Knots with g(E(K)) = 2 and g(E(K#K#K)) = 6 and Morimoto’s Conjecture

نویسنده

  • TSUYOSHI KOBAYASHI
چکیده

We show that there exist knots K ⊂ S with g(E(K)) = 2 and g(E(K#K#K)) = 6. Together with [5, Theorem 1.5], this proves existence of counterexamples to Morimoto’s Conjecture [10]. This is a special case of [6]. Let Ki (i = 1, 2) be knots in the 3-sphere S, and let K1#K2 be their connected sum. We use the notation t(·), E(·), and g(·) to denote tunnel number, exterior, and Heegaard genus respectively (we follow the definitions and notations given in [7]). It is well known that the union of a tunnel system for K1, a tunnel system for K2, and a tunnel on a decomposing annulus for K1#K2 forms a tunnel system for K1#K2. Therefore: t(K1#K2) ≤ t(K1) + t(K2) + 1. Since (for any knot K) t(K) = g(E(K))− 1 this gives: (1) g(E(K1#K2)) ≤ g(E(K1)) + g(E(K2)). We say that a knot K in a closed orientable manifold M admits a (g, n) position if there exists a genus g Heegaard surface Σ ⊂ M , separating M into the handlebodies H1 and H2, so that Hi ∩K (i = 1, 2) consists of n arcs that are simultaneously parallel into ∂Hi. It is known [10, Proposition 1.3] that if Ki (i = 1 or 2) admits a (t(Ki), 1) position then equality does not hold: g(E(K1#K2)) < g(E(K1)) + g(E(K2)). Morimoto proved that if K1 and K2 are m-small knots then the converse holds, and conjectured that this holds in general [10, Conjecture 1.5]: Conjecture 1 (Morimoto’s Conjecture). Given knots K1, K2 ⊂ S, g(E(K1#K2)) < g(E(K1)) + g(E(K2)) if and only if for i = 1 or i = 2, Ki admits a (t(Ki), 1) position. We denote the connected sum of n copies of K by nK. We prove: Date: February 2, 2008. 1991 Mathematics Subject Classification. 57M99.

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We show that there exist knots K ⊂ S with g(E(K)) = 2 and g(E(K#K#K)) = 6. Together with [5, Theorem 1.5], this proves existence of counterexamples to Morimoto’s Conjecture [10]. This is a special case of [6]. Let Ki (i = 1, 2) be knots in the 3-sphere S, and let K1#K2 be their connected sum. We use the notation t(·), E(·), and g(·) to denote tunnel number, exterior, and Heegaard genus respecti...

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تاریخ انتشار 2008