Closures of positive braids and the Morton-Franks-Williams inequality

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is an orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this result by using an interesting connection with Contact Topology. We also introduce a new technique to study the Homflypt polynomial for closures of positive braids, namely resolution trees whose leaves are simple braids. In terms of these simple resolution trees, we characterize closed positive braids for which the Morton-Franks-Williams inequality is strict. In particular, we determine explicitly the positive braid words on three strands whose closures have braid index three.

Proceedings of the International Billfish Symposium, Kailua-Kona, Hawaii, 9-12 August 1972 / Richard S. Shomura and Francis Williams, editors.

pt. 1

Proceedings of the International Billfish Symposium, Kailua-Kona, Hawaii, 9-12 August 1972 / Richard S. Shomura and Francis Williams, editors.

pt. 3

Letter from William Croswell to Samuel Williams, [1808]

Draft of a letter concerning a transaction related to Croswell's map.

Letters from William Croswell to the Harrison Gray Otis, 1824-1825

Two leaves containing drafts of brief notes that accompanied letters for Harvard Corporation between April 1824 and April 1825.

Letter from William Croswell to Harrison G. Otis, 1833 November 4

Two drafts of a letter requesting financial assistance.

Payment order for Jonathan Williams, 1755 March 3

Handwritten order to John Sale to pay scholarship funds to Deacon Jonathan Williams, signed by Thomas Foxcroft, Charles Chauncey, Thomas Waite, and Daniel Marsh.

Payment order for Robert Williams on behalf of William Bradford, 1757 August 16

Handwritten order to John Sale to pay scholarship funds to Robert Williams on behalf of his nephew William Bradford (Harvard AB 1760), signed by Thomas Foxcroft, Charles Chauncey, Thomas Waite, Jonathan Williams, and Daniel Marsh.

Receipt signed by Robert Williams, 1757 December 14

Handwritten receipt signed by Robert Williams acknowledging payment by John Sale of scholarship funds.