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.

[Form of Admission of Ephraim Morton (Harvard AB 1787)], 1784 March 15

Printed certificate of admission for undergraduate Ephraim Morton signed by President Joseph Willard. Includes a handwritten emendation to the text made by President Willard on February 17, 1784.

Letters from T. Morton Jones to William Tudor,

Three letters and one price circular for patent wrought iron nails. Correspondence includes details on nail design and prices, as well as gossip about mutual friends and associates.

Letters from T. Morton Jones to William Tudor,

Four letters regarding nail manufacture and the state of their business. Jones also recounts chance meetings with mutual associates, including a satirical account of one exchange.

Letters from T. Morton Jones to William Tudor,

Five letters, in which Jones recounts a club meeting and toasts and songs given, and provides updates on the business and descriptions of nail manufacture and design. He also comments on fashions of the day and includes an anectdote about a lecture he attended that was so dull, one audience member "fell fast asleep & tumbled down with a crash that startled everyone in the room." Letter dated 1814 March 24 is addressed to "Fanny" but also contains a message to Tudor from Jones.

Letters from T. Morton Jones to William Tudor,

Four letters regarding the nail business, including information on prices, designs, and types of metals. Also included is news of friends and a description of festivities on election day in Birmingham, and Jones’ thoughts on politics and the fate of Napoleon. The last letter regards an unsettled account related to the nail factory.

Letter from William Tudor to T. Morton Jones,

One letter responding to the letter from Jones dated 1819 October 22 about an unclosed account. The letter outlines details of Tudor’s engagement with Williams, Jones & Co. to run the nail factory.