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Detecting unknots via equational reasoning, I: Exploration
[article]
2014
arXiv
pre-print
Preserved Fulltext
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 ...
arXiv:1405.4211v1
fatcat:rquq5xbornazxbcxkixpe5hhq4
Detecting Unknots via Equational Reasoning, I: Exploration
[chapter]
2014
Lecture Notes in Computer Science
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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 ...
doi:10.1007/978-3-319-08434-3_7
fatcat:tuotcjxgdvgbbo5nxmlhojcgwq
Intelligent Explorations of the String Theory Landscape
[article]
2022
arXiv
pre-print
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Other applications of RL include the construction of quark mass models [85] , solving the conformal bootstrap equations [86] and learning to unknot [87] . ...
The reasons are multiple. ...
arXiv:2204.08073v2
fatcat:tqzcw7vmzzep5eny7kvb6vci6y
State-Sum Invariants of 4-Manifolds I
[article]
1994
arXiv
pre-print
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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. ...
arXiv:hep-th/9409167v1
fatcat:l4jv3mw6w5hxfaznda7qwrqo3u
Using Motion Planning for Knot Untangling
2004
The international journal of robotics research
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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. ...
doi:10.1177/0278364904045469
fatcat:mu6fwvndhbhd5of5a5kc3rblvi
HARD UNKNOTS AND COLLAPSING TANGLES
[chapter]
2011
Introductory Lectures on Knot Theory
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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. ...
doi:10.1142/9789814313001_0009
fatcat:55qkybou2rez7fib3lbqm5pheu
Tangled (up in) cubes
2007
Acta Crystallographica Section A Foundations of Crystallography
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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. ...
doi:10.1107/s0108767306052421
pmid:17301480
fatcat:coewvxskdfb3tozjivn2u5jwhe
Learning to Unknot
2021
Machine Learning: Science and Technology
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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. ...
doi:10.1088/2632-2153/abe91f
fatcat:opxkc5rgyrhytkmlw2lg2rjz7i
Hard Unknots and Collapsing Tangles
[article]
2014
arXiv
pre-print
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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. ...
arXiv:math/0601525v5
fatcat:dtlkdhmbo5eerojuikuwkf4dm4
The Theory of Quasiconformal Mappings in Higher Dimensions, I
[article]
2013
arXiv
pre-print
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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 [96] . ...
arXiv:1311.0899v1
fatcat:5nsxi5q3pvdfjfttdbuhnfsoiu
Writhe induced phase transition in unknotted self-avoiding polygons
[article]
2016
arXiv
pre-print
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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 ...
arXiv:1510.07558v3
fatcat:5ggllu5uoncyveccaddbn4mrp4
Elementary knot theory
[article]
2016
arXiv
pre-print
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It was shown recently by Kronheimer and Mrowka [47] 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 [78] . So, coupled with the fact that it is computable, this provides another unknot recognition algorithm. ...
arXiv:1604.03778v1
fatcat:4ptksoisobgc3l5idxky3kroxu
The rational Witt class and the unknotting number of a knot
[article]
2009
arXiv
pre-print
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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. ...
arXiv:0907.2275v1
fatcat:mllnryjyoregbdl2jqqe2kk6uy
On Rack Invariants Of Legendrian Knots
[article]
2017
arXiv
pre-print
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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. ...
arXiv:1706.07626v2
fatcat:3vpsk2fmqvgx5nonl4ztrwxmiq
Knots and solenoids that cannot be attractors of self-homeomorphisms of R^3
[article]
2018
arXiv
pre-print
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Thus e(K) has the natural neighbourhood basis {e(T i )} i≥i 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. ...
arXiv:1806.11151v1
fatcat:vpbstb4flzg4bp3o2ib4q7i7aq
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