On osculating framing of real algebraic links

TL;DR

This paper establishes a relationship between the maximal encomplexed writhe and the maximal self-linking number with respect to osculating plane framing in real algebraic links, linking geometric and topological invariants.

Contribution

It proves that the maximal encomplexed writhe corresponds exactly to the maximal self-linking number with osculating plane framing for real algebraic links.

Findings

Maximal encomplexed writhe is characterized by maximal self-linking number.

The osculating plane framing plays a key role in understanding link invariants.

The result connects algebraic, geometric, and topological properties of real algebraic links.

Abstract

For a real algebraic link in $RP^3$, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree.

Equations

$$F+B=d(d-2)-2h-2i.$$

$$\chi (X)+B+F+\chi (\mathbb{R}\tilde{C})=d.$$

$$d-\sum _{x\in C\cap {\mathbb{RP}}^2}(m(x)-r(x))=d^{\vee }-\sum _{x\in C^{\vee }\cap {{\mathbb{RP}}^2}^{\vee }}(m^{\vee }(x)-r^{\vee }(x))$$

$$b_p(L)=\sum _{q}s(q),\text{thus}|b_p(L)|\le h(C_p)$$

$$|\mathrm{osc}(L)|\le |\mathrm{osc}(L)-b_p(L)|+|b_p(L)|\le \frac{1}{2}f(C_p)+h(C_p)\le \frac{1}{2}d(d-2)$$

$$s(q)=+1\text{for any hyperbolic crossing q of Cp.}$$

$$2\mathrm{lk}(l_i,K_i)=a_i+2,2\mathrm{lk}({\text{l}}_i^{\prime },K_i)=a_i.$$

$$2\mathrm{osc}(K_i)=a_i(a_i+2)$$

$$2h\ge \sum _{i=1}^n(a_i+2)(a_i-1)+\sum _{i\ne j}|\mathrm{lk}(K_i,K_j)|.$$

$$h\le (d-1)(d-2)/2-g-1\le (d-1)(d-2)/2-n$$

$$d(d-2)$$

$$\le \sum _{i=1}^n{a}_i(a_i+2)+2h-\sum _{i=1}^n(a_i+2)(a_i-1)$$

$$=2h+2n+\sum _{i=1}^n{a}_i\le (d-1)(d-2)+\sum _{i=1}^n{a}_i.$$

$$\begin{split}2\operatorname{osc}(L)&=\sum_{i=0}^{g}\operatorname{osc}(K_{i})+\sum_{i\neq j}\operatorname{lk}(K_{i},K_{j})=\sum_{i=0}^{g}a_{i}(a_{i}+2)+\sum_{i\neq j}a_{i}a_{j}\\ &=\Big{(}\sum a_{i}\Big{)}^{2}+2\sum a_{i}=(d-2)^{2}+2(d-2)=d(d-2).\end{split}$$

Full text

On osculating framing of real algebraic links

Grigory Mikhalkin and Stepan Orevkov

Université de Genève, Section de Mathématiques, Battelle Villa, 1227 Carouge, Suisse.

grigory.mikhalkinunige.ch

Steklov Mathematical Institute, Gubkina 8, 119991, Moscow, Russia; IMT, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France.

orevkovmath.ups-tlse.fr

For a real algebraic link in $\mathbb{RP}^{3}$, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree.

††support: Research is supported in part by the SNSF-grants 159240, 159581, 182111 and the NCCR SwissMAP project (G.M.), and by RFBR grant No 17-01-00592a (S.O.)

1. Introduction and statement of the main result

By real algebraic curve in $\mathbb{RP}^{3}$ we mean a complex curve in $\mathbb{CP}^{3}$ invariant under complex conjugation. We use the same notation for a real curve and the set of its complex points and, if it is denoted by $A$, then $\mathbb{R}A$ stands for the set of real points which is called a real algebraic link if it is non-empty and $A$ is smooth. A real algebraic link is called maximally writhed or $MW_{\lambda}$-link if $|w_{\lambda}(L)|$ (a variation of Viro’s invariant [7]) attains the maximal possible value $(d-1)(d-2)/2-g$ where $d$ and $g$ is the degree and genus of $A$ respectively. We refer to [3] for a precise definition of $w_{\lambda}$.

In [3, Thm. 2] we proved that several topological and geometric invariants are maximized on $MW_{\lambda}$-links. In this paper we add one more item to this collection: we show that the self-linking number of $L$ with respect to the osculating framing attains its maximal value (for links of a given degree) if and only if $L$ is an $MW_{\lambda}$-link. The proof is very similar to that of the main theorem of [3]. Let us give precise definitions and statements.

