An Introduction to Knot Theory

By Louis H. Kauffman. 2001

1,009 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Combining self-supervised learning and imitation for vision-based rope manipulation (2017, background): “[6] developed a system for tying knots from visual observations that makes use of knot theory [25] to identify a sequence of knot states and then execute motor primitives to achieve these states.”
Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial (2011, background): “This is known to be one dimensional when the three colored points are admissibly colored, generated by a single trivalent vertex [Lic97].”
Regularizing and self-avoidance effects of integral Menger curvature (2010, neutral): “(See the book of Crowell and Fox, [8, pages 147-152] who prove that every simple arclength parametrized C 1 curve is ambient isotopic to a regular embedded polygon.)”
Polynomial Invariants for Torus Knots¶and Topological Strings (2000, background): “(5.3) is the unnormalized HOMFLY polynomial. The fact that the expansion in x starts with x−1 is a consequence of (4.5). Since the normalized HOMFLY polynomial is even under the exchange of t and t−1 [27], G(c) (1,0,···)(U) is odd, in agreement with the prediction. These two facts were already noted”
Virtual knots and links (2005, background): “Now the remaining part of the proof of Theorem 5 follows from the nontriviality of the connected sum in the classical case (see [ 3 ]).”
Symmetric unions and ribbon knots (2000, neutral): “5) we have the formula well-known from properties of the Kauffman polynomial (see for instance [17] p. 101).”
The head and tail conjecture for alternating knots (2011, methodology): “This is known to be one dimensional when the three colored points are admissibly colored, generated by a single trivalent vertex [12].”
Chern-Simons theory and topological strings (2004, neutral): “d in the canonical framing. Besidesprovidinginvariantsofthree-manifolds, ChernSimons theory also provides invariants of knots and links inside three-manifolds (for a survey of modern knot theory, see Lickorish (1998) and Prasolov and Sossinsky (1997)). Someexamplesofknotsandlinks aredepicted in Fig.”
An endomorphism of the Khovanov invariant (2002, methodology): “We follow [9] and [4] for basic notions in knot theory and graph theory, and [7] for notations and terminologies related to the Khovanov invariant.”
Knots, links and branes at large N (2000, background): “ unnormalized HOMFLY polynomial of the knot [3], the above result is in perfect agreement with the fact that the HOMFLY polynomial can be written in terms of even powers of the variable q−1 2 − q 1 2 [14][15]. For ℓ= 2, we have f (q,λ) = X Q X g≥0 q−1 2 Nb ,g,Q −q 1 2 Nb ,g,Q (q−1 2 − q 1 2 )2g−1λQ,”
Mahler measure, links and homology growth (2000, methodology): “, cN with each boundary ∂ci attached to 1-cells according to ri (see Chapter 11 of [Li]).”
The Word and Geodesic Problems in Free Solvable Groups (2008, neutral): “Here we recall some notions and results, referring to the books [10, 6, 29] for details.”
NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “See [5] or [14] for further explanation.”
Localization for Involutions in Floer Cohomology (2010, background): “The right-hand side (which, by definition, vanishes if b1(Nκ) > 0) is also (twice) the value of the Alexander polynomial at −1, known classically as the determinant of the knot [ Li , Cor.”
On knot Floer homology and cabling (2004, background): “The knot which is cabled is sometimes called the companion knot (see [7] for more details).”