New criteria and constructions of Brunnian links

Abstract

We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite families of Brunnian links are created and we establish a general way to construct new ones in bulk.

Hyperbolic Brunnian links

“…This is Case (1.0.0) in Lemma 6.10(1). Delete the ear E. Then using the approach in Example 5.3 in [5], we can show the asserted incredible circle in Proposition 6.11(1) does not exist. 6.3.…”

“…Then we are interested in how many Brunnian links in history, which, to our best knowledge, all appear in [6,8,20,1,3,2,15,12,10,18,16,5], are hyperbolic. Our methods work to detect s-primeness for all of them and untiedness for all except Fountains and Jade-pendants in [5]. Especially, we list some infinite series of Brunnian links:…”

“…Noting that the complement of U is smaller than the component of L, this proposition reduces the task of showing a disk is credible in application. In fact, the main content of [5] is to show disks are credible, which proposes two widely applicable methods, including several criteria and a general procedure.…”

“…From this example, we can see that this method is intuitive and efficient. It works conveniently to show all the Brunnian links in literatures, except Fountains and Jade-pendant in [5], are untied.…”

“…Extending [5], we will cope with these two tasks using geometric topology methods. [5] provides two intuitive methods, generally effective and performed by handwork, for detecting Brunnian property of a link if each component is known to be unknot. In Section 4, we will prove a universal criterion to detect whether a Brunnian link is untied.…”