With \(\mathbb{A}\) a nonempty subset of \(\{1,2\}\) and \(\mathbb{P}\) a nonempty set of propositional letters, let \(\mathcal{L_\mathbb{A}}\) be the set of propositional modal formulas built over \(\mathbb{P}\), with \([i]\), \(i\in\mathbb{A}\), as modal operators, and let \(L_\mathbb{A}\) be a normal modal logic over \(\mathcal{L_\mathbb{A}}\), that is, a subset of \(\mathcal{L_\mathbb{A}}\) closed under modus ponens and substitution instances, and containing all propositional tautologies, the axiom schema \(\bigl([i](\varphi\rightarrow\psi)\wedge[i]\varphi\bigr)\rightarrow[i]\psi\), \(i\in\mathbb{A}\), and the necessitation rule that from \(\varphi\) infers \([i]\varphi\), \(i\in\mathbb{A}\). Write \(L_1\), \(L_2\) and \(L_{1,2}\) for \(L_{\{1\}}\), \(L_{\{2\}}\) and \(L_{\{1,2\}}\), respectively. Given two subsets \(L\) and \(L'\) of \(\mathcal{L_\mathbb{A}}\), let \(L + L'\) denote the normal modal logic over \(\mathcal{L_\mathbb{A}}\) generated by \(L\cup L'\). The key notions of the paper are the following: {\parindent=0.4 cm \begin{itemize}\item[--] A set of \textit{interaction axioms} w.r.t.\ \((L_1,L_2)\) is a finite subset \(\Gamma\) of \(L_{1,2}\) for which there is no \(\chi\in L_1\cup L_2\) with \(\chi\leftrightarrow\bigwedge\Gamma\in L_1+L_2\). \item[--] Let \(x\in{\mathcal L}_1\) and a set \(\Gamma\) of interaction axioms w.r.t.\ \((L_1,L_2)\) be given. {\parindent=0.8 cm \begin{itemize}\item[--] \(x\) is \textit{characterised} by \(\Gamma\) iff \(L_1+\{x\}=(L_1+L_2+\Gamma)\cap{\mathcal L}_1\): when adding the interaction axioms to the base logics, one obtains exactly the theorems for \({\mathcal L}_1\) that would be obtained by only adding \(x\) to \(L_1\). \item[--] \(x\) is \textit{conservatively characterized} by \(\Gamma\) if \(L_1+\{x\}=(L_1+L_2+\Gamma)\cap{\mathcal L}_1\) and \(L_2 = (L_1 + L_2 + \Gamma)\cap{\mathcal L}_2\) both hold: no new theorems for \({\mathcal L}_2\) are obtained as `side effects'. \end{itemize}} \item[--] Let \(i\) and \(j\) be either 1 and 2 or 2 and 1. The modality \(\langle i\rangle\), that is, \(\neg[i]\neg\), is \textit{explicitly defined} in \(L_{i,j}\) in terms of the modality \(\langle j\rangle\) by a formula \(\mathrm{def}_i(p)\in\mathcal{L}_j\) iff \(\langle i\rangle p\leftrightarrow\mathrm{def}_i(p)\in L_{i,j}\). \end{itemize}} The aim of the paper is to demonstrate that in epistemic logic, axioms dealing with the notion of knowledge, which sometimes are hard to interpret intuitively, can be characterized in terms of understandable interaction axioms relating knowledge and belief, or knowledge and conditional belief. It is based on the following result: ``Assume that \(\langle 2\rangle\) is explicitly defined in \(L_1 + L_2 + \Gamma\) in terms of \(\langle 1\rangle\) by a formula \(\mathrm{def}_2(p)\in{\mathcal L}_1\) positive in \(p\) (that is, such that all occurrences of \(p\) in \(\mathrm{def}_2(p)\) written with only \(\neg\) and \(\wedge\) as Boolean operators are in the scope of an even number of negations). Then, the following are equivalent: {\parindent=6mm \begin{itemize}\item[--] \(x\) is characterized by \(\Gamma\); \item[--] (*) \(L_1+L_2+\Gamma=L1 +\{x,\langle 2\rangle p\leftrightarrow\mathrm{def}_2(p)\}\). \end{itemize}} Moreover, assume that \(\langle 1\rangle\) is also explicitly defined in \(L_1 + L_2 + \Gamma\) in terms of \(\langle 2\rangle\) by a formula \(\mathrm{def}_1(p)\in{\mathcal L}_2\) positive in \(p\). Then, the following are equivalent: {\parindent=6mm \begin{itemize}\item[--] \(x\) is conservatively characterized by \(\Gamma\); \item[--] (*) holds and \(L_1+L_2+\Gamma=L_2 +\{\langle 1\rangle p\leftrightarrow\mathrm{def}_1(p)\}\). '' \end{itemize}} The paper then examines interaction axioms from the epistemic logic literature, which connect the notions of belief or conditional belief with the notion of knowledge. It establishes results such as the following, where \(K\) and \(B\) are the knowledge and belief modal operators, respectively: ``Let \(I_1\), \(I_2\), \(I_3\), \(I_4\) and \(I_5\) denote \(Kp\rightarrow Bp\), \(Bp\rightarrow KBp\), \(Bp\rightarrow BKp\), \(Bp\leftrightarrow\langle K\rangle Kp\) and \(Kp\leftrightarrow(p\wedge Bp)\), respectively. \noindent Let \(T\), 4, \(D\) and 5 denote \(M\varphi\rightarrow\varphi\), \(M\varphi\rightarrow MM\varphi\), \(M\varphi\rightarrow \langle M\rangle\varphi\) and \(\neg M\varphi\rightarrow M\neg M\varphi\), respectively. Let \(S4_K\) and \(KD45_B\) be the smallest normal modal logic for \({\mathcal L}_K\) and \({\mathcal L}_B\) generated by the set of axioms \(\{T, 4\}\) and \(\{D, 4, 5\}\), with \(M\) replaced by \(K\) and \(B\), respectively. \noindent Then the sets \(\{I_1\}\), \(\{I_2\}\), \(\{I_3\}\), \(\{I_4\}\), \(\{I_5\}\) and \(\{I_1,I_2,I_3\}\) are sets of interaction axioms with respect to \((S4_K,KD45_B)\). ''