1. The problem is to find the bracket polynomial for the given link or knot diagram labeled as $L_3$. 2. The bracket polynomial, denoted $\langle L \rangle$, is a polynomial invariant of framed links defined by the rules: - $\langle \bigcirc \rangle = 1$ - $\langle \text{crossing} \rangle = A \langle \text{A-smoothing} \rangle + A^{-1} \langle \text{B-smoothing} \rangle$ - $\langle L \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle$ 3. To find the bracket polynomial, we recursively apply the smoothing rules to each crossing in $L_3$, replacing each crossing with a linear combination of simpler diagrams until only disjoint unions of circles remain. 4. Each smoothing contributes a factor of $A$ or $A^{-1}$, and each disjoint circle contributes a factor of $(-A^2 - A^{-2})$. 5. After fully resolving all crossings, sum all terms to get the bracket polynomial $\langle L_3 \rangle$. Since the exact diagram or crossings of $L_3$ are not provided, the explicit polynomial cannot be computed here. If you provide the crossing information or a diagram, I can compute the bracket polynomial step-by-step.