The special shadow-complexity of $\#_k(S^1\times S^3)$

TL;DR

This paper calculates the special shadow-complexity invariant for connected sums of multiple copies of S^1×S^3, showing it equals the number of summands plus one.

Contribution

It provides an exact computation of the special shadow-complexity for a family of 4-manifolds, expanding understanding of this invariant.

Findings

Special shadow-complexity of _k(S^1 S^3) is exactly k+1

The invariant distinguishes connected sums of S^1 S^3

Advances the study of shadow invariants in 4-manifold topology

Abstract

The special shadow-complexity is an invariant of closed $4$-manifolds defined by Costantino using Turaev's shadows. We show that for any positive integer $k$, the special shadow-complexity of the connected sum of $k$ copies of $S^1\times S^3$ is exactly $k+1$.