Turaev genus and alternating decompositions
We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.
1991 Mathematics Subject Classification:
The discovery of the Jones polynomial [Jon85] led to the resolution of the famous Tait conjectures. In particular, Kauffman [Kau87], Murasugi [Mur87], and Thistlethwaite [Thi88] use the Jones polynomial to prove that an alternating diagram of a link with no nugatory crossings has the fewest possible number of crossings. In Turaev’s [Tur87] alternate proof of this result, he associates a closed oriented surface to each link diagram , now known as the Turaev surface of . Let be a diagram of a non-split link with crossings, let be the Jones polynomial of , and let be the genus of the Turaev surface of . Turaev shows that
Thistlethwaite [Thi88] uses a decomposition of a link diagram into maximal alternating pieces to compute a lower bound on crossing number similar to Inequality (1.1). Consider a link diagram as -valent plane graph with over/under decorations at the vertices. An edge or face of should be understood to refer to an edge or face of the -valent plane graph. An edge of is called non-alternating if both of its endpoints are over-strands or both of its endpoints are under-strands. An edge is called alternating if one of its endpoints is an over-strand and the other is an under-strand. Mark each non-alternating edge of with two distinct points, and in each face of connect those marked points with arcs as depicted in Figure 1. This process results in a collection of pairwise disjoint simple closed curves . The pair is called the alternating decomposition of .
Thistlethwaite associates to a graph , which we call the alternating decomposition graph of , as follows. Suppose that is a connected link diagram, i.e. when is considered as a graph, it is a connected graph. If is an alternating diagram, then is a single vertex with no edges. Otherwise, the vertices of are in one-to-one correspondence with the curves of the alternating decomposition of . The edges of are in one-to-one correspondence with the non-alternating edges of . Let and be vertices of corresponding to curves and respectively. An edge of connects to if and only if the corresponding non-alternating edge of intersects both and . If is not a connected link diagram, then is the disjoint union of the alternating decomposition graphs of its connected components.
The plane embedding of induces an embedding of each component of onto a sphere, as described in Section 3. Since each component of can be embedded on a sphere, the graph is planar. Whenever we refer to with the sphere embeddings of its components induced by , we use the notation and call it the sphere embedding induced by . We also consider as an oriented ribbon graph of genus zero. See Section 3 for further discussion on oriented ribbon graphs. Each edge of can be labeled as “” or “” according to whether it corresponds to an over-strand edge of or an under-strand edge of respectively. Since the edges in each face of rotate between “” and “” edges, it follows that every face has an even number of edges in its boundary. Therefore is bipartite. Also, since every curve encloses a tangle, it follows that every vertex of has even degree. Proposition 3.3 below shows that a graph is an alternating decomposition graph if and only if it is planar, bipartite, and each vertex has even degree. See Section 3 for examples of alternating decompositions of link diagrams and their associated alternating decomposition graphs.
If has alternating decomposition curves , then an alternating region of is a component of that contains crossings of . As the name suggests, if one follows a strand inside of an alternating region of , then the crossings will alternate between over and under. Let be the number of alternating regions in the alternating decomposition of , and let be the number of edges in . Note that is also the number of non-alternating edges in . Thistlethwaite [Thi88] proves that if is a connected diagram of the link , then
Bae and Morton [BM03] use Thistlethwaite’s approach to study the extreme terms and the coefficients of the extreme terms in the Jones polynomial. Using combinatorial data from the planar dual of , a graph they call the non-alternating spine of , they recover Inequality (1.1) and show that it is a stronger bound than Inequality (1.2).
In this paper, we use Thistlethwaite’s alternating decompositions to study the Turaev surface of a link diagram. We show that the genus of the Turaev surface of a link diagram is determined by its alternating decomposition graph. If the Turaev surface is disconnected, then its genus refers to the sum of the genera of its connected components.
If and are link diagrams with isomorphic alternating decomposition graphs, then .
Champanerkar and Kofman [CK09] prove a version of Theorem 1.1 in the case where the two link diagrams are related by a rational tangle replacement. Lowrance [Low11] uses this special case to compute the Turaev genus of the -torus links and of many other closed -braids (see also [AK10]).
