Meander (mathematics)

In Mathematical, a meander is a configuration in the transversely being cut Plan ℝ ² formed by two simple plane curves. Intuitively, a meander can be seen like a road cutting a river through a certain number of bridges. One says open meander in the case or the two curves are isotopes has lines of the plan and meander closed in the case or a curve is closed and the other isotope has a line.

In the opened case, one can always find an isotopy which sends one of the two curves on a straight line L .

In the opened case, the number of points of crossings is a Integer positive N .

In the closed case, one can always find an isotopy which sends the noncompact curve on a straight line L .

In the closed case, the number of points of crossings is a Integer positive par 2n .

Two meanders are known as equivalent if they are isotopes in the plan ℝ ².

The meanders are objects difficult to count. One does not know a formula for the M_n number of meanders having N intersections.

One can color in black and white the areas of the plan determined by a meander while alternating.

Numbers meandric

The number of meanders distinct from order N is the number meandric M_n . The first fifteen numbers meandric are given below.

M ₁ = 1

M ₂ = 2

M ₃ = 8

M ₄ = 42

M ₅ = 262

M ₆ = 1.828

M ₇ = 13.820

M ₈ = 110.954

M ₉ = 933.458

M ₁₀ = 8.152.860

M ₁₁ = 73.424.650

M ₁₂ = 678.390.116

M ₁₃ = 6.405.031 050

M ₁₄ = 61.606.881 612

M ₁₅ = 602.188.541 928

Open meander

Being given a directed fixed line L in the Euclidean Plane $\ mathbb {R} ^2 \,$ , a open meander of order N is a directed curve which is not cut in $\ mathbb {R} ^2 \,$ which transversely cuts the line to N points for a certain positive entirety N . Two open meanders are known as equivalents if they are homeomorphic in the plan.

Examples

The open meander of order 1 cuts the line once:

The open meander of order 2 cuts the line twice:

Open numbers meandric

The number of open meanders distinct from order N is the open number meandric m_n . The first fifteen open numbers meandric are given below.

m ₁ = 1

m ₂ = 1

m ₃ = 2

m ₄ = 3

m ₅ = 8

m ₆ = 14

m ₇ = 42

m ₈ = 81

m ₉ = 262

m ₁₀ = 538

m ₁₁ = 1.828

m ₁₂ = 3.926

m ₁₃ = 13.820

m ₁₄ = 30.694

m ₁₅ = 110.954

Semi-meander

Being given a half-line R in the Euclidean Plane $\ mathbb {R} ^2 \,$ , a semi-meander of order N is a curve which is not cut in $\ mathbb {R} ^2 \,$ which transversely cuts the half-line to N points for a certain positive entirety N . Two semi-meanders are known as being equivalents if they are homeomorphic in the plan.

Examples

The semi-meander of order 1 cuts the half-line once:

The semi-meander of order 2 cuts the half-line twice:

Semi-méandriques numbers

The number of semi-meanders distinct from order N is the number semi-méandrique M_n (generally noted with a line above with the place a line below). The first fifteen semi-méandrique numbers are given below.

M ₁ = 1

M ₂ = 1

M ₃ = 2

M ₄ = 4

M ₅ = 10

M ₆ = 24

M ₇ = 66

M ₈ = 174

M ₉ = 504

M ₁₀ = 1.406

M ₁₁ = 4.210

M ₁₂ = 12.198

M ₁₃ = 37.378

M ₁₄ = 111.278

M ₁₅ = 346.846

Meandric properties of the numbers

There exists a injective Fonction numbers meandric towards the open numbers meandric:

M_n = m_ {2n - 1} \,

Each number meandric can be limited by semi-méandriques numbers:

M_n ≤ M_n ≤ M _{2 N}

For N > 1, the numbers meandric are even:

M_n \ equiv 0 \,

(MOD 2)

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