On the Question of Genericity of Hyperbolic Knots (original) (raw)

Hyperbolic knots are not generic

arXiv: Geometric Topology, 2019

We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings does not converge to 111 as nnn approaches infinity. Moreover, we show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime knots of nnn or fewer crossings does not converge to 000 as nnn approaches infinity.

Triple Crossing Number of Knots and Links

Journal of Knot Theory and Its Ramifications, 2013

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c3(K) for a variety of knots and links. We then use c3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to cn(K).

Hyperbolic links are not generic

arXiv: Geometric Topology, 2019

We show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime non-split links of nnn or fewer crossings does not converge to 000 as nnn approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of nnn or fewer crossings does not converge to 1 as nnn approaches infinity. We consider unoriented link types.

The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers

2009

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number.

THE NEXT SIMPLEST HYPERBOLIC KNOTS

Journal of Knot Theory and Its Ramifications, 2004

We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these "simple" hyperbolic knots have high crossing number. We also compute their Jones polynomials.

On the AJ conjecture for knots

Indiana University Mathematics Journal, 2015

We confirm the AJ conjecture [Ga2] that relates the A-polynomial and the colored Jones polynomial for hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. This extends the result of the first author in [Le2], who established the AJ conjecture for a large class of two-bridge knots, including all twist knots. Along the way, we explicitly calculate the universal SL 2 (C)-character ring of the knot group of the (−2, 3, 2n + 1)-pretzel knot, and show it is reduced for all integers n.

Hyperbolic Invariants of Knots and Links

Transactions of the American Mathematical Society, 1991

Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformai invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of 2, 3 and 4 components through 9 crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.

Multi-Crossing Numbers for Knots

2019

We study the projections of a knot K that have only n-crossings. The n-crossing number of K is the minimum number of n-crossings among all possible projections of K with only n-crossings. We obtain new results on the relation between n-crossing number and (2n− 1)-crossing number for every positive even integer n.

On the classification of knots

Proceedings of the American Mathematical Society, 1974

Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister’s table are the seven identified by Tait in 1884. Diagrams of the 165 prime 10 10 -crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.