The Turaev genus of an alternating decomposition graph , denoted , is defined to be where is a link diagram with alternating decomposition graph . Corollary 3.9 gives a recursive algorithm to compute without any reference to link diagrams. Theorem 1.1 coupled with our algorithm for computing show that the genus of the Turaev surface is determined by how the various alternating regions of are glued together along the non-alternating edges of . The recursive algorithm is at the core of our classification theorems.
A doubled path of length in is a subgraph of consisting of distinct vertices such that for each there are two distinct edges and in connecting vertices and and such that for . If is a graph with a doubled path consisting of vertices , then let be , the contraction of and from for some with . Then is called a doubled path contraction of . The inverse operation of lengthening a doubled path inside of is called a doubled path extension of . Two alternating decomposition graphs and are called doubled path equivalent if there is a sequence of doubled path contractions and extensions transforming into . Doubled path contraction/extension can make a graph non-bipartite (and hence not an alternating decomposition graph), but we do not require every graph in the sequence from to to be bipartite. Proposition 3.11 shows that if and are doubled path equivalent, then .
A graph is -edge connected for some positive integer if the graph remains connected whenever fewer than edges are removed. An alternating decomposition graph is called reduced if is a single vertex or every component of is -edge connected. In Section 3, we study the behavior of alternating decomposition graphs under connected sum. We show that for any link , there exists a diagram of with reduced alternating decomposition graph such that minimizes Turaev genus. The classification theorems characterize all reduced alternating decomposition graphs of a fixed Turaev genus.
Our main theorems give classifications of all reduced alternating decomposition graphs of Turaev genus one and two. A doubled cycle of length is the graph obtained from the cycle of length by doubling every edge.
A reduced alternating decomposition graph is of Turaev genus one if and only if is doubled path equivalent to , that is if and only if is a doubled cycle of even length.
The previous theorem implies that every Turaev genus one link has a diagram obtained by connecting an even number of alternating two-tangles into a cycle, as in Figure 2. Dasbach and Lowrance [DL16] use Theorem 1.2 to compute the signature of all Turaev genus one knots and to show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link has absolute value one.
A link is almost-alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram with a single crossing change (see [ABB92]). Abe and Kishimoto’s work [AK10] implies that all almost-alternating links have Turaev genus one. It is unknown whether there is a link with Turaev genus one that is not almost-alternating (see [Low15]). The following corollary shows another relationship between almost-alternating links and Turaev genus one links.
If is a link of Turaev genus one, then there is an almost-alternating link such that and are mutants of one another.
We present a similar classification theorem for reduced alternating decomposition graphs of Turaev genus two. However, instead of only one doubled path equivalence class, now there are five. Let and be two graphs. A one-sum is the graph obtained by identifying a vertex of with a vertex of . Let be an edge in connecting vertices and , and let be an edge in connecting vertices and . A two-sum is the graph obtained by identifying the triple with , and then deleting the edge corresponding to and . For example the two-sum of two three-cycles is a four cycle . Consider the following five classes of graphs, as depicted in Figure 3.
Let denote the disjoint union of the doubled cycles and .
Let be the graph obtained identifying a vertex of the doubled cycle with a vertex of .
Let be the graph obtained by identifying two paths of length in the cycle of length and the cycle of length . Furthermore, let be the graph with each edge doubled.
Let be the graph obtained by replacing two non-adjacent edges of the complete graph with doubled paths of lengths and respectively.
Let be the graph with one edge replaced by a doubled path of length . Let be the two-sum of and taken along the unique edge in each summand that is not contained in or adjacent to the doubled path.
The graphs in the above families are not necessarily bipartite (depending on their parameters). Informally, the subsequent theorem states that a reduced alternating decomposition graph has Turaev genus two if and only if it is in one of the above five families and it is bipartite. The precise statement uses doubled path equivalence.
A reduced alternating decomposition graph is of Turaev genus two if and only if is doubled path equivalent to one of the following five graphs:
Seungwon Kim [Kim15] has independently proved versions of Theorems 1.2 and 1.4. The following theorem shows that for each non-negative integer , there exists a similar classification of reduced alternating decomposition graphs of Turaev genus .
Let be a non-negative integer. There are a finite number of doubled path equivalence classes of reduced alternating decomposition graphs with Turaev genus .
