# Tie knots, random walks and topology

T. Fink, Y. Mao

*Physica A* 276, 109 (209)

#tieknots#randomwalks#statisticalphysics

The thirteen canonical knot classes and the corresponding most aesthetic knots.

Necktie knots are inherently topological structures; what makes them tractable is the particular manner in which they are constructed. This observation motivates a map between tie knots and persistent walks on a triangular lattice. The topological structure embedded in a tie knot may be determined by appropriately manipulating its projection; we derive corresponding rules for tie knot sequences. We classify knots according to their size and shape and quantify the number of knots in a class. Aesthetic knots are characterised by the conditions of symmetry and balance. Of the 85 knots which may be tied with conventional tie, we recover the four traditional knots and introduce nine new aesthetic ones. For large (though impractical) half-winding number, we present some asymptotic results.

#### Imaginary replica analysis of loopy regular random graphs

F. Lopez, T. Coolen

Sub. to *Journal of Physics A*

#### Degree-correlations in a bursting dynamic network model

F. Vanni, P. Barucca

*Journal of Economic Interaction and Coordination*

#### Phase transition creates the geometry of the continuum from discrete space

R. Farr, T. Fink

*Physical Review E*

#### Intelligently chosen interventions have potential to outperform the diode bridge in power conditioning

F. Liu, Y. Zhang, O. Dahlsten, F. Wang

*Scientific Reports *

123 / 123 papers