On T-positive links

Abstract: T-positive links form a subset of strongly quasipositive links that strictly contains the set of all non-split braid positive links. Analogous to Baader's characterisation of positive links as precisely the strongly quasipositive and homogeneous links, we show that T-positive links are precisely the strongly quasipositive links that are the closures of T-homogeneous braids. This complements previous characterizations of T-positive links by Rudolph and Banfield as links arising as boundaries of positive Hopf-plumbed baskets, or closures of staircase braids. We examine the behavior of T-positive links under cabling operations and connected sums, and demonstrate that all strongly quasipositive, fibered knots with at most 12 crossings are T-positive. Additionally, we compare T-positivity with other positivity notions for links and compile open questions.

• The paper proves that a link is T-positive exactly when it is strongly quasipositive and T-homogeneous, unifying Rudolph’s Hopf-plumbed baskets with Banfield’s staircase braids.
• The authors show that cables of staircase-braid knots remain T-positive when p ≥ 2 and q ≥ n, provide an explicit braid construction, and leave necessity of this bound open.
• A computational survey finds all 42 strongly quasipositive fibered knots with at most 12 crossings are T-positive, while structural examples establish strict separations from braid positivity and strong quasipositivity.

This paper by Bode and Truöl studies $\mathcal{T}$-positive links, a class introduced by Rudolph over two decades ago, situated strictly between non-split braid positive links and strongly quasipositive links. The central contribution is a new intrinsic characterization of $\mathcal{T}$-positivity in terms of $T$-homogeneity, complementing the previously known characterizations via positive Hopf-plumbed baskets (Rudolph) and staircase braids (Banfield). The paper also analyzes cabling behavior, verifies $\mathcal{T}$-positivity for all strongly quasipositive fibered knots with at most 12 crossings, and compiles implications and non-implications among various positivity notions.

Definitions and background

The framework rests on the Birman–Ko–Lee (BKL) band generators $a_{i,j}$ of the braid group $B_n$, which generalize the Artin generators ($a_{i,i+1} = \sigma_i$). A braid is strongly quasipositive if it is a product of positive BKL-generators. An espalier is a planar tree $T$ with $n$ vertices embedded in the lower half-plane with vertices at $(j,0)$; each edge $\mathcal{T}$0 determines a $\mathcal{T}$1-generator $\mathcal{T}$2, giving a subset $\mathcal{T}$3 of size $\mathcal{T}$4 of the $\mathcal{T}$5 BKL-generators. A braid is $\mathcal{T}$6-homogeneous if each $\mathcal{T}$7-generator appears at least once and only with exponents of one fixed sign; it is $\mathcal{T}$8-positive if all appearances are positive. A link is $\mathcal{T}$9-positive (respectively $T$0-homogeneous) if it is the closure of such a braid for some espalier.

Choosing the linear graph $T$1 recovers positive braids (with every Artin generator appearing at least once), which yields the strict inclusions

$T$2

Rudolph showed that $T$3-positive links are exactly the boundaries of positive Hopf-plumbed baskets—Seifert surfaces obtained from a disk by iteratively plumbing positive Hopf bands with all plumbing arcs inside the original disk—and hence are fibered. Banfield characterized them as closures of staircase braids: strongly quasipositive braids containing a positive power of the dual Garside element $T$4.

The main characterization theorem

The paper's main result states that $T$5-positive links are precisely the strongly quasipositive links that are $T$6-homogeneous. This directly parallels Baader's theorem that positive links are exactly the strongly quasipositive homogeneous links.

The proof proceeds as follows. Given a strongly quasipositive link $T$7 with a $T$8-homogeneous representative $T$9, the braided Seifert surface $\mathcal{T}$0 is shown to be an iterated Murasugi sum of Seifert surfaces for torus links $\mathcal{T}$1, where $\mathcal{T}$2 is the exponent sum of the generator corresponding to edge $\mathcal{T}$3 of the espalier. Since $\mathcal{T}$4 is a fiber surface, it realizes the maximal Euler characteristic and is therefore isotopic to any quasipositive Seifert surface coming from a strongly quasipositive braid closure of $\mathcal{T}$5. Rudolph's result that Murasugi sums preserve quasipositivity in both directions then forces each summand to be quasipositive, so $\mathcal{T}$6. If all $\mathcal{T}$7, the braid is already $\mathcal{T}$8-positive. If some $\mathcal{T}$9, the single negative band can be flipped to positive by an ambient isotopy rotating one component of the cut-open surface through angle $a_{i,j}$0 about an axis through the band; iterating handles all such indices. This argument relies on the uniqueness up to isotopy of the incompressible maximal-Euler-characteristic surface bounding a fibered link, and on Traczyk's signature criterion for strong quasipositivity of torus links $a_{i,j}$1.

As a corollary, non-split braid positive links are precisely the strongly quasipositive links that are closures of homogeneous braids—a statement the authors believe may be known to experts but had not appeared explicitly in the literature. The analogy also motivates a natural open question: whether almost positive links admit a similar characterization as strongly quasipositive "almost homogeneous" links.