This paper is organized as follows. In Section 2, we review background material on the Turaev surface and discuss its connections to other areas of knot theory. In Section 3, we give the algorithm to compute and prove Theorem 1.1. In Section 4, we classify alternating decomposition graphs of Turaev genus zero and show that all links have a Turaev genus minimizing diagram whose alternating decomposition graph is reduced. In Section 5, we prove the three main classification theorems (Theorems 1.2, 1.4, and 1.5).
The authors thank Sergei Chmutov, Oliver Dasbach, Nathan Druivenga, Charles Frohman, and Thomas Kindred for their helpful comments.
2. The Turaev surface
In this section, we give the construction of the Turaev surface of a link diagram and discuss its connections to other link invariants. For a more in depth summary, see Champanerkar and Kofman’s recent survey [CK14].
Each link diagram has an associated Turaev surface , constructed as follows. Figure 4 shows the and resolutions of a crossing in .
The collection of simple closed curves obtained by performing either an -resolution or a -resolution for each crossing of is a state of . Performing an -resolution for every crossing results in the all- state of . Similarly, performing a -resolution for every crossings results in the all- state of . Let and denote the number of components in the all- and all- states of respectively.
To construct the Turaev surface, we take a cobordism from the all- state of to the all- state of such that the cobordism consists of bands away from the crossings of and saddles in neighborhoods of the crossing, as depicted in Figure 5. Finally, to obtain , we cap off the boundary components of the cobordism with disks.
The Turaev surface is oriented, and we denote the genus of the Turaev surface of by . If the Turaev surface (or any oriented surface) is disconnected, then when we refer to its genus, we mean the sum of the genera of its connected components. Let be the number of split components of the diagram , i.e. the number of graph components of when is considered as a -valent graph whose vertices are the crossings. Also, let be the number of crossings of . It can be shown that
The Turaev genus of a link is the minimum genus of the Turaev surface of where is any diagram of , i.e.
Turaev [Tur87] constructs his surface in a slightly different, but equivalent way. Turaev’s construction allows us to see that a diagram of the link can be considered as a -valent graph simultaneously embedded on the sphere and the Turaev surface . First consider as embedded on a sphere . Then can be embedded into by replacing crossings of with suitably small balls where one strand passes over the other as in Figure 6.
We construct the Turaev surface of by first replacing each crossing of with the disk that is the intersection of the associated crossing ball and . Each alternating edge of is replaced with an untwisted band that lies completely in the projection sphere . Each non-alternating edge of is replaced with a twisted band. One arc on the boundary of the twisted band will be an arc in a component of the all- state of , and one arc on the boundary of the twisted band will be an arc in a component of the all- state of . The band can be twisted so that the arc corresponding to the all- state lies in the union of and its exterior, while the arc corresponding to the all- state lies in the union of and its interior. After replacing each crossing of with a band, the boundary of the resulting surface is the union of the all- state of and the all- state of . Moreover, the boundary components corresponding to the all- state lie in the union of and its exterior, and the boundary components corresponding to the all- state lie in the union of and its interior. Therefore, the boundary components of this surface can be capped off with disks embedded in , and the resulting surface is the Turaev surface . By projecting the link to in the crossing balls, one can consider the diagram to be embedded on both and the Turaev surface . See Figure 7.
The Turaev surface is a Heegaard surface in , that is is a union of two handelbodies.
The diagram is alternating on .
The Turaev surface is a sphere if and only if is a connected sum of alternating diagrams. Consequently, if and only if is alternating.
The complement is a collection of disks.
The above conditions do not completely characterize Turaev surfaces. Let be the minimal genus of Heegaard surface in on which the link has an alternating projection such that the complement of that projection to is a collection of disks. Adams [Ada94] studies knots and links where , and Balm [Bal13] studies the behavior of under connected sum. Lowrance [Low15] constructs a family of links where , but the Turaev genus is arbitrarily large. Armond, Druivenga, and Kindred [ADK15] show how to determine whether a surface satisfying the above conditions is a Turaev surface using Heegaard diagrams. Indeed, the Heegaard diagrams corresponding to Turaev surfaces of genus one first inspired Theorem 1.2 and the subsequent work in this paper.
Like many link invariants defined as minimums over all diagrams, there is no algorithm to compute the Turaev genus of a link. Instead, our computations rely on various bounds of Turaev genus. The first bound follows immediately from Inequality (1.2). We have
where is the minimum crossing number of . Several other bounds on Turaev genus come from link homologies.
