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The space writhes and signatures of polymer knots

Finn Thompson, Maria Maalouf, Alexander R. Klotz

TL;DR

The study investigates whether the space writhe of knotted polymers reflects intrinsic knot invariants. Using Langevin dynamics simulations of Kremer–Grest bead–spring chains, the authors show that the ensemble-averaged writhe for knots up to $10$ crossings nearly matches the ideal writhe, and that the correlation persists for more complex knots up to $38$ crossings where the writhe correlates strongly with the knot signature ($\text{corr} \approx 0.96$). The work also confirms writhe quantization tendencies and demonstrates that polymer knots can be used to probe topological invariants in 3D embeddings, while acknowledging that neither the predicted nor the ideal writhe perfectly captures the observed averages. Overall, the results provide evidence for a robust link between writhe and signature across a broad range of knot complexities and motivate a deeper theoretical understanding of this relationship.

Abstract

The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We perform Langevin dynamics simulations of knotted semiflexible polymers and measure their ensemble average space writhe. We show that for all knots up to 10 crossings, alternating and non-alternating, the average space writhe is almost equal to that of the tightest known configuration of the same knot, with minor differences. Using this equivalence, we show that for more complex knots with up to 38 crossings, the average space writhe is strongly correlated with the signature of the knot. This establishes that the connection between signature and space writhe holds at larger crossing numbers.

The space writhes and signatures of polymer knots

TL;DR

The study investigates whether the space writhe of knotted polymers reflects intrinsic knot invariants. Using Langevin dynamics simulations of Kremer–Grest bead–spring chains, the authors show that the ensemble-averaged writhe for knots up to crossings nearly matches the ideal writhe, and that the correlation persists for more complex knots up to crossings where the writhe correlates strongly with the knot signature (). The work also confirms writhe quantization tendencies and demonstrates that polymer knots can be used to probe topological invariants in 3D embeddings, while acknowledging that neither the predicted nor the ideal writhe perfectly captures the observed averages. Overall, the results provide evidence for a robust link between writhe and signature across a broad range of knot complexities and motivate a deeper theoretical understanding of this relationship.

Abstract

The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We perform Langevin dynamics simulations of knotted semiflexible polymers and measure their ensemble average space writhe. We show that for all knots up to 10 crossings, alternating and non-alternating, the average space writhe is almost equal to that of the tightest known configuration of the same knot, with minor differences. Using this equivalence, we show that for more complex knots with up to 38 crossings, the average space writhe is strongly correlated with the signature of the knot. This establishes that the connection between signature and space writhe holds at larger crossing numbers.
Paper Structure (6 sections, 10 equations, 4 figures)

This paper contains 6 sections, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Results and Discussion
  4. Conclusion
  5. Acknowledgements
  6. Appendix

Figures (4)

  • Figure 1: Left: A trefoil knot in three-dimensional space projected onto three perpendicular axes. Each projection has a different number of crossings, which may be positive (red) or negative (yellow). The difference between positive and negative crossings is the writhe of the projected diagram, and the space writhe is the average over all projections. Right: Three projections of a trefoil knot, each with different space writhe: a twisty one with space writhe 6.7, a tight one with a value of 3.42 (close to the predicted 24/7), and a polymer-like one with a value of 3.24.
  • Figure 2: Correlation between the ideal writhe of a knot and the average writhe of a polymer knot for alternating (black) and non-alternating (red) knots. The line represents equality.
  • Figure 3: Scatter plot of the deviation between the ideal and predicted writhe on the x-axis and between the polymer and predicted writhe on the y-axis, for alternating knots up to 7 crossings (black) and non-alteranting knots with 8 and 9 crossings (red). The diagonal line is y=x. The green star is from a sample of random stick trefoil knots.
  • Figure 4: Correlation between the signature of a knot and its average polymer writhe. Black points represent every knot between 3 and 10 crossings, red points represent selected alternating knots with 13-38 crossings, and blue points represent non-alternating knots with 20-29 crossings. The envelope from 1288 12-crossing knots klotz2024ropelength is shown for reference. Each set is shifted horizontally by a small amount for visual clarity.