Symmetries of embedded complete bipartite graphs

Transcription

1 Digital Loyola Maymount Univesity and Loyola Law School Mathematics Faculty Woks Mathematics Symmeties of embedded complete bipatite gaphs Eica Flapan Nicole Lehle Blake Mello Loyola Maymount Univesity, Matt Pittluck Xan Vongsathon Repositoy Citation Flapan, Eica; Lehle, Nicole; Mello, Blake; Pittluck, Matt; and Vongsathon, Xan, "Symmeties of embedded complete bipatite gaphs" (014). Mathematics Faculty Woks Recommended Citation E. Flapan, N. Lehle, M. Pittluck and X. Vongsathon, 014: Symmeties of embedded complete bipatite gaphs. Fundamenta Mathematicae, 6, 1-16, axiv: This Aticle - post-pint is bought to you fo fee and open access by the Mathematics at Digital Loyola Maymount Univesity and Loyola Law School. It has been accepted fo inclusion in Mathematics Faculty Woks by an authoized administato of Digital Commons@Loyola Maymount Univesity and Loyola Law School. Fo moe infomation, please contact digitalcommons@lmu.edu.

2 SYMMETRIES OF EMBEDDED COMPLETE BIPARTITE GRAPHS ERICA FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN axiv: v3 [math.gt] 3 Dec 014 Abstact. We chaacteize which automophisms of an abitay complete bipatite gaphk n,m can be induced by a homeomophismof some embedding of the gaph in S Intoduction Knowing the symmeties of a molecule helps to pedict its chemical behavio. Chemists use the point goup as a way to epesent the igid symmeties of a molecule. Howeve, molecules which ae lage enough to be flexible (such as long polymes) o have pieces which otate sepaately may have symmeties which ae not induced by igid motions. By modeling a molecule as a gaph Γ embedded in S 3, the igid and non-igid symmeties of the molecule can be epesented by automophisms of Γ which ae induced by homeomophisms of the pai (S 3,Γ). Diffeent embeddings of the same abstact gaph may have diffeent automophisms which ae induced by homeomophisms of the gaph in S 3. In fact, a given automophism of an abstact gaph may o may not be induced by a homeomophism of some embedding of the gaph. In paticula, it was shown in [F1] that a cyclic pemutation of fou vetices of the complete gaph on six vetices K 6 cannot be induced by a homeomophism of S 3, no matte how the gaph is embedded in S 3. In geneal, we ae inteested in which automophisms of a gaph can be inducedbyahomeomophism ofsomeembedding ofthegaphins 3.Flapan [F] answeed this question fo the family of complete gaphs K n. Now we do the same fo the family of complete bipatite gaphs K n,m. This is an inteesting family of gaphs to conside because it was shown in [FNPT] that fo evey finite subgoup G of Diff + (S 3 ), thee is an embedding Γ of some complete bipatite gaph such that the goup of all automophisms of Γ which ae induced by oientation peseving homeomophisms of S 3 is 1991 Mathematics Subject Classification. 57M15, 57M5, 05C10. Key wods and phases. topological symmety goups, spatial gaphs, bipatite gaphs. This eseach was suppoted in pat by NSF gant DMS

3 ERICA FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN isomophic to G. By contast, it was shown in [FNT] that this is not the case fo the complete gaphs. We pove the following Classification Theoem which detemines pecisely which automophisms of a complete bipatite gaph can be induced by a homeomophism of (S 3,Γ) fo some embedding Γ of the gaph in S 3. The theoem is divided into two pats, accoding to whethe the homeomophism inducing a paticula automophism is oientation peseving o evesing. The automophisms ae descibed by thei fixed vetices and thei cycle stuctue. Note that in this pape we use the tem cycle to efe to a cycle in the pemutation on the vetices of a gaph induced by an automophism of the gaph, not to a cycle in the gaph in the usual gaph-theoetic sense. Classification Theoem. Let m,n > and let ϕ be an ode automophism of a complete bipatite gaph K n,m with vetex sets V and W. Thee is an embedding Γ of K n,m in S 3 with an oientation peseving homeomophism h of (S 3,Γ) inducing ϕ if and only if all vetices ae in -cycles except fo the fixed vetices and exceptional cycles explicitly mentioned below (up to intechanging V and W): (1) Thee ae no fixed vetices o exceptional cycles. () V contains one o moe fixed vetices. (3) V and W each contain at most fixed vetices. (4) j and V contains some j-cycles. (5) = lcm(j,k), and V contains some j-cycles and k-cycles. (6) = lcm(j,k), and V contains some j-cycles and W contains some k-cycles. (7) V and W each contain one -cycle. (8) is odd, V and W each contain one -cycle, and V contains some -cycles. (9) ϕ(v) = W and V W contains one 4-cycle. Thee is an embedding Γ of K n,m in S 3 with an oientation evesing homeomophism h of (S 3,Γ) inducing ϕ if and only if is even and all vetices ae in -cycles except fo the fixed vetices and exceptional cycles explicitly mentioned below (up to intechanging V and W): (10) ϕ(v) = V and thee ae no fixed vetices o exceptional cycles. (11) = and all vetices of V and at most vetices of W ae fixed. (1) V contains at most fixed vetices, and one of the following is tue. (a) W contains one -cycle.