Khovanov [Kho00] constructs a categorification of the Jones polynomial, now known as Khovanov homology. Khovanov homology is a bi-graded -module with homological grading and quantum grading , and one may write as a direct sum over its bi-graded summands . Define
Champanerkar, Kofman, and Stoltzfus [CKS07] show that
A link diagram is adequate if the number of components in the all- (respectively all-) state is strictly greater than the number of components in every state containing exactly one -resolution (respectively exactly one -resolution). A link is adequate if it has an adequate diagram. Khovanov [Kho03] studies the Khovanov homology of adequate links, and Abe [Abe09] proves that Inequality (2.2) is tight when is adequate.
Ozsváth and Szabó [OS04] and independently Rasmussen [Ras03] construct a categorification of the Alexander polynomial of a knot , called knot Floer homology. Knot Floer homology is also a bi-graded -module with homological (or Maslov) grading and Alexander grading , and one may write as a direct sum over its bi-graded summands . Define
Lowrance [Low08] shows that
Essentially all known computations of the Turaev genus of a link rely on some inequality among (2.2) through (2.6). Finding a new method for computing the Turaev genus remains a challenging open question.
3. Alternating decomposition graphs
Throughout this section, we assume that is a link diagram, is the alternating decomposition graph of , and is the graph with the sphere embedding induced by . We begin the section with some examples.
Figure 8 shows a diagram of the knot from Rolfsen’s table, along with its alternating decomposition curves . Since the alternating decomposition of has two curves that both intersect the same four non-alternating edges of , it follows that the alternating decomposition graph of is , the graph with two vertices and four parallel edges between them. In this example, and since is non-alternating, it follows that .
Figure 9 shows a connected link diagram with a disconnected alternating decomposition graph . The alternating decomposition graph is disconnected when has an alternating region with more than one boundary component. In this case, the alternating decomposition graph is , the disjoint union of two doubled two cycles. The disjoint union of two copies of the diagram from Figure 8 also has as its alternating decomposition graph.
The embedding of into the plane induces an embedding of each component of the alternating decomposition graph onto a sphere. Each curve of the alternating decomposition of is incident to two regions, precisely one of which contains crossings of . In the examples of Figures 8 and 9, the alternating regions with crossings are shaded, and the regions without crossings are unshaded. If and are different boundary curves of the same alternating region, then their associated vertices belong to different components of . Let be the curves of the alternating decomposition graph associated to all of the vertices of a particular component of . One may consider the diagram as being embedded on the sphere , and thus the curves are also embedded on . The embedding of this component of onto the sphere is obtained by considering the vertex associated to to be the disk with boundary containing the alternating region incident to . This disk may contain other curves from the alternating decomposition of , but these other curves are associated to a different component of . The edges of this component are the segments of the non-alternating edges of that go between two curves of the alternating decomposition of . Thus each component of has an induced embedding onto a sphere.
Thistlethwaite [Thi88] proved that if is an alternating decomposition graph of some link diagram, then is planar, bipartite, and each vertex of has even degree. Our first result of this section is the converse.
Let be a planar, bipartite graph such that each vertex of has even degree. Then is the alternating decomposition graph of some link diagram . Moreover, may be chosen to be adequate.
Fix a planar embedding for . For each vertex in , choose an alternating tangle with endpoints along the boundary. Each tangle must contain at least one crossing, and each face of the tangle can only meet the boundary circle in at most one arc. Assign to each endpoint the sign “” or “” based on whether the strand emanating from that point is the over-strand or the under-strand, respectively, of the first crossing it meets. The signs “” and “” will alternate around the boundary of . Since is bipartite, the edges of can also be assigned “” or “” in such a way that the signs alternate around each vertex in the planar embedding. Replace with in the planar embedding of so that each endpoint of an arc in and the edge of which it gets connected to have the same sign. This produces a link diagram with the property that the non-alternating arcs exactly correspond to the edges of .