Let $L$ be an oriented link in a rational homology 3-sphere. A framing of $L$ is a continuous 1-dimensional subbundle of the normal bundle of $L$ or, equivalently, a continuous field (defined on $L$) of 2-dimensional planes tangent to $L$. Given a framed oriented link $L$, its self-linking number is defined as follows. Let $F$ be an embedded annulus or Möbius band with core $L$, tangent to the framing. Then the self-linking number is $\frac{1}{2}\operatorname{lk}(L,\partial F)$ where the boundary $\partial F$ of $F$ is oriented so that $[\partial F]=2[L]$ in $H_{1}(F)$.

For an oriented link $L$ in $\mathbb{RP}^{3}$, the osculating framing is the framing defined by the field of osculating planes. We denote the self-linking number of $L$ with respect to this framing by $\operatorname{osc}(L)$. If $L$ is a non-oriented link and $O$ an orientation of $L$, we use the notation $\operatorname{osc}(L,O)$ which is self-explained.

Recall that a smooth irreducible real algebraic curve $A$ is called an $M$-curve if $\mathbb{R}A$ has $g+1$ connected components where $g$ is the genus of $A$. In this case $\mathbb{R}A$ divides $A$ into two halves. The boundary orientation on $\mathbb{R}A$ induced by any of these halves is called a complex orientation. The main result of the paper is the following.

Theorem 1

Let $L=\mathbb{R}A$ be an irreducible real algebraic link of degree $d\geq 3$ and $O$ be an orientation of $L$. Then:

**Remark. ** In the space of real algebraic links of a given degree and genus we can distinguish three kinds of “walls”. The walls of the first kind correspond to curves with a double point with real local branches. When crossing such walls, both invariants $w_{\lambda}(L)$ and $\operatorname{osc}(L)$ are changed by $\pm 2$. The walls of the second kind correspond to curves with a real double point with complex conjugate local branches. When crossing such walls, $w_{\lambda}(L)$ does change but $\operatorname{osc}(L)$ does not. The third kind of wall corresponds to curves which have a local branch parametrized by $t\mapsto(t,t^{3}+o(t^{3}),t^{4}+o(t^{4}))$ in some affine chart. When crossing such a wall, $w_{\lambda}(L)$ does not change but $\operatorname{osc}(L)$ does. So, in general, the invariants $w_{\lambda}(L)$ and $\operatorname{osc}(L)$ are more or less independent. Nevertheless, Theorem 1 implies that the chamber where they have maximal value is bounded only by the walls of the first kind – common for the both invariants.

2. A variant of Klein’s formula for the number of real inflection points

Let $C\in\mathbb{P}^{2}$ be a nodal real irreducible algebraic curve. It may have three types of nodes: real nodes with real local branches of $C$, real nodes with imaginary local branches of $C$, or non-real nodes (coming in conjugate pairs). Denote the number of nodes of each type with $h$, $e$, and $i$ respectively.

A real flex is a local real branch of $C$ with the order of tangency $\omega$ to its tangent line greater than 1 (i.e. the local intersection number is $\omega+1\geq 3$). The multiplicity of a real flex is $\omega-1$. In an affine chart of $\mathbb{P}^{2}$ a flex corresponds to a critical point of the Gauss map. It is easy to see that the multiplicity of a flex equals to the multiplicity of the corresponding critical point. Thus a multiple flex can be thought of as $\omega-1$ ordinary flexes collected at the same point. We denote with $F$ the number of flexes counted with multiplicities.

A solitary real bitangent is a real line $L\subset\mathbb{P}^{2}$ which is tangent to $C$ at a non-real point (and thus also at the complex conjugate point). The multiplicity of $L$ is the sum of the orders $\omega$ over all local branches of $C\setminus\mathbb{RP}^{2}$ tangent to $L$. We denote with $B$ the number of solitary real bitangents counted with multiplicities. Clearly, $B$ is an even number.

Lemma \lemFlex

(Klein’s formula [1] for nodal curves). For a nodal real irreducible curve of degree $d$ in $\mathbb{P}^{2}$ we have

[TABLE]

Demonstration Proof

As in [6], we use additivity of the Euler characteristic $\chi$ to derive Klein’s formula. Let $\nu:\tilde{C}\to C$ be the normalization. The space of all real lines in $\mathbb{P}^{2}$ is homeomorphic to $\mathbb{RP}^{2}$, and thus has the Euler characteristic 1. For a real line $L$ the set $\nu^{-1}(L)$ consists of $d$ distinct points unless $L$ is tangent to $C$. Each tangency decreases the size of this set by $\omega$.

