Knot theory in modern chemistry

Horner, Kate Elizabeth; Miller, Mark A.; Steed, Jonathan W.; Sutcliffe, Paul

doi:10.1039/c6cs00448b

Cited by 79 publications

(55 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A knot is an embedding of a circle in the 3-sphere S 3 . These objects play important roles in a wide range of fields including particle physics, statistical mechanics, molecular biology, chemistry, sailing, and art [1][2][3][4][5][6]. Figure 1 depicts several well known knots.…”

Section: Introductionmentioning

confidence: 99%

Deep learning the hyperbolic volume of a knot

2019

View full text Add to dashboard Cite

An important conjecture in knot theory relates the large-N , double scaling limit of the colored Jones polynomial J K,N (q) of a knot K to the hyperbolic volume of the knot complement, Vol(K). A less studied question is whether Vol(K) can be recovered directly from the original Jones polynomial (N = 2). In this report we use a deep neural network to approximate Vol(K) from the Jones polynomial. Our network is robust and correctly predicts the volume with 97.6% accuracy when training on 10% of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial.B.6) The network can be straightforwardly implemented in Mathematica 11.3.0.0 [19] with the command 2 KnotNet = NetChain[{DotPlusLayer[100], ElementwiseLayer[LogisticSigmoid], DotPlusLayer[100], ElementwiseLayer[LogisticSigmoid], SummationLayer[]}, "Input" -> {18}];2 In Mathematica 12.0.0.0, the DotPlusLayer command is replaced by LinearLayer.

show abstract

Section: Introductionmentioning

confidence: 99%

Deep learning the hyperbolic volume of a knot

2019

View full text Add to dashboard Cite

show abstract

“…For example, poly(N-isopropylacrylamide) (pNIPAM) gels in water gradually deswell under heating until ∼32 • C. Beyond this, due to a change in solvent nature from good to poor, they abruptly expel most of their solvent [9]. In fact, there is a first-order phase transition ‡ Note that we do not consider here polymer rings that may be topologically linked to form rigid "Olympic" gels [1,4] or other knotted polymer networks [5,6].…”

Section: Introductionmentioning

confidence: 99%

Swelling thermodynamics and phase transitions of polymer gels

et al. 2019

View full text Add to dashboard Cite

We present a pedagogical review of the swelling thermodynamics and phase transitions of polymer gels. In particular, we discuss how features of the volume phase transition of the gel's osmotic equilibrium is analogous to other transitions described by mean-field models of binary mixtures, and the failure of this analogy at the critical point due to shear rigidity. We then consider the phase transition at fixed volume, a relatively unexplored paradigm for polymer gels that results in a phase-separated equilibrium consisting of coexisting solventrich and solvent-poor regions of gel. Again, the gel's shear rigidity is found to have a profound effect on the phase transition, here resulting in macroscopic shape change at constant volume of the sample, exemplified by the tunable buckling of toroidal samples of polymer gel. By drawing analogies with extreme mechanics, where large shape changes are achieved via mechanical instabilities, we formulate the notion of extreme thermodynamics, where large shape changes are achieved via thermodynamic instabilities, i.e. phase transitions.

show abstract

“…We investigate the entanglement probability and types of self-entanglements formed on linear and circular chains, with entanglements on the linear chain providing a framework for understanding entanglements on its circular counterpart. By treating a self-entangled circular chain as a link of two components, we can use topological descriptors from knot theory-the Alexander-Briggs knot notation, Dowker-Thistlethwaite code, and linking number [28]-to characterize self-entanglements on circular chains.…”

Section: Introductionmentioning

confidence: 99%

Self-entanglement of a tumbled circular chain

Soh

Gengaro

Klotz

et al. 2019

Phys. Rev. Research

View full text Add to dashboard Cite

The spontaneous knotting of linear chains has been well studied, but little attention has been given to the self-entanglement of chains with more complex topologies. In this work, we perform experiments with granular chains that undergo tumbling motion to investigate the self-entanglement of circular chains, which lack the chain ends essential for forming knots. We study the entanglement probability and types of self-entanglements formed on linear and circular chains, using the well-studied self-entanglements on a linear chain to frame our understanding of self-entanglements on a circular chain. We describe a characterization method that views a self-entangled circular chain as a link of two components and use it to characterize the self-entanglements on circular chains with known topological descriptors from knot theory. Our experimental results show that an increase in circular chain length leads to an increase in entanglement probability and entanglement complexity until a plateau is reached, similar to the trends observed with linear chains. By examining the formation pathway of several self-entanglements, we infer a general mechanism for the self-entanglement of circular chains.

show abstract

Knot theory in modern chemistry

Cited by 79 publications

References 55 publications

Deep learning the hyperbolic volume of a knot

Deep learning the hyperbolic volume of a knot

Swelling thermodynamics and phase transitions of polymer gels

Self-entanglement of a tumbled circular chain

Contact Info

Product

Resources

About