To make the link diagram adequate, appropriate tangles must be chosen for the . Choosing the tangles shown in Figure 10 will produce an adequate link diagram. This is because the circles in the all- and all- resolutions come in two types: Those completely contained in one of the tangles, and those that pass through multiple tangles. Each crossing is either between two distinct circles of the first type, or between a circle of the first type and a circle of the second type. Specifically, each crossing is always between two distinct circles. Thus if one crossing is changed from the -resolution to the -resolution in the all- state (or vice-versa in the all- state), then the number of circles will decrease by one. ∎
Abe [Abe09] proves that if is adequate, then minimizes Turaev genus, that is . Consequently, we have the following corollary.
Let be a planar, bipartite graph such that each vertex has even degree. Then there is a link diagram whose alternating decomposition graph is such that .
An oriented ribbon graph is a graph cellularly embedded in an oriented surface . The genus of an oriented ribbon graph is the genus of . We often visualize the vertices of an oriented ribbon graph as round disks and the edges of an oriented ribbon graph as rectangular bands attached on opposite ends to the round vertices. The sphere embedding of an alternating decomposition graph is a ribbon graph embedded on a disjoint union of spheres. From , we construct another ribbon graph such that the genus of is equal to . The ribbon graph has the same vertices and edges as . To obtain from a half-twist is applied to each edge band of . We say that is the twisted embedding of the alternating decomposition graph . See Figure 11. The operation of twisting some edges in a ribbon graph has been recently studied by Ellis-Monaghan and Moffatt under the name “partial petrials” [EMM13].
Let be the twisted embedding of the alternating decomposition graph of a link diagram . The genus of is .
Each vertex in corresponds to a curve in the alternating decomposition of . Suppose a collection of curves bound an alternating region in the alternating decomposition of , and let be their corresponding vertices in . The region is a surface of genus zero with boundary components. The vertices all lie in different components of . Consider the vertices as disks. Form the connected sum by identifying disks inside of vertices . What was a collection of disks is now a single planar surface with boundary components, just like . Repeat this process for each collection of curves that bound an alternating region to form the surface .
We partially construct the Turaev surface as follows. Consider as embedded on a sphere sitting inside of . Replace crossings of with round disks, and replace all edges of with either flat or twisted bands according to whether the edge is alternating or non-alternating. The boundary components of the resulting surface correspond to the union of the all- and all- states of . If one such boundary component lies completely in (i.e. each arc in the component contained in an edge band is contained in a flat edge band), then cap that boundary component off with a disk as follows. If the boundary component corresponds to a component of the all- state, the interior of the disk should be contained inside , and if the boundary component corresponds to a component of the all- state, the interior of the disk should be contained outside . The resulting surface is , and so . ∎
Proposition 3.5 implies that the genus of the Turaev surface of is determined by the sphere embedding of its alternating decomposition graph . Hence we define to be for any diagram with sphere embedding of its alternating decomposition graph . We give a recursive algorithm to compute without referring to the link diagram . Our recurrence depends on the following lemma.
Let be a sphere embedding of a connected, alternating decomposition graph , and suppose the number of edges in is nonzero.
Either contains a face bounded by exactly two edges or contains at least four vertices of degree two.
Either contains a pair of parallel edges or contains at least four vertices of degree two.
The degree of a face is defined to be the number of edges in its boundary. Suppose that has no face of degree two and three or fewer vertices of degree two. Since every vertex in has even degree, it follows that the other vertices of have degree at least four. Let , , and denote the number of vertices, edges, and faces of respectively. Also, let and be the vertex and face sets of . The handshaking lemma implies
Thus . Since is bipartite, all of its faces have even degree, and since has no face of degree two, the handshaking lemma applied to the planar dual of implies
Thus Now since is connected and planar, its Euler characteristic is two. Therefore, we have
which is a contradiction. Therefore must have at least four vertices of degree two. The second statement follows immediately from the first. ∎
For any graph (or oriented ribbon graph), let denote the number of connected components in . If is an edge in incident to vertices and , then the contraction of , denoted is the graph obtained by identifying the vertices and and deleting the edge . Any graph that can be obtained from via a sequence of edge contractions and edge or vertex deletions is called a minor of . The sphere embedding of a graph induces a sphere embedding on any of its minors. If is bipartite, then is also bipartite. If is bipartite and , then is also bipartite. In the following proposition, whenever a set of edges is deleted or contracted, the induced sphere embedding on the subgraph is assumed. Proposition 3.7 gives a recursive algorithm to compute .
Let be a sphere embedding of an alternating decomposition graph .
