A pair of distinguished generators of the fundamental group $\pi_1(\partial(S^3 \setminus K))$ of the boundary torus of a knot complement are usually called the "meridian" and "longitude". However, this terminology has always seemed a bit odd to me: in geography a meridian is a line of longitude, so shouldn't the curve it intersects be a "latitude"? Does anyone know the origin of this language?
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asked Jul 7, 2021 at 18:00
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3$\begingroup$ Maybe because it goes along the knot... Besides, geographical "latitude" is not a curve, it is rather a number, namely, the angle measure along the geographical meridian from the equator to the given point. $\endgroup$– მამუკა ჯიბლაძეCommented Jul 7, 2021 at 18:48
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1$\begingroup$ The lines that complement the meridians are called the parallels $\endgroup$– Manfred WeisCommented Jul 7, 2021 at 18:55
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2$\begingroup$ Sometimes "latitude" is also used to mean "line of constant latitude" even though "parallel" is apparently more accurate. $\endgroup$– Calvin McPhail-SnyderCommented Jul 7, 2021 at 18:56
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2$\begingroup$ I have no proof, but a plausible explanation for the naming may be that the osculating planes of the knot as a smooth spatial curve is interpreted as the equatorial plane of the osculating sphere whose "geographic" coordinates are adoptet $\endgroup$– Manfred WeisCommented Jul 7, 2021 at 19:15
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$\begingroup$ You know chicken salad is not the opposite of tuna, salmon is the opposite of tuna, because salmon swim against the current, and the tuna swim with it. $\endgroup$– markvsCommented Jul 8, 2021 at 18:33
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2 Answers
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There is a fundamental asymmetry between latitude and longitude on a sphere, whereas on a torus, there is a symmetry between the two generators. This symmetry could motivate the use of nearly synonymous words.
The terminology may originate with the paper On the homology invariants of knots by H. Seifert (Quart. J. Math. Oxford (2) 1 (1950), 23–32). The torus $T$ of interest to Seifert is the boundary of a closed tubular neighborhood $V$ of a knot in $\mathbb{R}^3$. Seifert says that a Jordan curve on $T$ that bounds on $V$ (respectively, on $\mathbb{R}^3 - V + T$) but not on $T$ is called a meridian (respectively, a longitudinal circuit). Seifert phrases these definitions in a way that highlights the symmetry, which makes the use of near synonyms seem natural.
answered Jul 8, 2021 at 0:08
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2$\begingroup$ In fairness, I should say that Seifert's use of the term longitudinal circuit (rather than longitude, as later authors seem to have settled on) is also consistent with Michael Hardy's suggestion; perhaps the idea is that a longitudinal circuit runs through all the possible values of the longitude. $\endgroup$ Commented Jul 8, 2021 at 13:11
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If a vector points northward, then it generates a meridian of longitude. Following that curve keeps one at the same longitude. A vector pointing eastward points in the direction of a curve whose points are at different longitudes. You mark the points on that curve with their longitudes.
Suppose a person's income were a function of the person's height and you want to graph that function. Label the horizontal axis "height" and the vertical axis "income". The axis labeled "height" is a line along which the height varies. It is not a line of constant height. Likewise the thing labeled "longitude" is a curve along which the longitude varies, not a curve of constant longitude.
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$\begingroup$ But you wouldn't call the axis labeled height "a height", whereas it seems that the thing labelled longitude is called "a longitude". $\endgroup$– LSpiceCommented Sep 23, 2021 at 3:33
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$\begingroup$ @LSpice : I think I need more context to say anything about your last comment. $\endgroup$ Commented Sep 23, 2021 at 4:10