4 BIPARTITE SYMMETRIES 3 (b) V contains some -cycles. (c) V may contain some -cycles, is odd, and all vetices of W ae in cycles. (d) W contains at most one -cycle, is odd, and all non-fixed vetices of V ae in cycles. (13) 4, ϕ(v) = W and V W contains at most two -cycles. We will begin by poving the necessity of the conditions in the Classification Theoem, and then povide constuctions to show that each of the cases listed can actually occu.. Necessity of the Conditions The following esult allows us to focus ou attention on finite ode homeomophisms of embeddings of ou gaphs in S 3. Automophism Theoem. [F] Let ϕ be an automophism of a 3-connected gaph which is induced by a homeomophism f of (S 3,Γ 1 ) fo some embedding Γ 1 of the gaph in S 3. Then ϕ is induced by a finite ode homeomophism h of (S 3,Γ ) fo some possibly diffeent embedding Γ of Γ 1 in S 3. Futhemoe, h is oientation evesing if and only if f is oientation evesing. If n,m, then it is easy to see that all of the automophisms of K n,m canbeinducedbyahomeomophismofsomeembeddingofk n,m ins 3.Thus fothe emainde of thepape, we focusonk n,m with n,m >. Inthis case, K n,m is 3-connected, and hence we can apply the Automophism Theoem. It follows that any automophism which is induced by a homeomophism fo some embedding of K n,m is induced by a finite ode homeomophism fo some (possibly diffeent) embedding of the gaph. Thus we begin with some esults about finite ode homeomophisms of S 3. The following is a special case of a well-known esult of P. A. Smith. Smith Theoy. [Sm] Let h be a non-tivial finite ode homeomophism of S 3. If h is oientation peseving, then fix(h) is eithe the empty set o is homeomophic to S 1. If h is oientation evesing, then fix(h) is homeomophic to eithe S 0 o S. Lemma 1. Let h be a finite ode homeomophism of S 3 which is fixed point fee. Then thee ae at most two cicles which ae the fixed point set of some powe of h less than.

5 4ERICA FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN Poof. Suppose that some powe of h has non-empty fixed point set. Then by Thuston s Obifold Theoem [BLP], h is conjugate to an oientation peseving isomety g of S 3. Now g can be extended to an oientationpeseving isomety ĝ of R 4 fixing the oigin; i.e. an element of SO(4). Evey element of SO(4) is the composition of two otations about pependicula planes in R 4 (see [D], fo example), and it is easy to show that these ae the only planes fixed by any powe of ĝ. It follows that thee ae at most two cicles in S 3 which ae the fixed point set of some powe of g. Finally, since h is conjugate to g, thee ae at most two cicles which ae the fixed point set of some powe of h. Lemma. Let h be an ode homeomophism of S 3 which is fixed point fee such that fo some minimal k < j <, A = fix(h k ) and B = fix(h j ) ae distinct cicles. Then the following ae tue: (1) A B =. () h(a) = A and h(b) = B. (3) The points in A, B, and S 3 (A B) ae in k, j, and -cycles espectively. (4) = lcm(k,j). Poof. Suppose, fo the sake of contadiction, that thee exists some x A B. Then x is fixed by both h k and h j. Since k is the smallest powe of h with a nonempty fixed point set, we have that k j, and so A B. But this is impossible since A and B ae distinct cicles. Thus condition (1) holds. Let x A, then h k (x) = x. Hence h k (h(x)) = h(h k (x)) = h(x). So h(x) is fixed by h k. It follows that h(x) A. Thus condition () holds. Obseve that k is the smallest powe of h that fixes any point of S 3. Thus all of the points in A have ode k unde h. Similaly, j is the smallest powe of h that fixes any point of S 3 A, and hence of B. Thus all of the points in B have ode j unde h. Finally, it follows fom Lemma 1 that evey point of S 3 (A B) has ode unde h, and hence condition (3) holds. Obseve that h lcm(k,j) fixes A B and hence must be the identity. Thus divides lcm(k,j). But we also know that k and j both divide since fix(h k ) and fix(h j ) ae non-empty. So lcm(k,j) divides. It follows that condition (4) holds. Now we conside complete bipatite gaphs. Thoughout the pape, we will use V and W to denote the vetex sets of a complete bipatite gaph

