100 Are there any very hard unknots? 2011-01-27T09:35:29.600

50 Tying knots with reflecting lightrays 2010-09-15T11:56:24.910

49 Kirby calculus and local moves 2010-07-07T22:58:42.290

45 Prime numbers as knots: Alexander polynomial 2010-07-09T21:42:40.880

42 Can knot diagrams be monotonically simplified using under moves? 2014-10-08T21:39:08.660

36 Knot security (When to trust your life with a knot) 2012-09-17T22:18:42.543

33 In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$? 2011-04-27T12:26:49.713

33 Applications of arithmetic topology to number theory 2016-07-23T23:21:16.650

32 What is the three-dimensional hyperbolic volume of a four-manifold? 2011-07-08T19:14:14.463

31 Applications of knot theory 2010-12-03T22:23:17.700

29 probabilistic knot theory 2009-10-30T04:54:27.067

29 Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks? 2010-05-18T23:29:48.223

29 Random knot on six vertices 2011-02-09T19:51:38.467

29 Is there a "knot theory" for graphs? 2011-10-30T00:46:32.147

28 The Jones polynomial at specific values of $t$. 2014-06-30T05:20:55.397

28 Metrics on the 3-sphere with knotted geodesics 2015-07-24T22:40:26.647

27 Does this knot invariant distinguish trefoil chiralities? 2012-04-30T12:59:34.327

27 What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka) 2016-02-27T09:02:12.870

26 Intuitive explanation for the use of matrix factorizations in knot theory 2010-07-08T11:09:36.277

25 Complexity of random knot with vertices on sphere 2011-02-05T16:27:17.010

23 What would the slice-ribbon conjecture imply? 2009-11-28T14:51:43.720

23 How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. 2010-08-18T10:20:40.913

22 Why is it so hard to implement Haken's Algorithm for knot theory? 2011-01-25T19:00:55.613

21 Why is the Alexander polynomial a quantum invariant? 2009-12-27T02:00:16.350

21 Proof of the Reidemeister theorem 2010-02-13T23:35:18.103

21 What is the knot associated to a prime? 2011-01-01T19:39:51.010

21 Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials? 2015-08-19T21:40:27.183

20 How are the Conway polynomial and the Alexander polynomial different? 2010-07-07T22:17:42.227

20 Complete knot invariant? 2010-08-15T19:21:13.337

20 Are there moves between Reidemeister moves? 2010-11-29T03:45:11.700

20 Utility of virtual knot theory? 2010-12-01T13:29:01.783

20 Concordance and homology cobordism 2015-08-18T18:32:41.817

19 Is the 4x5 chessboard complex a link complement? 2010-08-26T18:41:38.630

19 Propagation of an error in the LMO invariant? (Revision: I don't think LMO is wrong!) 2010-10-14T15:39:14.737

19 Is anything known about this braid group quotient? 2011-05-11T15:04:59.597

19 Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group? 2013-01-21T15:53:56.580

19 Topological description of inverting a knot 2014-09-05T21:48:33.410

18 Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants 2010-08-27T02:20:28.893

18 What is a path in K-theory space? 2010-09-01T12:43:41.693

18 Classification of tangles? 2014-08-16T06:15:01.873

18 Human Knot game 2015-05-10T15:17:13.460

17 Does the super Temperley-Lieb algebra have a Z-form? 2009-10-29T17:46:09.173

17 Can surfaces be interestingly knotted in five-dimensional space? 2011-01-20T21:53:41.673

17 How many definitions are there of the Jones polynomial? 2011-06-13T23:37:50.283

17 What is the state of the art for algorithmic knot simplification? 2013-10-07T00:35:26.863

16 Are there piecewise-linear unknots that are not metrically unknottable? 2009-12-08T00:29:39.740

16 topological "milnor's conjecture" on torus knots. 2010-02-17T00:25:45.573

16 What's about "quantum modular forms"? 2011-01-04T15:48:51.933

16 Visualize Fourth Homotopy Group of $S^2$ 2013-04-26T12:13:47.297

16 Random rings linked into one component? 2013-04-27T17:56:13.007

16 Looking for "large knot" examples 2013-09-04T20:07:22.453

15 HOMFLY and homology; also superalgebras 2009-10-22T06:48:37.027

15 can you fool SnapPea? 2009-11-10T20:39:04.913

15 Who thought that the Alexander polynomial was the only knot invariant of its kind? 2010-04-01T00:36:06.800

15 Semidirect product decomposition of the Borromean rings group 2013-06-06T10:29:21.933

15 Can infinitely many alternating knots have the same Alexander polynomial? 2014-10-27T09:29:10.093

15 How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras? 2015-05-25T10:57:00.773

14 Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid? 2010-04-12T22:58:04.487

