approximately 1.151 mph or 0.514 m/s). [1][2] The ISO standard symbol for the knot is kn. [3] The same symbol is preferred by the Institute of Electrical and Electronics Engineers (IEEE), while kt is also common, especially in aviation, where it is the...

2003年9月26日 · This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over the space of prime knots.

2019年7月31日 · Abstract: By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we ...

2007年1月1日 · Given a long knot f ∈ K ˆ, we denote the component of K ˆ containing f by K ˆ f. • We say f is a connected-sum of f 1, …, f n if there exists f ̃ ∈ K ˆ f with f ̃ = κ n (L 1, L 2, …, L n, f 1, f 2, …, f n), for some n, (L 1, L 2, …, L n) ∈ C 2 (n) and f …

2023年11月9日 · As an application, we give a full description of the homotopy type (away from 2 and up to the concordance embedding stable range) of the space of long knots of codimension at least 3.

2024年12月19日 · We’ll walk you step-by-step through how to tie some of the most basic knots, from the classic overhand knot to the fancy-sounding Prusik knot. Next time you need to tie a knot, you’ll have plenty of options at the ready. Tie an overhand knot if …

A long knot is a smooth embedding f: R ,→ Rd that coincides with a fixed linear embedding outside of a compact subset of R. The space of long knots, equipped with the weak C∞-topology, will be denoted by Emb(R,Rd). One similarly defines the space of long immersions of R into Rd, and we will denote this space by Imm(R,Rd).

This list of knots includes many alternative names for common knots and lashings. Knot names have evolved over time, and there are many conflicting or confusing naming issues. The overhand knot, for example, is also known as the thumb knot. The figure-eight knot is also known as the Savoy knot or the Flemish knot.

Locke's portrayal of primary cord, pendents and susidiaries, and of the three types of information-bearing knots that appear in standard ancient quipus: the overhand knot (i, j, 1 j), the figure-eight knot (g, h, 1 h), and the long knots tied as an overhand knot but casting the end through the loop from 2 to 9 times (2, ..., 9; 4 is shown ...