A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabil...A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabilization problem,reducibilities,minimal Heegaard splitting and Heegaard distance.展开更多
Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-ma...Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M＇ has a Heegaard splitting with sufficiently high distance.展开更多
Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 ...Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.展开更多
For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut ope...For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut open along these tori,one resulting piece is hyperbolic while the others are small Seifert fibered spaces;(3)it provides a substantial result to the rank versus genus problem.These generalize a result in Qiu and Zou(2019).展开更多
We introduce a method to compute the girth of knots, defined by Herne^ndez and Lin, using the Jones and Brandt-Lickorish-Millett-Ho polynomial. We determine the girth of all knots up to 10 crossings.
基金This work was partially supported by Science and Technology Commission of Shanghai Municipality(STCSM),(18dz2271000)and NSFC(12131009)。
文摘A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabilization problem,reducibilities,minimal Heegaard splitting and Heegaard distance.
文摘Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M＇ has a Heegaard splitting with sufficiently high distance.
文摘Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.
基金supported by National Natural Science Foundation of China(Grant Nos.12131009 and 12326601)Science and Technology Commission of Shanghai Municipality(Grant No.22DZ2229014)。
文摘For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut open along these tori,one resulting piece is hyperbolic while the others are small Seifert fibered spaces;(3)it provides a substantial result to the rank versus genus problem.These generalize a result in Qiu and Zou(2019).
文摘We introduce a method to compute the girth of knots, defined by Herne^ndez and Lin, using the Jones and Brandt-Lickorish-Millett-Ho polynomial. We determine the girth of all knots up to 10 crossings.
