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### Detecting unknots via equational reasoning, I: Exploration [article]

Andrew Fish, Alexei Lisitsa
2014 arXiv   pre-print
We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology.  ...  The theorem proving approach utilises equational reasoning, whilst the model finder searches for a minimal size counter-model.  ...  Equational Reasoning and Untangling Unknots Recall Proposition 1: a knot diagram D is a diagram of the unknot if and only if E iq (D) ∧ i=1...n−1 (a i = a i+1 ) , where denotes derivability in the equational  ...

### Detecting Unknots via Equational Reasoning, I: Exploration [chapter]

Andrew Fish, Alexei Lisitsa
2014 Lecture Notes in Computer Science
We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology.  ...  The theorem proving approach utilises equational reasoning, whilst the model finder searches for a minimal size counter-model.  ...  Equational Reasoning and Untangling Unknots Recall Proposition 1: a knot diagram D is a diagram of the unknot if and only if E iq (D) ∧ i=1...n−1 (a i = a i+1 ) , where denotes derivability in the equational  ...

### Intelligent Explorations of the String Theory Landscape [article]

Andrei Constantin
2022 arXiv   pre-print
Other applications of RL include the construction of quark mass models  , solving the conformal bootstrap equations  and learning to unknot  .  ...  The reasons are multiple.  ...

### State-Sum Invariants of 4-Manifolds I [article]

Louis Crane, Louis H. Kauffman, David N. Yetter
1994 arXiv   pre-print
In general, the handlesliding followed by the encirclement lemma shows that x 2 = N z + = N z − where z ± is the value of a ±1-framed unknot labelled with i∈Z(C) dim(i)i.  ...  There is reason to hope that the entire picture in 4D can be rendered as algebraic as the lower dimensional cases.  ...

### Using Motion Planning for Knot Untangling

Andrew M. Ladd, Lydia E. Kavraki
2004 The international journal of robotics research
Applications to Computational Topology Our planner can be used for stick number calculation of a knot where unknot detection is a special case.  ...  Detecting the existence of zeroes for clr(K(t)) for t ∈ [0, 1] is the self-intersection detection problem for knot manipulation.  ...

### HARD UNKNOTS AND COLLAPSING TANGLES [chapter]

Louis H. Kauffman, Sofia Lambropoulou
2011 Introductory Lectures on Knot Theory
Recognizing unknots is a starting point for all of knot theory.  ...  In such a case we say that the loop is unknotted. So it is not absurd if we sometimes ask, Is this knot unknotted? The question is not only non-contradictory, it is often hard.  ...  Finally, another approach to detecting unknots is the use of invariants of knots and links.  ...

### Tangled (up in) cubes

S. T. Hyde, G. E. Schröder-Turk
2007 Acta Crystallographica Section A Foundations of Crystallography
Evidently, an infinity of tangled cubic nets are realizable via this operation. Generic examples are not isomorphic to the unknotted pcu net.  ...  A reasonable starting point is to consider the simpler situation of a finite graph and knottings thereof.  ...

### Learning to Unknot

James Halverson, Sergei Gukov, Piotr Sulkowski, Fabian Ruehle
2021 Machine Learning: Science and Technology
We study the UNKNOT problem of determining whether or not a given knot is the unknot.  ...  Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions  ...  unknotting via reinforcement learning.  ...

### Hard Unknots and Collapsing Tangles [article]

Louis H. Kauffman, Sofia Lambropoulou
2014 arXiv   pre-print
unknots" by Henrich and Kauffman.  ...  The paper uses these results in studying processive DNA recombination, finding minimal size unknot diagrams, generalizing to collapses to knots as well as to unknots, and in finding unknots with arbirarily  ...  Finally, another approach to detecting unknots is the use of invariants of knots and links.  ...

### The Theory of Quasiconformal Mappings in Higher Dimensions, I [article]

Gaven J. Martin
2013 arXiv   pre-print
Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following  ...  These quantities can be used to detect the size of sets in a similar fashion to Hausdorff dimension.  ...  This takes the form d S (G, I n ) ∞ + d S (H, I n ) ∞ ≤ M < ∞, (8.6) where d S is the metric of S(n) and I n is the n × n identity matrix. This metric is discussed in Wolf's book  .  ...

### Writhe induced phase transition in unknotted self-avoiding polygons [article]

Eduardo Dagrosa, Aleksander L Owczarek, Thomas Prellberg
2016 arXiv   pre-print
Recently it has been argued that weighting the writhe of unknotted self-avoiding polygons can be related to possible experiments that turn double stranded DNA.  ...  Acknowledgments One of the authors, ED, gratefully acknowledges the financial support of the University of Melbourne via its Melbourne International Research Scholarships scheme.  ...  Financial support from the Australian Research Council via its support for the Centre of Excellence for Mathematics and Statistics of Complex Systems and the Discovery Projects scheme (DP160103562) is  ...

### Elementary knot theory [article]

Marc Lackenby
2016 arXiv   pre-print
It was shown recently by Kronheimer and Mrowka  that a more refined invariant, Khovanov homology, does detect the unknot.  ...  One of the earliest results about Heegaard Floer homology was that it detects the unknot  . So, coupled with the fact that it is computable, this provides another unknot recognition algorithm.  ...

### The rational Witt class and the unknotting number of a knot [article]

Stanislav Jabuka
2009 arXiv   pre-print
Our results are mainly concerned with unknotting number one but we also address, somewhat more marginally, the case of higher unknotting numbers.  ...  In many cases we obtain new lower bounds and in some cases explicit values for their unknotting numbers.  ...  Using this observation, Theorem 1.1 implies that ϕ(L i ) = ϕ(L i+1 ) ⊕ − 2 det L i+1 det L i ⊕ −2 i = 1, ..., n Adding these last n equations immediately yields the result of Corollary 1.4.  ...

### On Rack Invariants Of Legendrian Knots [article]

Dheeraj Kulkarni, T. V. H. Prathamesh
2017 arXiv   pre-print
These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot.  ...  We also present the results from the experiments on Legendrian unknots involving auto-mated theorem provers, and describe how they led to our current formulation.  ...  Since a 0 , a 1 , a 3 , . . . , a 9 |a i+1 = a n i for all 0 ≤ i ≤ 9 a 2 a 1 Figure 8. minimal Legendrian unknot They have the same number of cusps, in the front projection with no crossings.  ...

### Knots and solenoids that cannot be attractors of self-homeomorphisms of R^3 [article]

Héctor Barge, J. J. Sánchez-Gabites
2018 arXiv   pre-print
Thus e(K) has the natural neighbourhood basis {e(T i )} ii 0 all of whose members are unknotted; that is, e(K) is unknotted. (ii) We reason by contradiction.  ...  (i) (setting e := h −1 ). We reason by contradiction.  ...
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