6 BIPARTITE SYMMETRIES 5 K n,m. We begin with the following well known esult about automophisms of complete bipatite gaphs. Fact. Let ϕ be a pemutation of the vetices of K n,m. Then ϕ is an automophism of K n,m if and only if ϕ eithe intechanges V and W o setwise fixes each of V and W. We will use the above Lemmas to obtain necessay conditions on finite ode homeomophisms of complete bipatite gaphs embedded in S 3. Lemma 3. Let n,m >. Suppose that Γ is an embedding of K n,m in S 3 with a finite-ode homeomophism h of (S 3,Γ). If fix(h) = S, then V o W is entiely contained in fix(h). Poof. Since fix(h) = S, it follows fom Smith Theoy that h is oientation evesing, and hence intechanges the components A and B of S 3 fix(h). Suppose fo the sake of contadiction that neithe V no W is entiely contained in fix(h). Since neithe V no W is contained in fix(h) and A and B ae intechanged by h, thee ae vetices v of V and w of W such that v A and w B. Now the edge vw intesects fix(h), and hence h fixes a point of vw. Thus h intechanges w and v. Fom the above Fact we know that h(v) = V o h(v) = W. Hence we must have h(v) = W. It follows that no vetices of the gaph ae fixed. Since n >, without loss of geneality, thee is a pai of distinct vetices v 1 and v of V which ae contained in A. Then some h(v ) B and h(v ) is in W. Howeve, the edge v 1 h(v ) intesects fix(h) and hence, h(v 1 ) = h(v ). But this is impossible since h is a bijection. Thus fix(h) contains one of the vetex sets. Fixed Vetices Lemma. Let m,n >. Suppose that Γ is an embedding of K n,m in S 3 with an ode homeomophism h of (S 3,Γ) which fixes at least one vetex if h is oientation peseving and at least 3 vetices if h is oientation evesing. Then all non-fixed vetices ae in -cycles and one of the following holds (up to intechanging V and W): (1) If h is oientation peseving, then eithe h fixes no vetices of W, o h fixes at most vetices of each of V and W. () If h is oientation evesing, then h has ode, and h fixes all vetices of V and at most vetices of W. Poof. Fist suppose that h is oientation peseving. If h fixes at least 3 vetices of V, then h cannot fix any vetices of W, since othewise fix(h) = S 1 would contain a K 3,1 gaph. Thus condition (1) is satisfied.

7 6ERICA FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN Next suppose that h is oientation evesing. Since at least 3 vetices ae fixed by h, by Smith Theoy fix(h) = S. Hence h has ode. Also, by Lemma 3, without loss of geneality fix(h) contains all of V, which includes at least 3 vetices. If fix(h) also contained moe than vetices of W, then fix(h) = S would contain the nonplana gaph K 3,3. Thus condition () is satisfied. In eithe case, by Smith Theoy all non-fixed vetices ae in -cycles. Oientation Revesing Lemma. Let m,n >. Suppose that Γ is an embedding of K n,m in S 3 with an ode oientation evesing homeomophism h of (S 3,Γ) which fixes at most vetices. Then the fixed vetices ae in a single vetex set V, and one of the following holds, with all emaining non-fixed vetices in -cycles: (1) Thee ae no fixed vetices o exceptional cycles, and h(v) = V. () W contains one -cycle. (3) V contains some -cycles. (4) V W contains at most two -cycles, is even, and h(v) = W. (5) V may contain some -cycles, is odd, and all vetices of W ae in -cycles. (6) W contains at most one -cycle, is odd, and all non-fixed vetices of V ae in -cycles. Poof. Obseve that since h is oientation evesing, must be even. We will obseve that if is odd, then h(v) = V. Towads contadiction, suppose is odd, and h(v) = W. Then h is also oientation-evesing, intechanges V and W and divides V W into -cycles (v i w i ). But then h must fix a point on each edge v i w i ; since m,n >, this means fix(h ) = S. But, by Lemma 3, this contadicts the fact that h does not fix any vetices. Hence h(v) = V. In paticula, if = then h(v) = V, and conditions (1) and/o (3) follows tivially. So we assume that >. Since n,m 3 and no moe than vetices ae fixed by h, both V and W must contain non-fixed vetices. Thus by Lemma 3, fix(h) cannot be homeomophic to S. Now it follows fom Smith Theoy that h has pecisely fixed points. Thus h cannot fix one vetex fom each vetex set, since that would foce h to fix evey point on the edge between the fixed vetices. So without loss of geneality, all fixed vetices ae in V. Since h is oientation peseving and has non-empty fixed point set, fix(h ) = S 1. By applying Smith Theoy to h we see that all points not fixed by h ae in -cycles unde h, and hence ae eithe in -cycles o