Consider the space $X=\{(p,L)\mid p\in C,\ L\ni p\}$, where $L\subset\mathbb{RP}^{2}$ is a real line. From the observation above we deduce

[TABLE]

Note that $\chi(\mathbb{R}\tilde{C})=0$ and $\chi(X)=\chi(\nu^{-1}(C\setminus\mathbb{R}C))=\chi(\tilde{C})-2e$, as each point of $\mathbb{R}C$ lifts to a circle in $X$ while $\chi(S^{1})=0$. The lemma now follows from the adjunction formula $\chi(\tilde{C})=3d-d^{2}+2e+2h+2i$.

Remark 2.2. Lemma 2.1 can be also obtained as an almost immediate consequence from Schuh’s generalization [5] of another Klein’s formula

[TABLE]

(see [6, Thm. 6.D] for a proof via Euler characteristics) combined with the class formula $d^{\vee}=d(d-1)-2e-2h-2i$. Here $C^{\vee}$ is the dual curve, $d^{\vee}$ is its degree, $m(x)$ and $r(x)$ (resp. $m^{\vee}(x)$ and $r^{\vee}(x)$) are the multiplicity and the number of real local branches of $C$ (resp. of $C^{\vee}$) at $x$.

3. Proof of the main theorem

Let $L=\mathbb{R}A$ be a smooth irreducible real algebraic link of degree $d$ endowed with an orientation $O$. Let $\Cal{U}$ be the set of points $p$ in $\mathbb{RP}^{3}\setminus L$ such that the projection of $L$ from $p$ is a nodal curve.

Fix a point $p\in\Cal{U}$. Let $C_{p}=\pi_{p}(A)$ where $\pi_{p}:\mathbb{P}^{3}\setminus\{p\}\to\mathbb{P}^{2}$ is the linear projection from $p$. Consider the field of tangent planes to $L$ passing through $p$, (so-called blackboard framing). Let $b_{p}(L)$ be the self-linking number with respect to it. We have

[TABLE]

where $q$ runs the hyperbolic (i. e., with real local branches) double points of $C_{p}$, $h(C_{p})$ is the number of them, and $s(q)$ is the sign of the crossing at $q$ in the sense of knot diagrams. The difference $|\operatorname{osc}(L)-b_{p}(L)|$ is bounded by one half of the number of those points where the osculating plane passes through $p$. This is the number of real flexes of $C_{p}$ which we denote by $f(C_{p})$. We have $f(C_{p})\leq d(d-2)-2h(C_{p})$ by Lemma 2.1. Thus

[TABLE]

which is Part (a) of Theorem 1.

Now suppose that $|\operatorname{osc}(L)|=d(d-2)/2$. Then for any choice of $p\in\Cal{U}$ we have the equality sign everywhere in (2), in particular, we have the equality sign in (1), i.e., all crossings are of the same sign, say, positive:

[TABLE]

By Lemma 2.1, the equality sign in the last inequality of (2) implies that all flexes of $C_{p}$ are ordinary for any choice of $p\in\Cal{U}$. This implies that $L$ has non-zero torsion at each point. Indeed, otherwise there exists a real plane $P$ which has tangency with $L$ of order greater than $3$. It is easy to check that $\Cal{U}$ has non-empty intersection with any plane, thus we can choose a point $p\in\Cal{U}\cap P$, and then $C_{p}$ would have a $k$-flex with $k>3$. Moreover, the positivity of all crossings for any generic projection implies that the torsion is everywhere positive (cf. the proof of [2, Prop. 1]).

Similarly to [2, 3], we derive from these conditions that the real tangent surface $TL$ (the union of all real lines in $\mathbb{RP}^{3}$ tangent to $L$) is a union of (non-smooth) embedded tori. Indeed, suppose that two tangent lines cross. Let $P$ be the plane passing through them (any plane passing through them if they coincide) and let $\ell$ be the line passing through the two tangency points. Let $p$ be a generic real point on $\ell$. Then $C_{p}$ has two real local branches at the same point such that each of them is either singular or tangent to the line $\pi_{p}(P)$. Since $L$ has non-zero torsion, all singular branches of $C_{p}$ are ordinary cusps. Then we can find a generic point close to $p$ such that the projection from it does not satisfy (3).

