On 2-adjacency relation of two-bridge knots and links. (English) Zbl 1147.57012
Summary: We give a necessary condition for a two-bridge knot or link \(S(p,q)\) to be 2-adjacent to another two-bridge knot or link \(S(r,s)\). In particular, we show that if the trivial knot or link is 2-adjacent to \(S(p,q)\), then \(S(p,q)\) is trivial, that if \(S(p,q)\) is 2-adjacent to its mirror image, then \(S(p,q)\) is amphicheiral, and that for a prime integer \(p\), if \(S(p,q)\) is 2-adjacent to \(S(r,s)\), then \(S(p,q)=S(r,s)\) or \(S(r,s)=S(1,0)\).
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
References:
| [1] | DOI: 10.1017/S0305004104007741 · Zbl 1064.57008 · doi:10.1017/S0305004104007741 |
| [2] | DOI: 10.1016/S0166-8641(97)00238-1 · Zbl 0926.57004 · doi:10.1016/S0166-8641(97)00238-1 |
| [3] | DOI: 10.1090/S0002-9939-98-04181-1 · Zbl 0887.57007 · doi:10.1090/S0002-9939-98-04181-1 |
| [4] | Montesinos, Ann. of Math. Stud. 84 pp 227– (1975) |
| [5] | Kalfagianni, Pacific J. Math. 228 pp 251– (2006) |
| [6] | DOI: 10.1142/S0218216505003956 · Zbl 1076.57010 · doi:10.1142/S0218216505003956 |
| [7] | DOI: 10.1112/S0024609301008980 · Zbl 1027.57004 · doi:10.1112/S0024609301008980 |
| [8] | DOI: 10.2307/1971311 · Zbl 0633.57006 · doi:10.2307/1971311 |
| [9] | DOI: 10.3836/tjm/1202136680 · Zbl 1146.57013 · doi:10.3836/tjm/1202136680 |
| [10] | DOI: 10.3836/tjm/1166661876 · Zbl 1104.57007 · doi:10.3836/tjm/1166661876 |
| [11] | DOI: 10.1016/j.topol.2003.08.001 · Zbl 1045.57002 · doi:10.1016/j.topol.2003.08.001 |
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