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

2014
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arXiv
*
pre-print

We

arXiv:1405.4211v1
fatcat:rquq5xbornazxbcxkixpe5hhq4
*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*...##
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Detecting Unknots via Equational Reasoning, I: Exploration
[chapter]

2014
*
Lecture Notes in Computer Science
*

We

doi:10.1007/978-3-319-08434-3_7
fatcat:tuotcjxgdvgbbo5nxmlhojcgwq
*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*...##
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Intelligent Explorations of the String Theory Landscape
[article]

2022
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arXiv
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pre-print

Other applications of RL include the construction of quark mass models [85] , solving the conformal bootstrap

arXiv:2204.08073v2
fatcat:tqzcw7vmzzep5eny7kvb6vci6y
*equations*[86] and learning to*unknot*[87] . ... The*reasons*are multiple. ...##
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State-Sum Invariants of 4-Manifolds I
[article]

1994
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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

arXiv:hep-th/9409167v1
fatcat:l4jv3mw6w5hxfaznda7qwrqo3u
*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. ...##
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Using Motion Planning for Knot Untangling

2004
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The international journal of robotics research
*

Applications to Computational Topology Our planner can be used for stick number calculation of a knot where

doi:10.1177/0278364904045469
fatcat:mu6fwvndhbhd5of5a5kc3rblvi
*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. ...##
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HARD UNKNOTS AND COLLAPSING TANGLES
[chapter]

2011
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Introductory Lectures on Knot Theory
*

Recognizing

doi:10.1142/9789814313001_0009
fatcat:55qkybou2rez7fib3lbqm5pheu
*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. ...##
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Tangled (up in) cubes

2007
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Acta Crystallographica Section A Foundations of Crystallography
*

Evidently, an infinity of tangled cubic nets are realizable

doi:10.1107/s0108767306052421
pmid:17301480
fatcat:coewvxskdfb3tozjivn2u5jwhe
*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. ...##
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Learning to Unknot

2021
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Machine Learning: Science and Technology
*

We study the

doi:10.1088/2632-2153/abe91f
fatcat:opxkc5rgyrhytkmlw2lg2rjz7i
*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. ...##
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Hard Unknots and Collapsing Tangles
[article]

2014
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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. ...

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

2013
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arXiv
*
pre-print

Reshetnyak and subsequently

arXiv:1311.0899v1
fatcat:5nsxi5q3pvdfjfttdbuhnfsoiu
*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] . ...##
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Writhe induced phase transition in unknotted self-avoiding polygons
[article]

2016
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arXiv
*
pre-print

Recently it has been argued that weighting the writhe of

arXiv:1510.07558v3
fatcat:5ggllu5uoncyveccaddbn4mrp4
*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 ...##
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Elementary knot theory
[article]

2016
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arXiv
*
pre-print

It was shown recently by Kronheimer and Mrowka [47] that a more refined invariant, Khovanov homology, does

arXiv:1604.03778v1
fatcat:4ptksoisobgc3l5idxky3kroxu
*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. ...##
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The rational Witt class and the unknotting number of a knot
[article]

2009
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arXiv
*
pre-print

Our results are mainly concerned with

arXiv:0907.2275v1
fatcat:mllnryjyoregbdl2jqqe2kk6uy
*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. ...##
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On Rack Invariants Of Legendrian Knots
[article]

2017
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arXiv
*
pre-print

These rack invariants do not result in a complete invariant, but

arXiv:1706.07626v2
fatcat:3vpsk2fmqvgx5nonl4ztrwxmiq
*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. ...##
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Knots and solenoids that cannot be attractors of self-homeomorphisms of R^3
[article]

2018
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arXiv
*
pre-print

Thus e(K) has the natural neighbourhood basis {e(T

arXiv:1806.11151v1
fatcat:vpbstb4flzg4bp3o2ib4q7i7aq
*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. ...
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