8 BIPARTITE SYMMETRIES 7 -cycles unde h. So all non-fixed vetices of Γ ae in -cycles, -cycles, o -cycles unde h. Suppose at least one vetex of V is fixed and some vetex w W is in a -cycle. Then h fixes these vetices of W togethe with any vetices of V which ae fixed by h. If W contained a second -cycle then fix(h ) would contain a K 1,4 gaph, and if V contained a -cycle then fix(h ) would contain a K 3, gaph. Both cases ae impossible since fix(h ) = S 1. Hence the -cycle in W is the only -cycle in the gaph. So, if thee is a fixed vetex and no vetices of Γ ae in -cycles (o = 4), then eithe condition () o (3) is satisfied depending on whethe o not W contains a -cycle. Suppose thee ae no fixed vetices in Γ. Then the vetex sets V and W ae intechangeable. As above, if fix(h ) contains moe than vetices of V, then it cannot contain any vetices of W since fix(h ) = S 1. Thus eithe thee ae at most two -cycles in V W and these ae the only -cycles in Γ, o all of the -cycles of Γ ae in V. If thee ae -cycles (v 1 v ) and (w 1 w ) thenfix(h )must beexactlythese fouvetices andtheedges between them. But this set contains the two points of fix(h); since h does not fix a vetex it must fix a point on an edge, and hence must intechange the endpoints of this edge. But then h(v) = W, which contadicts the assumption that h(v 1 ) = v. So, if thee ae -cycles involving both V and W, we must have that h(v) = W, and hence must be even. Thus eithe thee ae at most two -cycles in V W, is even and h(v) = W, o all of the -cycles of Γ ae in V and h(v) = V. In paticula, if = 4 o no vetices of Γ ae in -cycles, then one of conditions (1), (3) o (4) is satisfied. So we can assume that some vetex of Γ is in an -cycle and 4. Then the vetices in the -cycle ae not fixed by h. But if is even, then fix(h ) fix(h ) = S 1, so the fixed point sets ae the same. Hence must be odd and 6. Thus h fixes at least 3 vetices fom a -cycle. Since h is oientation evesing, this implies that fix(h ) = S. Now by Lemma 3, fix(h ) contains all of the vetices in one of the vetex sets. Let X denote the vetex set contained in fix(h ) and let Y denote the othe vetex set. Hence all of the non-fixed vetices of X ae in -cycles. Suppose that some vetex y Y is also in a -cycle. Then y, h(y), and h (y) ae distinct vetices in fix(h ). Since X contains at least 3 vetices, this would imply that fix(h ) contains a K 3,3 gaph. As this is impossible, no vetex in Y is in an -cycle. In paticula, V and W cannot both contain -cycles. Now if W is contained in fix(h ) then condition (5) is satisfied. If thee ae no

9 8ERICA FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN fixed vetices, then V and W ae intechangeable, and hence again condition (5) is satisfied. If thee is at least one fixed vetex and V is contained in fix(h), then condition (6) is satisfied since in this case W contains at most one -cycle. The situation is somewhat moe complicated if h is oientation peseving and fixes no vetices of Γ. In this case, by Lemma 1 one o two distinct cicles (but no moe) could be fixed by diffeent powes of h. This gives many possibilities depending on whethe these cicles contain vetices fom the sets V and/o W. Oientation Peseving Lemma. Let m, n >. Suppose that Γ is an embedding of K n,m in S 3 with an ode oientation peseving homeomophism h of (S 3,Γ) which fixes no vetices. If some powe of h less than then fixes a vetex, then one of the following holds (up to intechanging V and W), with all emaining vetices in -cycles: (1) If thee is pecisely one cicle fix(h j ) containing a vetex and j < is minimal, then one of the following holds. (a) V may contain some j-cycles. (b) fix(h j ) contains vetices v 1,v V and w 1,w W such that eithe j = and h induces (v 1 v )(w 1 w ) o j = 4 and h induces (v 1 w 1 v w ). () If thee ae two cicles fix(h j ) and fix(h k ) containing vetices and j,k < ae minimal, then one of the following holds. (a) V contains some j-cycles and k-cycles. (b) V contains some j-cycles and W contains some k-cycles. (c) j =, fix(h ) contains v 1,v V and w 1,w W such that h induces (v 1 v )(w 1 w ), and V contains some -cycles with odd. Poof. By ou hypotheses, some powe of h less thanfixes a vetex, though h itself fixes no vetices. Since h is oientation peseving it follows fom Smith Theoy that h is fixed point fee. FistsupposethattheeisacicleA = fix(h j )whichcontains v 1 V and w 1 W and j is minimal. Since h is fixed point fee and h(a) = A, we know that h otates the cicle A. Thus A contains the same numbe of vetices fom each of the vetex sets V and W, and since A must also contain the edges between vetices this numbe must be at least (o h would not send edges to edges). Thus A must consist of vetices v 1, w 1, v, w togethe with the edges between them, and j = o 4. If no vetex is fixed by any othe

