Let the circle act on a compact almost complex manifold M. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. By the fixed point data we mean a collection of the multisets of the weights at the fixed points. First, if dimM=2\documentclass[12pt]{minimal}
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\begin{document}$$\dim M=2$$\end{document}, then M is a disjoint union of rotations on two 2-spheres. Second, if dimM=4\documentclass[12pt]{minimal}
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\begin{document}$$\dim M=4$$\end{document}, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if dimM=6\documentclass[12pt]{minimal}
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\begin{document}$$\dim M=6$$\end{document}, we prove that six types occur for the fixed point data; CP3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {CP}^3$$\end{document} type, complex quadric in CP4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {CP}^4$$\end{document} type, Fano threefold type, S6∪S6\documentclass[12pt]{minimal}
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\begin{document}$$S^6 \cup S^6$$\end{document} type, blow up of a fixed point of a rotation on S6\documentclass[12pt]{minimal}
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\begin{document}$$S^6$$\end{document} type, and unknown type that might possibly be realized as a blow up of S2\documentclass[12pt]{minimal}
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\begin{document}$$S^2$$\end{document} inside a manifold like S6\documentclass[12pt]{minimal}
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\begin{document}$$S^6$$\end{document}. When dimM=6\documentclass[12pt]{minimal}
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\begin{document}$$\dim M=6$$\end{document}, we recover the result by Ahara (J Fac Sci Univ Tokyo Sect IA Math 38(1):47–72, 1991) in which the fixed point data is determined if furthermore Todd(M)=1\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Todd}(M)=1$$\end{document} and c13(M)[M]≠0\documentclass[12pt]{minimal}
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\begin{document}$$c_1^3(M)[M] \ne 0$$\end{document}, and the result by Tolman (Trans Am Math Soc 362(8):3963–3996, 2010) in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.