If is a collection of isolated vertices, then .
Suppose that contains a face bounded by exactly two edges and . Let , and let . If , then is a sphere embedding of an alternating decomposition graph and . If , then both and are sphere embeddings of alternating decomposition graphs and .
Suppose that contains a vertex of degree two, incident to edges and . Let . Then is a sphere embedding of an alternating decomposition graph, and .
(1) Let be the disjoint union of alternating diagrams. Then and is isolated vertices. Thus .
(2) Deleting or contracting two edges from a graph embedded on a disjoint union of spheres results in a graph embedded on a disjoint union of spheres. Moreover, since and bound a face, they are incident to the same two vertices. Hence all vertices of and have even degree. Since is obtained from by deleting two edges, it follows that is bipartite. Also, since and are parallel, it follows that if the deletion of and increases the number of components in , then is bipartite. Thus is a sphere embedding of an alternating decomposition graph, and if , then is a sphere embedding of an alternating decomposition graph.
Let , , and be the twisted embeddings of , , and respectively. Define to be the number of components of where is the surface on which is embedded. Note that is also the number of boundary components of . Similarly define and .
We have , , and . If is an oriented ribbon graph, then its genus is
Both and have the same underlying graph , and so they have the same number of components. A similar statement holds for and . If , then
and if , then
Also, if , then can be obtained from by taking a connected sum along the two vertices incident with and in . Hence .
(3) As in the previous case, contracting two edges from a graph embedded on a disjoint union of spheres leads to a graph embedded on a disjoint union of spheres. Let and be the two vertices adjacent to , and let be the vertex in corresponding to vertices and in . If , then the degree of is , which is even. If , then , which is also even. All other vertices in have the same degree as their corresponding vertices in . Also, the bipartition of the vertices of induces a bipartition of the vertices of . Thus is a sphere embedding of an alternating decomposition graph.
Let and be the twisted embeddings associated to and , respectively. Then and . If , then and, and if , then and . Hence ∎
As the following theorem shows, the Turaev genus of the sphere embedding of the alternating decomposition graph does not depend on its embedding at all.
Let and be sphere embeddings of the same alternating decomposition graph . Then .
We proceed by induction on the number of edges in . If has no edges, then both and are embeddings of a disjoint union of vertices. Hence .
Suppose that has edges and that any two embeddings of an alternating decomposition graph with fewer than edges have the same Turaev genus. Suppose that has a vertex of degree two incident to edges and . Since has the same underlying graph as , the same statement holds for , that is the vertex in has degree two and is incident to edges and . Set , , and . By Proposition 3.7, we have that and . Since and are sphere embeddings of the same graph , the inductive hypothesis implies that . Therefore .
Now suppose that does not have a vertex of degree two. By Lemma 3.6, has a face bounded by exactly two edges, say and . Let . Then Proposition 3.7 implies that if , then , and if , then . Since and have the same underlying graph , the edges and are parallel in , but do not necessarily bound a face of degree two. Let .
The twisted embedding is obtained from by adding the two twisted edges corresponding to and . The twisted edges and contain four boundary arcs that are pieces of boundary components of . Fix one of the boundary arcs and fix an endpoint of that boundary arc. As one travels along the boundary of starting from the fixed endpoint, one of the other seven endpoints of boundary arcs of and must be encountered first. The planarity of lets us rule out four of those endpoints. Furthermore, each edge in corresponds to a non-alternating edge in some link diagram . The two boundary arcs of that edge correspond to a segment in a component of the all- state of and a segment in a component of the all- state of . In particular, two boundary arcs of the same edge must belong to different components of the boundary of the twisted embedding of the associated alternating decomposition graph. This rules out one more of the endpoints as being the next endpoint encountered. There are two remaining cases, each depicted in Figure 12.
The four boundary arcs of and lie in exactly two components of the boundary of . Moreover, if the twisted edges and are removed, then the two boundary components containing boundary arcs of and are transformed into two boundary components of the twisted embedding . Since no other boundary components of are changed by deleting and , it follows that . Since and , it follows that if , then , and if , then . The embedded graphs and have the same underlying graph, and hence the inductive hypothesis implies that . Deleting and from increases the number of components if and only if deleting and from increases the number of components. Therefore , and the desired result is proven. ∎