Cabling and staircase braids

Using Banfield's staircase characterization, the paper proves a sufficient condition for cables of $a_{i,j}$2-positive knots to remain $a_{i,j}$3-positive: if $a_{i,j}$4 is the closure of a staircase braid $a_{i,j}$5 and $a_{i,j}$6, $a_{i,j}$7, then the cable knot $a_{i,j}$8 is represented by a staircase braid on $a_{i,j}$9 strands. The proof gives an explicit algorithm: the $B_n$0-cable of each BKL-generator decomposes into $B_n$1 BKL-generators, and the cabled dual Garside element produces $B_n$2 after an isotopy, provided $B_n$3 residual negative $B_n$4-twists are cancelled by the $B_n$5 added positive twists—hence the threshold $B_n$6.

The authors note that Hedden's work implies $B_n$7 is strongly quasipositive and fibered whenever $B_n$8, so $B_n$9 is necessary; they ask whether $a_{i,i+1} = \sigma_i$0 is also necessary. The positive trefoil $a_{i,i+1} = \sigma_i$1 shows the condition is sharp in at least one instance. Notably, explicit strongly quasipositive braid representatives for cable knots appear not to have been documented before; the proof supplies them algorithmically.

Low crossing numbers

A computational result establishes that all 42 strongly quasipositive, fibered knots with at most 12 crossings are $a_{i,i+1} = \sigma_i$2-positive, and consequently all 33 positive fibered knots with at most 12 crossings are as well. Of these, 17 are braid positive, five have braid index 3 (handled by a proposition showing that strongly quasipositive fibered 3-braid closures always admit staircase representatives, via Stoimenow's classification), and the remaining 20 are verified case-by-case through either explicit staircase braid words or genus-two/three positive Hopf-plumbed basket descriptions computed with SnapPy and Sage. This means no counterexample to the paper's central open question exists below 13 crossings or at braid index 2 or 3.

Further observations

Several structural results round out the picture:

• Connected sums: the connected sum of closures of $a_{i,i+1} = \sigma_i$3- and $a_{i,i+1} = \sigma_i$4-homogeneous (or $a_{i,i+1} = \sigma_i$5/$a_{i,i+1} = \sigma_i$6-positive) braids is the closure of a $a_{i,i+1} = \sigma_i$7-homogeneous (resp. positive) braid, where $a_{i,i+1} = \sigma_i$8 is the vertex-connected sum of espaliers. More generally, any braided Stallings plumbing preserves these properties. However, arbitrary Murasugi sums do not: Misev constructed infinitely many positive Hopf plumbings per genus that are not baskets, hence not $a_{i,i+1} = \sigma_i$9-positive.
• Visual primeness: whereas positive and homogeneous braid diagrams are visually prime (Cromwell; Feller–Lewark–Orbegozo), the paper exhibits a $T$0-positive braid diagram whose closure is $T$1 but whose dual graph has no length-2 loop, so the diagram admits no decomposition circle. Thus $T$2-positive diagrams need not be visually prime.
• Unknotting number: the $T$3-positive knot $T$4 satisfies $T$5, providing a concrete instance relevant to Stoimenow's conjecture that $T$6 for positive fibered knots.
• Positive trefoil plumbings: the classes are incomparable. The knot $T$7 is $T$8-positive but not a positive trefoil plumbing, while infinitely many positive trefoil plumbings (distinguished by Alexander polynomials, against finitely many baskets per genus) are not $T$9-positive.
• Strictness of inclusions: infinitely many $n$0-positive knots are not braid positive (the family $n$1 of Kegel–Lewark–Manikandan–Misev–Mousseau–Silvero), and infinitely many strongly quasipositive fibered knots are not $n$2-positive—for example, the $n$3-cable of the trefoil, whose fiber surface does not deplumb a Hopf band by Melvin–Morton.

Two flowcharts summarize all known implications and non-implications among positivity notions, including conditional arrows for the open questions.

Limitations and open questions

The paper leaves several questions explicitly open. Foremost is whether there exist positive, fibered knots that are not $n$4-positive; the low-crossing-number and low-braid-index results rule out counterexamples in those regimes, but no general obstruction or construction is given. The converse direction of the cabling criterion (whether $n$5 being a staircase closure forces $n$6) remains unresolved except for the trefoil example. It is also unknown whether infinitely many knots are $n$7-positive but not positive trefoil plumbings, and whether almost positive links admit a homogeneity-based characterization analogous to the main theorem. On the technical side, the step-by-step verification of the Murasugi sum decomposition in Claim 1 is left partly to the reader, and the classification of the 20 braid-index-4 knots relies on computer-assisted verification in SnapPy and Sage rather than a uniform theoretical argument.

Conclusion

The paper consolidates the theory of $n$8-positive links by proving that they coincide with strongly quasipositive $n$9-homogeneous links, thereby completing a triad of characterizations alongside Rudolph's basket description and Banfield's staircase braids. The result yields, as special cases, clean statements for braid positive and positive links, and the accompanying computations establish that within the 12-crossing census, $(j,0)$0-positivity is indistinguishable from the conjunction of strong quasipositivity and fiberedness. The remaining separation questions—particularly whether positive fibered knots are always $(j,0)$1-positive—define the natural next targets in this line of inquiry.