Let $K_{1},\dots,K_{n}$ be the connected components of $L$, and let $TK_{i}$ be the connected component of $TL$ that contains $K_{i}$ (the union of real lines tangent to $K_{i}$). The same arguments as in [3, Lemma 4.12] show that, for some positive integers $a_{1},\dots,a_{n}$, there exist real lines $\ell_{i}$, $\ell^{\prime}_{i}$, $i=1,\dots,n$, such that (for suitable choice of the orientations) the linking numbers of their real loci $l_{i}=\mathbb{R}\ell_{i}$ and $l^{\prime}_{i}=\mathbb{R}\ell^{\prime}_{i}$ with the components of $L$ are:

[TABLE]

Moreover, each $TK_{i}$ splits $\mathbb{RP}^{3}$ into two solid tori $U_{i}$ and $V_{i}$ such that $l_{i}\subset U_{i}$, $l^{\prime}_{i}\subset V_{i}$, the homology classes $[l_{i}]_{U}$ and $[l^{\prime}_{i}]_{V}$ generate $H_{1}(U_{i})$ and $H_{1}(V_{i})$ respectively, and we have $[K_{i}]_{U}=a_{i}[l_{i}]_{U}$ and $[K_{i}]_{V}=(a_{i}+2)[l^{\prime}_{i}]_{V}$. It follows that

[TABLE]

(the linking number of $K_{i}$ with its small shift disjoint from $TL$). Indeed, if $K_{i}$ is parametrized by $t\mapsto r(t)$ and the torsion is non-zero, then $TK_{i}$ has a cuspidal edge along $K_{i}$ and a small shift of $K_{i}$ in the direction of the vector field $\ddot{r}$ is disjoint from $TK_{i}$ (see Figure 1). A priori this argument proves (5) up to sign only. However the positivity of the torsion implies that $\operatorname{osc}(K_{i})$ is positive.

\botcaption

Figure 1 \endcaption

If $L$ is connected (i. e., $n=1$), it remains to note that then the condition $2\operatorname{osc}(K_{1})=d(d-2)$ implies $(a_{1}+2)a_{1}=d(d-2)$, hence $a_{1}=d-2$. Thus $L$ satisfies Condition (v) of [3, Thm. 1] which concludes the proof that $L$ is an $MW$-knot.

If $L$ is not necessarily connected, we argue as follows. By Murasugi’s result [4, Prop. 7.5] (see also [3, Prop. 1.2]), the number of crossings of any projection of $K_{i}$ is at least $(a_{i}+2)(a_{i}-1)/2$. Hence, for $h=h(C_{p})$, we have

[TABLE]

On the other hand, if we choose $p$ on a line passing through a pair of complex conjugate points, then $C_{p}$ has at least one elliptic double point (i. e., a real double point with complex conjugate local branches), whence by the genus formula we obtain

[TABLE]

(the second inequality in (7) is the Harnack’s bound). Hence

[TABLE]

Thus $\sum a_{i}\geq d-2$ and we conclude that $L$ is an $MW_{\lambda}$-link. This fact follows from [3, Prop. 1.1] (which implies that $\operatorname{ps}(L)=\sum a_{i}$) combined with [3, Thm. 2] (which claims, in particular, that $L$ is an $MW_{\lambda}$-link as soon as $\operatorname{ps}(L)\geq d-2$). Here we denote with $\operatorname{ps}(L)$ the plane section number of $L$. It is a topological invariant of a link in $\mathbb{RP}^{3}$ defined in [3] as the minimal number of intersection points with a generic plane where the minimum is taken over the isotopy class of the link.

Let us show that $O$ is a complex orientation of $L$. It is easy to see that the plane section number is at most $d-2$ for any algebraic link of degree $d$. Indeed, it is enough to consider a small shift of a non-osculating tangent plane in a suitable direction. Thus the inequality in $\operatorname{ps}(L)=\sum a_{i}\geq d-2$ is in fact an equality. It follows that the equality is attained in all the inequalities used in the proof, in particular, we have $|\operatorname{lk}(K_{i},K_{j})|=\operatorname{lk}(K_{i},K_{j})$ for $i\neq j$. Since all components of an $MW_{\lambda}$-link endowed with a complex orientation are positively linked (see [3]), we are done. This completes the proof of the “only if ” part of (b).

To prove the “if ” part of (b), we notice that by [3, Thm. 3 and §4.4], any $MW_{\lambda}$-link $L$ of degree $d$ and genus $g$ is a union of $g+1$ knots $K_{0}\cup\dots\cup K_{g}$ and $\operatorname{lk}(K_{i},K_{j})=a_{i}a_{j}$, $i\neq j$, for some positive integers $a_{0},\dots,a_{g}$ with $a_{0}+\dots+a_{g}=d-2$. Furthermore, the torsion of $L$ is everywhere positive and each knot $K_{i}$ is arranged on its tangent surface $TK_{i}$ as described above, thus (5) holds for each $i$, and we obtain

[TABLE]