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Monte Carlo simulations are used to study the conformational behavior of asemiflexible polymer confined to cylindrical and conical channels. The channelsare sufficiently narrow that the conditions for the Odijk regime are marginallysatisfied. For cylindrical confinement, we examine polymers with a single knotof topology $3_1$, $4_1$, or $5_1$, as well as unknotted polymers that arecapable of forming S-loops. We measure the variation of the free energy $F$with the end-to-end polymer extension length $X$ and examine the effect ofvarying the polymer topology, persistence length $P$ and cylinder diameter $D$on the free energy functions. Similarly, we characterize the behavior of theknot span along the channel. We find that increasing the knot complexityincreases the typical size of the knot. In the regime of low $X$, where theknot/S-loop size is large, the conformational behavior is independent ofpolymer topology. In addition, the scaling properties of the free energy andknot span are in agreement with predictions from a theoretical modelconstructed using known properties of interacting polymers in the Odijk regime.We also examine the variation of $F$ with position of a knot in conicalchannels for various values of the cone angle $\alpha$. The free energydecreases as the knot moves in a direction where the cone widens, and it alsodecreases with increasing $\alpha$ and with increasing knot complexity. Thebehavior is in agreement with predictions from a theoretical model in which thedominant contribution to the change in $F$ is the change in the size of thehairpins as the knot moves to the wider region of the channel.

Free Energy of a Knotted Polymer Confined to Narrow Cylindrical and Conical Channels