Rasmussen invariants of Whitehead doubles and other satellites

TL;DR

This work determines how Rasmussen invariants behave on satellites with wrapping number two, deriving an exact formula for the F_2 invariant in winding-number-zero satellites: there exists a unique $\vartheta_2(K)$ with $s_2(P(K)) = s_2(P_{-\vartheta_2(K)}(U))$ for all such patterns $P$. The authors introduce a new concordance invariant $\vartheta_2$, prove its additivity under connected sum, and connect $s_2$ to a multicurve framework built from Bar-Natan homology, enabling a reduction to pattern-only data twisted by a rational tangle. They also compare this to the Ozsváth–Szabó invariant $\tau$, establish geometric applications (including a $\mathbb{Z}^2$-summand in concordance), and develop a general theory for patterns with winding number $\pm 2$ via $\vartheta_\nu$-rationality. The paper further explores extensions to other coefficients, conjectures about the relationship between $\vartheta_\nu$ and $\varepsilon$, and provides extensive computations for small knots, including alternating and torus knots, highlighting linear independence among $\tau$, $s_c$, and $\vartheta_c$ across several characteristics. Overall, the work advances both the computational toolkit and conceptual framework for understanding satellite operations in knot concordance via Rasmussen-type invariants and multicurve technology.

Abstract

We prove formulae for the $\mathbb{F}_2$-Rasmussen invariant of satellite knots of patterns with wrapping number 2, using the multicurve technology for Khovanov and Bar-Natan homology developed by Kotelskiy, Watson, and the second author. A new concordance homomorphism, which is independent of the Rasmussen invariant, plays a central role in these formulae. We also explore whether similar formulae hold for the Ozsváth-Szabó invariant $τ$.

Figures (16)

• Figure 1: An application of \ref{['thm:main']}: Plugging a suitable pattern tangle $T_P$ into the grey discs in (a) and (b) results in the two knots $P(T_{2,3})$ and $P_{-4}(U)$, respectively, where $T_{2,3}$ denotes the right-handed trefoil. Some possibilities for $T_P$ are shown in (c)--(g). For a fixed $P$, those two knots have the same Rasmussen invariant $s_2$ according to \ref{['thm:main']} since $\vartheta_2(T_{2,3})=4$.
• Figure 2: The behaviour of the functions $t\mapsto \nu(P_t(K))$ for wrapping number 2 patterns in the case $\vartheta_\nu(P,K)\neq\infty$
• Figure 3: Some tangles for patterns with winding number $\pm2$
• Figure 4: Diagrams of some rational tangles $Q_{p/q}$
• Figure 5: (a) The union and (b) the sum of two Conway tangles $T_1$ and $T_2$

Theorems & Definitions (160)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Example 2.4
• Proposition 2.6
• proof
• Proposition 2.7