10 BIPARTITE SYMMETRIES 9 powe of h less than, then condition (1b) is satisfied. Suppose that some vetex not on A is fixed by h k with k minimal such that j < k <. Since h k does not fix any point of A, j cannot divide k. In paticula, B = fix(h k ) cannot contain vetices fom both V and W (since then k = o 4 by the same agument, and j would divide k). Thus without loss of geneality, B only contains vetices of V. Since h(b) = B, we must have h(v) = V. Thus h must induce (v 1 v )(w 1 w ). Now it follows fom Smith Theoy that k is odd and fom Lemma that k =. Hence condition (c) is satisfied. Thus we assume that no powe of h less than simultaneously fixes vetices fom each of V and W. Now by Lemmas 1 and, it is easy to check that one of conditions (1a), (a), o (b) is satisfied. 3. Realizing Automophisms of Complete Bipatite Gaphs We will povide constuctions to show each case of the Classification Theoem is possible. Ou constuctions poceed by fist embedding the vetices of K n,m so that thee is an isomety of S 3 that acts on them as desied, and then embedding the edges using the following Edge Embedding Lemma, so that the esulting embedding of K n,m is setwise invaiant unde g, and g induces ϕ on K n,m. Recall that a gaph H is a subdivision of a gaph G if H is the esult of adding distinct vetices to the inteios of the edges of G. Edge Embedding Lemma. [FMNY] Let V and W denote the vetex sets of K n,m and let γ be a subdivision of K n,m constucted by adding vetices Z. Assume the vetices of γ ae embedded in S 3 so that an isomety g of S 3 of ode induces a faithful action on γ. Let Y denote the union of the fixed point sets of all the g i, fo i <. Suppose the following hypotheses hold fo adjacent pais of vetices in V W Z: (1) If a pai is pointwise fixed by g i and g j fo some i,j <, then fix(g i ) = fix(g j ). () No pai is intechanged by any g i. (3) Any pai that is pointwise fixed by some g i with i < bounds an ac in fix(g i ) whose inteio is disjoint fom V W Z (Y fix(g i )). (4) Evey pai is contained in the closue of a single componentof S 3 Y Then thee is an embedding of the edges of γ in S 3 such that the esulting embedding of γ is setwise invaiant unde g. Note that if g is an isomety, then Y only sepaates S 3 if fix(g i ) = S fo some i. So, by Lemma 3, condition (4) of the Edge Embedding Lemma

