Closures of T-homogeneous braids are real algebraic

TL;DR

The authors address the Benedetti–Shiota conjecture by proving that a large class of fibered links are real algebraic. They develop and leverage Rampichini diagrams to encode P-fibered braids, and use inner-loop insertions and odd symmetry to construct explicit loops of polynomials whose zeros realize the target braids as links of isolated singularities. A key advance is producing semiholomorphic real polynomial maps with isolated singularities whose links match closures of T-homogeneous braids and braids with positive Garside factors, thereby broadening the catalog of known real algebraic links. The results yield concrete corollaries for homogeneous braids and provide a path to representing any link as part of a real algebraic pair, with implications for the topology of polynomial mappings in four real dimensions.

Abstract

A link in $S^3$ is called real algebraic if it is the link of an isolated singularity of a polynomial map from $\mathbb{R}^4$ to $\mathbb{R}^2$. It is known that every real algebraic link is fibered and it is conjectured that the converse is also true. We prove this conjecture for a large family of fibered links, which includes closures of T-homogeneous (and therefore also homogeneous) braids and braids that can be written as a product of the dual Garside element and a positive word in the Birman-Ko-Lee presentation. The proof offers a construction of the corresponding real polynomial maps, which can be written as semiholomorphic functions. We obtain information about their polynomial degrees.

Figures (12)

• Figure 1: The motion in the complex plane of the critical values $v_j(t)$ of the loop of polynomials $g_t$ corresponding to a P-fibered geometric braid.
• Figure 2: The band generator $a_{2,5}$ in $\mathbb{B}_6$.
• Figure 3: a) The singular leaves of the singular foliation on $D_t$ induced by $\arg(g_t)$ and the definition of the segments $A_i$. Small circles are roots of $g_t$. b) The singular leaf containing $c_k(t)$. The transposition associated to $c_k(t)$ and $v_k(t)=g_t(c_k(t))$ is $(i\to j)$.
• Figure 4: A Rampichini diagram. The transpositions at $t=0$ are $\tau_1=(1\to 4)$, $\tau_2=(1\to3)$ and $\tau_3=(1\to2)$. A band word for the fiber $F_{\varphi=2\pi-\varepsilon}$ is given by $a_{1,2}a_{3,4}^{-1}a_{1,4}$.
• Figure 5: The inner loop $\gamma_3$.

Theorems & Definitions (16)

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