In the absence of other ideas, I'll risk posting something that perhaps satisfies (2), and could be arranged to satisfy (1):

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          [![OverUnder][1]][1]

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If indeed this satisfies (2), then perhaps its length could approach $2 \pi$, by moving the under/over crossings close together, and closer to the top of the diagram.

In the absence of other ideas, I'll risk posting something that perhaps satisfies (2), and could be arranged to satisfy (1):

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          [![OverUnder][1]][1]

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If indeed this satisfies (2), then perhaps its length could approach $2 \pi$, by moving the over/under crossings close together, and closer to the top of the diagram.

In the absence of other ideas, I'll risk posting something that perhaps satisfies (2), and could be arranged to satisfy (1):

---

          [![OverUnder][1]][1]

---

If indeed this satisfies (2), then perhaps its length could approach $2 \pi$.

In the absence of other ideas, I'll risk posting something that perhaps satisfies (2), and could be arranged to satisfy (1):

---

          [![OverUnder][1]][1]

---

If indeed this satisfies (2), then perhaps its length could approach $2 \pi$, by moving the under/over crossings close together, and closer to the top of the diagram.