11 ERICA 10 FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN is equivalent to saying that if fix(g i ) = S then eithe V fix(g i ) o W fix(g i ). We will fist conside automophisms ϕ of K n,m which can be ealized by oientation-peseving isometies of S 3. The oles of V and W can be evesed in all the following lemmas. Notice that the conditions of each lemma ae numbeed to match the coesponding conditions in the Classification Theoem. We fist show that the fist thee cases of the Classification Theoem ae ealizable by otations in S 3, as long as the automophism fixes V and W setwise. Rotations Lemma. Let m,n > and let ϕ be an ode automophism of K n,m with vetex sets V and W. Suppose that ϕ(v) = V and all vetices ae in -cycles except fo the fixed vetices and exceptional cycles explicitly mentioned below (up to intechanging V and W): (1) Thee ae no fixed vetices o exceptional cycles. () V contains one o moe fixed vetices. (3) V and W each contain at most fixed vetices. Then ϕ is ealizable by a otation of S 3 of ode. Poof. Let g be a otation of S 3 by π aound a cicle X. Then the ode of g is. Embed all the fixed points of ϕ on X. In the case whee V and W each contain fixed vetices, altenate vetices fom V and W aound X. Embed the emaining vetices in paiwise disjoint -cycles of g in S 3 X. Thus g induces ϕ on V W. We now check that the conditions of the Edge Embedding Lemma ae satisfied. Since g(v) = V and g(w) = W, condition () is satisfied. Notice that fo evey i < we have that fix(g i ) = X, and X does not sepaate S 3, so conditions (1) and (4) ae satisfied. Futhemoe, if ϕ fixes eithe 1 o vetices in each of V and W, thee is a collection of acs in X fom each of the fixed points of V to each of the fixed points of W such that the inteio of each ac is disjoint fom V W. Hence condition (3) is met. Thus by the Edge Embedding Lemma, thee is an embedding of K n,m such that ϕ is ealized by g. The next 6 cases of the Classification Theoem ae ealizable by glide otations in S 3, along with the automophisms in case (1) that intechange the vetex sets V and W. Recall that a glide otation is the composition of two otations about linked cicles such that each otation fixes the axis of the othe otation setwise.

12 BIPARTITE SYMMETRIES 11 Glide Rotations Lemma. Let m,n > and let ϕ be an ode automophism of K n,m with vetex sets V and W. Suppose that thee ae no fixed vetices, and all vetices ae in -cycles except fo the exceptional cycles explicitly mentioned below (up to intechanging V and W): (1) ϕ(v) = W and thee ae no exceptional cycles. (4) j and V contains some j-cycles. (5) = lcm(j,k), and V contains some j-cycles and k-cycles. (6) = lcm(j,k), and V contains some j-cycles and W contains some k-cycles. (7) V and W each contain one -cycle. (8) is odd, V and W each contain one -cycle, and V contains some -cycles. (9) ϕ(v) = W and V W contains one 4-cycle. Then ϕ is ealizable by a glide otation of S 3 of ode. Poof. Let X and Y be geodesic cicles in S 3 which ae the intesections of S 3 with pependicula planes though the oigin in R 4. We define a glide otationg ineach case asfollows (we split case (1)into two pats, depending on whethe is odd o even; since ϕ(v) = W, we know must be even): (1a) ( is odd) g is the composition of a otation by 4π aound X and a otation by π aound Y. (1b) ( is even) g is the composition of a otation by π aound X and a otation by π aound Y. (4) g is the composition of a otation by π aound X and a otation by j π aound Y. (5) g is the composition of a otation by π aound X and a otation by j π k aound Y. (6) same as case (5). (7) g is the composition of a otation by π aound X and a otation by π aound Y. (8) g is the composition of a otation by π aound X and a otation by 4π aound Y. (9) same as case (1b). Obseve that in case (7) is even since thee is a -cycle. In case (8) is odd by assumption, and in case (9) is divisible by 4 since thee is a 4-cycle (so is even). So in each case, the ode of g is. We begin by embedding the vetices of K n,m in each case. In cases (4), (5) and (6), g Y has ode j, and we embed all of the j-cycles of V unde ϕ in Y as paiwise disjoint j-cycles of g. In cases (7) and (8), g Y has ode,

