FIG. 1. Knotted bead-spring polymer: Starting configuration with N=16384 beads; after 6 reduction steps (N=265); final 
configuration after 15 iterations (N=8) with the knotted (trefoil) region circled in red; and magnified.
 (enhanced online) 
Capturing knots in polymers 
Peter Virnau, Mehran Kardar  
Department of Physics, MIT, Cambridge, MA 
02139-4307, USA 
Yacov Kantor 
School of Physics and Astronomy, Tel Aviv 
University, 69978 Tel Aviv, Israel 
(received, published) 
[DOI: 10.1063/1.2130690]  
Visualizing topological properties is a particularly 
challenging task. Although algorithms can usually 
determine if a loop contains a knot, finding its 
exact location is difficult (and not necessarily 
well-defined).
     Here, we apply a reduction method by Koniaris 
and Muthukumar
, which was originally proposed 
to simplify polymers before calculating knot 
invariants. We start with one end and consider 
consecutive triangles formed by three adjacent 
monomers. If the triangle is not crossed by any of 
the remaining bonds, the particle in the middle is 
removed. Going back and forth between both ends 
we proceed until the configuration cannot be 
reduced any further (see Fig.1). 
    Although the method is not perfect (sometimes 
entangled, but unknotted regions remain), it 
provides us with a valuable impression on the 
typical number of knots, their respective location 
and sizes
     This work was supported by the DFG grant 
Vi237/1. 
 P. Virnau, Y. Kantor, and M. Kardar, J. Am. Chem. Soc., 
in press (2005). 
 W. G. Taylor, Nature 406, 916 (2000). 
 K. Koniaris and M. Muthukumar, J.Chem.Phys. 95, 2873 
(1991). 
 Pictures and movie were generated using the VMD 
visualization package; see W. Humphrey, A. Dalke, and K. 
Schulten,, J. Molec. Graphics 14, 33 (1996).  
Copyright (2005) American Institute of Physics. 
This article may be downloaded for personal use 
only. Any other use requires prior permission of 
the author and the American Institute of Physics.  
The following article appeared in the Gallery of Images in 
Chaos 15, 041103 (2005) 
and may be found at 
http://chaos.aip.org/chaos/gallery/toc_Dec05.jsp 
This version also contains a movie of the algorithm.

  This paper visualizes a knot reduction algorithm

FIG. 1. Knotted bead-spring polymer: Starting configuration with N=16384 beads; after 6 reduction steps (N=265); final 
configuration after 15 iterations (N=8) with the knotted (trefoil) region circled in red; and magnified.
 (enhanced online) 
Capturing knots in polymers 
Peter Virnau, Mehran Kardar  
Department of Physics, MIT, Cambridge, MA 
02139-4307, USA 
Yacov Kantor 
School of Physics and Astronomy, Tel Aviv 
University, 69978 Tel Aviv, Israel 
(received, published) 
[DOI: 10.1063/1.2130690]  
Visualizing topological properties is a particularly 
challenging task. Although algorithms can usually 
determine if a loop contains a knot, finding its 
exact location is difficult (and not necessarily 
well-defined).
     Here, we apply a reduction method by Koniaris 
and Muthukumar
, which was originally proposed 
to simplify polymers before calculating knot 
invariants. We start with one end and consider 
consecutive triangles formed by three adjacent 
monomers. If the triangle is not crossed by any of 
the remaining bonds, the particle in the middle is 
removed. Going back and forth between both ends 
we proceed until the configuration cannot be 
reduced any further (see Fig.1). 
    Although the method is not perfect (sometimes 
entangled, but unknotted regions remain), it 
provides us with a valuable impression on the 
typical number of knots, their respective location 
and sizes
     This work was supported by the DFG grant 
Vi237/1. 
 P. Virnau, Y. Kantor, and M. Kardar, J. Am. Chem. Soc., 
in press (2005). 
 W. G. Taylor, Nature 406, 916 (2000). 
 K. Koniaris and M. Muthukumar, J.Chem.Phys. 95, 2873 
(1991). 
 Pictures and movie were generated using the VMD 
visualization package; see W. Humphrey, A. Dalke, and K. 
Schulten,, J. Molec. Graphics 14, 33 (1996).  
Copyright (2005) American Institute of Physics. 
This article may be downloaded for personal use 
only. Any other use requires prior permission of 
the author and the American Institute of Physics.  
The following article appeared in the Gallery of Images in 
Chaos 15, 041103 (2005) 
and may be found at 
http://chaos.aip.org/chaos/gallery/toc_Dec05.jsp 
This version also contains a movie of the algorithm.
