The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology

By Curtis T. McMullen. 2002

105 citations recorded by the source corpus.

Let M be a connected, compact, orientable 3-manifold with b1(M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kk A ≤ kk T between the Alexander norm on H 1 (M, Z), defined in terms of the Alexan- der polynomial, and the Thurston norm, defined in terms of the Eu- ler characteristic of embedded surfaces. (A similar result holds when b1(M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bdcb05ffe487089842a610d526647946fe8b4098

Citations in the corpus

• Floer homology of surgeries on two-bridge knots (2002, unknown): “e sign of the associated intersection point u, which is (−1)gr(u) = (−1)ǫx(u). Strictly speaking, the Alexander polynomial is the gcd of da1wβ1 and da2wβ1, but an argument of McMullen (Theorem 5.1 of [4]) shows that when the presentation of π 1 comes from a Heegard splitting, da1wβ1 and da2wβ1 agree”
• Grid Homology for Knots and Links (2015, unknown): “In this section we recall the construction of the Thurston norm and refer the reader to [217] for a detailed account of this beautiful story; see also [19, 20, 21, 140].”
• Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm (2002, unknown): “McMullen defined the Alexander norm of a cohomology class of a 3-manifold via the Alexander polynomial and proved that it is a lower bound for the Thurston norm [Mc].”
• Floer homology and knot complements (2003, unknown): “e the gcd of all the polynomials P k= det(d x iw j) j6= k When the presentation comes from a Heegaard splitting, however, all of the P k’s divide each other, so we are done. (See, e.g, Theorem 5.1 of [16].)FLOER HOMOLOGY AND KNOT COMPLEMENTS 21 4. The Alexander Filtration In this section, we conside”
• Twisted Alexander polynomials detect fibered 3-manifolds (2008, unknown): “Note that McMullen [McM02] showed that when the class φ is primitive, the condition ∆N,φ 6= 0 implies that there exists a connected Thurston normminimizing surface Σ dual to φ.”
• The decategorification of sutured Floer homology (2009, unknown): “A well-known theorem of McMullen [22] says that Δ(Y ) gives a lower bound on the Thurston norm of Y .”
• The Thurston norm and twisted Alexander polynomials (2012, unknown): “The proof is essentially given in [32] and we provide it for the reader’s convenience as it is helpful in understanding the proof of Theorem 1.”
• The Thurston norm, fibered manifolds and twisted Alexander polynomials (2005, unknown): “In [FK05] we introduce twisted Alexander norms (similar to McMullen’s Alexander norm [Mc02]) which are well–suited to study the Thurston norm of link complements.”
• Closed surface bundles of least volume (2010, unknown): “For background and definitions, see McMullen’s original paper [17].”
• Link Floer homology and the Thurston norm (2006, unknown): “For the case of knots in the three-sphere, this invariant is a bigraded Abelian group, whose graded Euler characteristic is the Alexander polynomial.”
• A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, unknown): “ote that if γ: π1(N) → GL(Z,1) is the trivial representation, then we just obtain McMullen’s Alexander norm. The following is proved in [FK08a, Theorem 3.1], but we also refer to the work of McMullen [McM02], Turaev [Tu02a], [Tu02b, Section 6], Harvey [Ha05] and Vidussi [Vi99, Vi03]. The main idea o”