13 ERICA 1 FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN andwe embed the -cycles of V and W in Y as altenating paiwise disjoint -cycles of g. In case (9), g Y has ode 4, and we embed the 4-cycle in Y with v s and w s altenating. In cases (5) and (6), g X has ode k, and we similaly embed all of the k-cycles of V o W unde ϕ in X. In case (8), g X has ode, and we embed all of the -cycles of V unde ϕ in X. Finally, in all the cases we embed the -cycles of V and W in S 3 (X Y). Thus g induces ϕ on V W. We will now use the Edge Embedding Lemma to embed the edges of K n,m. We will fist conside case (1a), which is a bit moe complex than the othes. In this case we need to fist embed the midpoints of all the edges that ae inveted by any g i fo i <. Since all the cycles ae length, this is only possible if i =. Futhemoe, since is odd, g invets n edges of K n,n (since ϕ(v) = W, we have n = m in this case). Fo each edge vw which is inveted by ϕ we add a vetex z vw at the midpoint of the edge of the abstact gaph K n,n. Denote this set of n vetices by Z. Obseve that the vetices of Z ae in -cycles unde ϕ. Then we define H = K n,n Z, so H is a subdivision of K n,n. Now g Y has ode. We embed the -cycles of vetices z vw unde ϕ as a -cycle of g in Y. Thus g induces ϕ on V W Z. We now check that the conditions of the Edge Embedding Lemma ae satisfied fo H. If any pai of adjacent vetices of H is fixed by g i, then g i must fixavetexofv W,butnoneofthesevetices aeembeddedinx Y. So conditions (1) and (3) ae tivially satisfied. Since fix(g i ) S fo all i, condition (4) is satisfied. Fo evey pai of vetices v V and w W such that g (v) = w, thee exists a vetex z vw Z such that z vw fix(g ) = X, so v and w ae not adjacent in H. Hence condition () is satisfied. Thus by the Edge Embedding Lemma, thee is an embedding of H that is setwise invaiant unde g. Finally, we delete the embedded midpoint vetices of ou embedding of H to obtain an embedding of K n,n such that ϕ is ealized by g. We now check that the conditions of the Edge Embedding Lemma ae satisfied fo the othe cases. In cases (4) (8) we have that g(v) = V and g(w) = W, hence condition () is satisfied. Fo cases (1b) and (9) we obseve that ϕ i invets an edge of K n,n if and only if i is odd and ϕ has a vetex cycle of length i. But the only vetex cycles in these case ae length 4 o, and is even. So none of the edges ae inveted, and condition () is againsatisfied. Since no g i with i < pointwise fixes X Y, if two points ae fixed by g i then they aeeithe bothin X o bothin Y. The only cases when vetices of both V and W ae embedded in the same cicle is when both V

14 BIPARTITE SYMMETRIES 13 and W have a single -cycle embedded in Y, o when a 4-cycle is embedded iny. If g i andg j bothfixsuch a pai, foi,j <, then fix(g i ) = fix(g j ) = Y, so condition (1) is satisfied. Also, since the 4 vetices in each case altenate between v s and w s, condition (3) is satisfied. Othewise, vetices of W and V ae not both embedded in the same cicle. Hence g i fo i < does not simultaneously fix both a v V and a w W, and conditions (1) and (3) follow tivially. Since fix(g i ) S fo evey i <, condition (4)is satisfied. Thus by the Edge Embedding Lemma, thee is an embedding of K n,m such that ϕ is ealized by g. We now conside the automophisms that can be ealized by oientationevesing isometies of S 3. Case (11) of the Classification Theoem can be ealized by a eflection. Reflections Lemma. Let m,n > and let ϕ be an automophism of K n,m of ode that fixes all of the vetices of V and at most vetices of W, with the emaining vetices of W patitioned into -cycles. Then ϕ is ealizable by a eflection of S 3. Poof. Let g be a eflection of S 3 though a sphee S. Embed all the fixed vetices of ϕ on S, and embed the -cycles of ϕ in S 3 S as paiwise disjoint -cycles of g. Thus g induces ϕ on V W. We now check that the conditions of the Edge Embedding Lemma ae satisfied by the embedded vetices V W of K n,m. By constuction we have that g(v) = V and g(w) = W, hence condition () is satisfied. Since the ode of g is, condition (1) of the Edge Embedding Lemma is met. Since K,n is a plana gaph, thee is a collection of disjoint acs in S connecting the vetices of V to the (at most ) fixed vetices of W in S, hence condition (3) is satisfied. Futhemoe, since V fix(g), condition(4) is satisfied. Thus by the Edge Embedding Lemma, thee is an embedding of K n,m such that ϕ is ealized by g. Cases (10), (1) and (13) of the Classification Theoem can be ealized by impope otations. An impope otation of S 3 is the composition of a eflection though a geodesic -sphee S and a otation about a geodesic cicle X that intesects S pependiculaly in two points. Impope Rotations Lemma. Let m, n > and let ϕ be an ode (whee is even) automophism of K n,m with vetex sets V and W. Suppose that all vetices ae in -cycles except fo the fixed vetices and exceptional cycles explicitly mentioned below (up to intechanging V and W):

