Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down

Yang, Guo Qiu; Lei, Feng Chun

doi:10.1007/s10114-011-9021-2

Cited by 5 publications

(4 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…And with the weaker assumptions, we obtain the stronger conclusion that the minimal Heegaard splitting of M is in some sense unique. Hence we remark that the situations here are quite different from those in [2], [8] and [17], and the arguments there are not applicable to the main cases here.…”

Section: Introductioncontrasting

confidence: 59%

“…Lemma 2.9 ( [8], [17]). Let N be a compact orientable 3-manifold which is not a compression body, and F = ∂N.…”

Section: Lemma 23 An Incompressible Surface F In a Compression Bodymentioning

confidence: 99%

“…Some sufficient conditions for tunnel number of knots not to go down under connected sum are given in [17]. When both A 1 and A 2 are non-separating on the corresponding boundary component, there are some sufficient conditions for the equality g(M) = g(M 1 ) + g(M 2 ) to hold, see [2], [8].…”

Section: Introductionmentioning

confidence: 99%

“…Note that papers [8] and [17] only consider the case that both A 1 and A 2 are non-separating. In [2], the bound of the Heegaard distance for the additivity to hold is "d(S i ) ≥ 2(g(M 1 ) + g(M 2 )), i = 1, 2", in other words, the bound of d(S 1 ) (d(S 2 ), resp.)…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

Li¹,

Lei²,

Yang³

2014

MATH. SCAND.

View full text Add to dashboard Cite

Let $M_{i}$ be a compact orientable 3-manifold, and $A_{i}$ an incompressible annulus on a component $F_i$ of $\partial M_i$, $i=1,2$. Suppose $A_{1}$ is separating on $F_{1}$ and $A_{2}$ is non-separating on $F_{2}$. Let $M$ be the annulus sum of $M_1$ and $M_2$ along $A_1$ and $A_2$. In the present paper we show that if $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}}W_{i}$ with Heegaard distance $d(S_{i})\geq2g(M_{i})+5$ for $i=1,2$, then $g(M)=g(M_{1})+g(M_{2})$. Moreover, when $g(F_{2})\geq 2$, the minimal Heegaard splitting of $M$ is unique up to isotopy.

show abstract

Section: Introductioncontrasting

confidence: 59%

“…Lemma 2.9 ( [8], [17]). Let N be a compact orientable 3-manifold which is not a compression body, and F = ∂N.…”

Section: Lemma 23 An Incompressible Surface F In a Compression Bodymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

Li¹,

Lei²,

Yang³

2014

MATH. SCAND.

View full text Add to dashboard Cite

show abstract

Heegaard genera of high distance are additive under annulus sum

Yang

Lei

2010

Topology and its Applications

View full text Add to dashboard Cite

Let M i be a compact orientable 3-manifold, and A i a non-separating incompressible annulus on ∂ M i , i = 1, 2. Let h : A 1 → A 2 be a homeomorphism, and M = M 1 ∪ h M 2 the annulus sum of M 1 and M 2 along A 1 and A 2 . In the present paper, we show that if M i has a Heegaard splitting V i ∪ S i W i with distance d(S i ) 2g(M i ) + 3 for i = 1, 2, then g(M) = g(M 1 ) + g(M 2 ). Moreover, if g(F i ) 2, i = 1, 2, then the minimal Heegaard splitting of M is unique.

show abstract

The tunnel number and the cutting number with constituent handlebody-knots

Murao

2021

Topology and its Applications

View full text Add to dashboard Cite

Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down

Cited by 5 publications

References 25 publications

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

Heegaard genera of high distance are additive under annulus sum

The tunnel number and the cutting number with constituent handlebody-knots

Contact Info

Product

Resources

About