Wirtinger systems of generators of knot groups

Abstract

We define the Wirtinger number of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson's Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial technique for obtaining strong upper bounds on bridge numbers. This technique has so far allowed us to add the bridge numbers of approximately 50,000 prime knots of up to 14 crossings to the knot table. As another application, we use the Wirtinger number to show there exists a universal constant C with the property that the hyperbolic volume of a prime alternating link L is bounded below by C times the bridge number of L.

Bridge numbers and meridional ranks of knotted surfaces and welded knots

“…FIGURE 2. In a k-partially colored diagram D, if there is a crossing where locally, the incoming strand is assigned the color blue and the overstrand is assigned the color blue, a coloring move can be performed to extend the coloring to the outgoing strand so that we have a (k + 1)partial coloring of D. [5]. The idea is to define the minimal number of arcs of a diagram needed to complete the Wirtinger algorithm to present π 1 (S 3 \ N(K).…”

“…The bridge number provides a comprehensible exhaustion of all knots; indeed, 2-bridge knots are the simplest of knots in many ways, and their classification by Schubert was a triumph of early knot theory [37]. For suitable classes of knots, the bridge number can be used to estimate other geometric invariants such as hyperbolic volume [5,Theorem 1.5], distortion [6,Theorem 1.1], ropelength [12,Theorem 5], and total curvature [29,Corollary 3.2]. Cappell and Shaneson's Meridional Rank Conjecture posits that β (K) is equal to the meridional rank µ(K), the minimal number of meridians needed to generate π 1 (S 3 \K).…”

“…We will show that if a BBKM knot K admits a surjective homomorphism to a Coxeter group C(∆), performing appropriate local operations on a diagram of K involving insertions of virtual crossings still gives a surjective homomorphism to a Coxeter group C(∆ ) with the same rank, but possibly with different weights on the edges of the graph ∆ . The Wirtinger number is a combinatorial upper bound for the bridge number of a classical knot, introduced by Blair, Kjuchukova, Velazquez, and Villanueva in [5]. In that work they prove that the Wirtinger number is in fact equal to the bridge number.…”

“…In general, Purcell and Zupan showed that for any g, b ≥ 1, there does not exist C such that for any hyperbolic knot Cβ g (K) ≤ vol(K), where β g is the genus g-bridge number [32]. Blair et al showed that the analogue of Jørgensen and Thurston's result does hold if one restricts to the collection of prime alternating knots in S 3 with respect to the genus 0 Heegaard splitting, and set the universal constant C = 1 6 v tet , where v tet ≈ 1.01 is the volume of a regular ideal tetrahedron [5,Theorem 1.5]. We will show that C = 1 6 v tet can still function as a universal constant for larger families of knots.…”

Coxeter groups and meridional rank of links

“…We have slightly simplified the original definition of k-colorability, which makes use of k different colors. Multiple colors are needed in the proof of the Main Theorem of[4], but they are of no help to us here. The modified definition has no effect on the value of ω(D).…”