Coherent band pathways between knots and links

Abstract

We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.

Distance one lens space fillings and band surgery on the trefoil knot

“…Figure 10. Coherent band surgery is better understood than the non-coherent case (see for example [37,38,13,14,21,64]). If a coherent band surgery decreases the maximal Euler characteristic of an oriented surface without closed components bounding the link, it is known that the band can be isotoped to lie onto a taut Seifert surface [62], [35,Theorem 1.6].…”

Nullification numbers of knots with up to 10 crossings

“…However, the nullification operation can be seen in several different perspectives which have frequent appearance in the literature [4, 5, 10-12, 15-21, 24, 30]. The most common operation considered in the above literature that is equivalent to nullification is called coherent band surgery [4,5,12,[16][17][18][19]21]. In particular, let L be an oriented link, and b : I × I → R 3 an embedding such that b(I × I) ∩ L = b(I ×∂I), where I is a closed interval.…”

“…knot n θ (K) low up knot n θ (K) low up 4 1 + θ σ s,∆ 7 5 3 + θ σ u,∆ 7 6 1 + θ σ u,∆ 7 7 1 + θ g * u,∆ 8 1 1 + θ g * u,∆ 8 2 3 + θ σ u,∆ 8 3 1 + θ g * s,∆ 8 4 1 + θ σ u±,∆ 8 5 3 + θ σ u,∆ 8 6 1 + θ σ u±,∆ 8 7 1 + θ σ u,∆ 8 8 1 + |θ − 2| =,σ r,∆ 8 9 1 + |θ − 2| =,σ u,r,∆ 8 10 1 + θ σ u±,∆ 8 11 1 + θ σ u,∆ 8 12 1 + θ g * u±,∆ 8 13 1 + θ g * u,∆ 8 14 1 + θ σ u,∆ 8 15 3 + θ σ u,∆ 8 16 1 + θ σ u±,∆ 8 17 1 + θ g * u,∆ 8 18 1 + θ g * u±,∆ 8 19 5 + θ σ u,∆ 8 20 1 + |θ − 2| =,σ u,r,∆ 8 21 1 + θ σ u,∆ 9 1 7 + θ σ u,∆ 9 2 1 + θ σ u,∆ 9 3 5 + θ σ u,∆ 9 4 3 + θ σ u,∆ 9 5 1 + θ σ s,∆ 9 6 5 + θ σ u,∆ 9 7 3 + θ σ u,∆ 9 8 1 + θ σ u±,∆ 9 9 5 + θ σ u,∆ 9 10 3 + θ σ s,∆ 9 11 3 + θ σ u,∆ 9 12 1 + θ σ u,∆ 9 13 3 + θ σ s,∆ 9 14 1 + θ g * u,∆ 9 15 1 + θ σ u±,∆ …”

Recent advances on the non-coherent band surgery model for site-specific recombination

“…This process has been the focus of numerous experimental and theoretical studies, e.g. [42,33,83,12,13,85].…”

“…While the coherent band surgery model for site-specific recombination has received much attention in the literature, (e.g. [44,12,13,19,83]), non-coherent band surgery is less-understood and is intrinsically difficult from an analytical perspective. This is due to the implicit existence of a non-orientable surface induced by the banding.…”