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• Unknotting problem (category Knot theory)
  co-NP. Knot Floer homology of the knot detects the genus of the knot, which is 0 if and only if the knot is an unknot. A combinatorial version of knot Floer
  11 KB (1,220 words) - 21:16, 6 February 2024
• Link group (category Knot invariants)
  invariants, and in fact they (and their products) are the only rational finite type concordance invariants of string links; (Habegger Masbaum). The number of linearly
  9 KB (1,196 words) - 13:58, 6 February 2024
• Khovanov homology (category Knot invariants) (section The relation to link (knot) polynomials)
  [ø] = 0 → Z → 0, where ø denotes the empty link. [O D] = V ⊗ [D], where O denotes an unlinked trivial component. [D] = F(0 → [D0] → [D1]{1} → 0) In the
  11 KB (1,333 words) - 15:33, 6 February 2024
• Quadrisecant (category Knot theory) (section Knots and links)
  was extended to knots in suitably general position and links with nonzero linking number, and later to all nontrivial tame knots and links. Pannwitz proved
  17 KB (2,017 words) - 20:31, 6 February 2024
• Ropelength (category Knot invariants) (section Dependence on crossing number)
  }[/math] denotes the crossing number. There exist knots and links, namely the [math]\displaystyle{ (k,k-1) }[/math] torus knots and [math]\displaystyle{
  6 KB (733 words) - 20:48, 6 February 2024
• Book embedding (section Planarity and outerplanarity)
  book crossing number of a graph is also NP-hard, because of the NP-completeness of the special case of testing whether the 2-page crossing number is zero
  65 KB (7,764 words) - 21:35, 6 February 2024
• Graph theory (section Physics and chemistry)
  problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between
  52 KB (6,469 words) - 18:45, 8 February 2024
• Tangle (mathematics) (category Knot theory) (section Rational and algebraic tangles)
  theory. Tanglement puzzle Conway, J. H. (1970). "An Enumeration of Knots and Links, and Some of Their Algebraic Properties". in Leech, J.. Computational
  8 KB (994 words) - 17:43, 6 February 2024
• Möbius strip (section Polyhedral surfaces and flat foldings)
  }[/math]-plane and is centered at [math]\displaystyle{ (0, 0, 0) }[/math]. The same method can produce Möbius strips with any odd number of half-twists
  85 KB (9,635 words) - 13:43, 6 February 2024
• Wirtinger presentation (category Knot theory) (section Wirtinger presentations of high-dimensional knots)
  \rang. }[/math] Knot group Rolfsen, Dale (1990), Knots and links, Mathematics Lecture Series, 7, Houston, TX: Publish or Perish, ISBN 978-0-914098-16-4 ,
  4 KB (467 words) - 04:50, 27 June 2023
• Dowker–Thistlethwaite notation (category Knot theory) (section Uniqueness and counting)
  order of traversal (each crossing is visited and labelled twice), with the following modification: if the label is an even number and the strand followed crosses
  4 KB (408 words) - 22:18, 6 February 2024
• Möbius energy (category Knot theory) (section Knot invariant)
  [math]\displaystyle{ \gamma_0(t)=(\cos t, \sin t, 0) }[/math] denote a unit circle. We have [math]\displaystyle{ |\gamma_0(x)-\gamma_0(y)|^2={\left(2\sin\tf
  21 KB (3,389 words) - 15:01, 6 February 2024
• Lists of mathematics topics (category Outlines of mathematics and logic) (section External links and references)
  mathematics." Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are
  21 KB (2,590 words) - 13:37, 6 February 2024
• Arc diagram (section Minimizing crossings)
  semicircle per edge and no crossings, it is also NP-hard to find an arc diagram of this type that minimizes the number of crossings. This crossing minimization
  20 KB (2,362 words) - 20:05, 6 February 2024
• Petal projection (category Knot theory) (section Generalization to links)
  description: Form of knot diagram In knot theory, a petal projection of a knot is a knot diagram with a single crossing, at which an odd number of non-nested
  5 KB (619 words) - 22:20, 16 November 2021
• List of knot theory topics (category Knot theory) (section Knots, links, braids)
  Bridge number Crosscap number Crossing number (knot theory) Hyperbolic volume (knot) Kontsevich invariant Linking number Milnor invariants Racks and quandles
  6 KB (752 words) - 16:13, 4 August 2021
• Tunnel number (category Knot invariants)
  "Tunnel number one knots satisfy the Poenaru conjecture", Topology and Its Applications 18 (2–3): 235–258, doi:10.1016/0166-8641(84)90013-0 . Scharlemann
  3 KB (338 words) - 17:17, 8 February 2024
• Fox n-coloring (category Knot theory) (section Number of colorings)
  of 3-Manifolds and Related Topics", Prentice-Hall, NJ, 1961, pp. 120–167. MR0140099 Ralph H. Fox, Metacyclic invariants of knots and links, Canadian Journal
  7 KB (1,014 words) - 11:42, 10 August 2021
• Average crossing number (category Knot theory)
  Ernst, Claus (2001). "The Crossing Numbers of Thick Knots and Links". in Jorgr Alberto Calvo. Physical Knots: Knotting, Linking, and Folding Geometric Objects
  4 KB (513 words) - 21:43, 6 February 2024
• List of unsolved problems in mathematics (section Number theory)
  conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree The Albertson conjecture: the crossing number can be lower-bounded
  186 KB (18,657 words) - 04:25, 9 March 2024

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