14 What do tangles teach us about braids? 2014-08-03T08:23:03.940

14 Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold? 2015-08-11T11:27:00.513

14 Classification of knots by geometrization theorem 2016-01-16T17:21:36.537

13 Knots that unknot in a manifold 2009-12-12T19:57:34.613

13 What are the points of Spec(Vassiliev Invariants)? 2010-01-06T20:00:29.937

13 How to motivate the skein relations? 2010-04-05T17:29:50.937

13 Are homological knot invariants of finite type? 2010-09-13T16:08:01.753

13 Fox differential calculus and the Alexander invariant of a link 2011-11-13T20:22:08.860

13 Obtain any 3-manifold from repeating surgeries on knots in $S^3$ 2015-03-21T21:29:59.240

13 Does Lackenby's polynomial bound on knot moves imply polynomial mixing in "Quantum Money From Knots?" 2017-12-23T21:32:02.280

12 Slice knots and exotic $\mathbb R^4$ 2010-10-18T13:00:14.887

12 Random Reidemeister moves to unknot 2011-10-09T00:54:19.823

12 Why "Categorify"? Relating to link/knot homologies... 2012-01-08T21:10:05.960

12 Integer matrices with a strange divisibility property 2012-05-16T19:23:38.267

12 Representing SU(3) with 3 ropes in 3 dimensions 2013-09-03T13:47:34.273

12 Kontsevich integral : state of the art 2013-10-10T21:07:02.633

12 What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether? 2014-01-21T03:54:56.613

12 $S^3$ as cyclic branched cover of itself 2016-08-12T10:48:04.283

12 Structure of Gordian graph of knots 2016-11-20T05:50:02.507

12 Elementary proof that knot complements are path-connected 2016-11-24T22:00:18.570

11 slice-ribbon for links (surely it's wrong) 2010-01-14T01:13:55.563

11 Is there a non-trivial knot with trivial Homfly polynomial? 2010-05-24T06:27:04.947

11 complexity of counting homomorphisms 2010-05-31T16:17:07.427

11 Is there a procedure for obtaining all knots in S^3? 2010-08-04T06:06:18.617

11 Algorithm for detecting ribbon or slice links? 2011-02-19T03:25:45.540

11 What is the metamathematical interpretation of knot diagrams? 2012-01-30T04:12:22.870

11 On connection between Knot theory and Operator algebra 2012-05-23T20:10:06.170

11 How much do homological knot invariants improve the classification problem of knots? 2012-09-09T15:32:21.363

11 Descriptive Complexity of Knot Equivalence 2015-04-17T16:48:35.553

11 A fun game related to knot theory 2015-08-10T16:58:58.650

11 Distance between two knots 2015-11-27T14:53:31.107

11 Can Khovanov homology have arbitrarily large torsion? 2017-04-18T20:14:20.607

11 Gap in Przytycki's computation of the skein module of links in a handlebody? 2017-08-08T08:28:00.757

11 How hard is it to guess Kuperberg's certificate of knottedness? 2018-01-08T02:17:18.970

10 Ramified covers of 3-torus 2009-11-14T18:33:44.893

10 Computing the structure of the group completion of an abelian monoid, how hard can it be? 2010-02-03T07:24:35.683

10 Categorical Schur's Lemma 2010-11-10T22:17:59.623

10 Do the results of (1/n)-surgery determine the link?... 2011-02-22T02:36:21.277

10 Traces on Hecke algebras and the Jones polynomial 2011-11-13T23:15:37.690

10 Is there a periodic table for knots? 2011-12-03T17:02:54.163

10 Self-tightening knot 2012-01-08T13:01:49.927