Squeezed Knots

TL;DR

This work investigates squeezed knots—those arising as slices of genus-minimizing cobordisms between a positive torus knot $T^+$ and a negative torus knot $T^-$. It develops a robust framework combining ribbon surfaces, quasihomogeneity, and quasipositive theory to certify squeezedness for large knot families (notably most knots up to $10$ crossings) and to exhibit their concordance subgroup structure. Importantly, it shows that obstructions to squeezedness predominantly come from quantum/Khovanov-type invariants: refined Rasmussen invariants in Lipshitz-Sarkar and Sarkar-Scaduto-Stoffregen theory, along with Schütz's integral Rasmussen, and $S_n$-gimel data from Khovanov-Rozansky homology, tightly constraining squeezedness and enabling obstructions for explicitly non-squeezed knots such as $9_{42}$, $10_{132}$, and $10_{136}$. The paper also introduces a squeezing framework that formalizes invariants as metric-space maps with cobordism-distance bounds, clarifying how various invariants behave on squeezed knots (often matching $g_4$ or $s$-based predictions). Overall, the results highlight the power of quantum invariants in knot cobordism problems, establish that squeezed knots form a substantial concordance subgroup, and open directions for discovering deeper non-squeezed phenomena among knot families beyond small-crossing examples.

Abstract

Squeezed knots are those knots that appear as slices of genus-minimizing oriented smooth cobordisms between positive and negative torus knots. We show that this class of knots is large and discuss how to obstruct squeezedness. The most effective obstructions appear to come from quantum knot invariants, notably including refinements of the Rasmussen invariant due to Lipshitz-Sarkar and Sarkar-Scaduto-Stoffregen involving stable cohomology operations on Khovanov homology.

Figures (9)

• Figure 1: We draw a schematic of a genus-minimizing knot cobordism $\Sigma \subset S^3 \times [-1,1]$ between the positive torus knot $T^+$ and the negative torus knot $T^-$. The knot $K$ appears a slice of this knot cobordism and hence $K$ is a squeezed knot.
• Figure 2: On the left is the positive trefoil knot, and on the right is the negative trefoil knot. The knot in the middle is the Figure Eight knot, which is obtained from either trefoil by the addition of two oriented $1$-handles. Hence we have exhibited the Figure Eight knot as a slice of a genus $2$ (which is minimal) cobordism between the positive and negative trefoils.
• Figure 3: A strongly quasihomogeneous knot with slice genus $2$, squeezed between the positive and the negative trefoil.
• Figure 4: We show a $1$-manifold of intersection in an immersed surface in the $3$-sphere. Such self-intersections (where the preimage of the locus of the intersection consists of an arc with both boundary points on the boundary of the surface, and an arc in the interior of the surface) are called ribbon self-intersections.
• Figure 5: A motivational picture for the definition of quasihomogeneous. The green disc together with the yellow discs and the yellow ribbons should form a quasinegative surface, while the green disc together with the blue discs and the blue ribbons should form a quasipositive surface. The blue ribbons do not intersect the yellow discs, while the yellow ribbons do not intersect the blue discs. We refer to the union of all the discs and ribbons as a quasihomogeneous surface.

Theorems & Definitions (60)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• Remark 1.5
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• lemma 2.5