15 ERICA 14 FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN (10) ϕ(v) = V and thee ae no fixed vetices o exceptional cycles. (1) ϕ(v) = V, V contains at most fixed vetices, and one of the following is tue. (a) W contains one -cycle. (b) V contains some -cycles. (c) V may contain some -cycles, is odd, and all vetices of W ae in cycles. (d) W contains at most one -cycle, is odd, and all non-fixed vetices of V ae in cycles. (13) 4, ϕ(v) = W and V W contains at most two -cycles. Then ϕ is ealizable by an impope otation of S 3 of ode. Poof. Let S be a geodesic sphee in S 3 and X be a geodesic cicle which intesects S pependiculaly at exactly two points. In cases(10),(1a),(1b) and (13), we define an impope otation g as the composition of a eflection in S with a otation by π aound X. In cases (1c) and (1d) (whee is odd), we define g as the composition of a eflection in S with a otation by 4π aound X. In each case, g has ode. We will now embed the vetices of K n,m. LetF = fix(g) = X S, so F =.Ineach case, embed thefixed vetices of ϕ (if any) as points of F. If =, then ϕ(v) = V and we embed the emaining vetices as -cycles in S 3 (S X); in this case, the conditions of the Edge Embedding Lemma ae easily veified. So we will assume >. Obseve g X has ode. In each case, we embed any -cycles unde ϕ in X F as paiwise disjoint -cycles of g (in cases (1a) and (1d), the embedded vetices of W will altenate with any vetices of V embedded in F); in case (13), we also altenate vetices fom V and W aound X. Next obseve that in cases (1c) and (1d) g S has ode. In these cases, we embed the -cycles unde ϕ in S F as paiwise disjoint -cycles of g. Finally, we embed the -cycles unde ϕ as paiwise disjoint -cycles of g in S 3 (S X). Thus g induces ϕ on V W. We now check that the conditions of the Edge Embedding Lemma ae satisfied. We fist conside cases (10) and (1); case (13) is slightly moe complex. In cases (10) and (1) we have that g(v) = V and g(w) = W, hence condition () is satisfied. Since no g i with i < pointwise fixes S X, if vetices ae fixed by g i then they ae both in S o both in X. In each of cases (10) and (1a) (1d), all the vetices embedded in X F ae fom the same vetex set and all the vetices embedded in S F ae fom the same vetex set. So we only need to conside the cases when thee ae vetices of

16 BIPARTITE SYMMETRIES 15 V embedded in F and vetices of W in eithe X F (cases (1a) and (1d)) o S F (case (1c)). In cases (1a) and (1d), any g i (with i < ) that fixes an adjacent pai has fixed point set X, so condition (1) is satisfied; since thee ae at most two vetices of V and W embedded in X, altenating between V and W, condition (3) is also satisfied. In case (1c), any g i (with i < ) that fixes an adjacent pai has fixed point set S, so condition (1) is satisfied; since K,n is plana thee thee is a collection of disjoint acs connecting the vetices of W in S F to the vetices of V in F, so condition (3) is also satisfied. The only cases when fix(g i ) = S ae cases (1c) and (1d), when i =. In these cases, eithe V S o W S, so condition 4 is satisfied. Thus by the Edge Embedding Lemma, thee is an embedding of K n,m such that ϕ is ealized by g. Finally, we conside case (13). In this case we need to fist embed the midpoints of any edges that ae inveted by any g i fo i <. Since is even, ϕ(v) = V, so this will not happen fo any of the vetices in the -cycles; it emains to conside the -cycles. Fo each -cycle (vw) we add a vetex z vw at the midpoint ofthe edge vw in the abstact gaphk n,n (since ϕ(v) = W, we have n = m). Denote this set of vetices by Z; note that Z. Then we define H = K n,n Z, so H is a subdivision of K n,n. The vetices of Z ae fixed by ϕ, so we embed them as points in F (these points ae available, since ϕ doesn t fix any vetices of V W). Thus g induces ϕ on V W Z. We now check that the conditions of the Edge Embedding Lemma ae satisfied fo H. If any pai of adjacent vetices of H is fixed by g i, then g i must fix X, so condition (1) is satisfied. Since the vetices of V and W altenate aound X (with at most one vetex of Z in between each pai), condition (3) is satisfied. Since fix(g i ) S fo all i, condition (4) is satisfied. Fo evey pai of vetices v V and w W such that g(v) = w, thee exists a vetex z vw Z such that z vw fix(g) = F, so v and w ae not adjacent in H. Hence condition () is satisfied. Thus by the Edge Embedding Lemma, thee is an embedding of H that is setwise invaiant unde g. Finally, we delete the embedded midpoint vetices of ou embedding of H to obtain an embedding of K n,n such that ϕ is ealized by g. 4. Conclusion Poof of the Classification Theoem. The poof follows immediately fom ou lemmas. Let ϕ be an automophism. If fix(ϕ) 1, then ϕ is only ealizable by an oientation-peseving homeomophism if it falls

17 ERICA 16 FLAPAN, NICOLE LEHLE, BLAKE MELLOR, MATT PITTLUCK, AND XAN VONGSATHORN into case () o (3), by the Fixed Vetices Lemma. In these cases, ϕ is ealizable by a otation. If fix(ϕ) = 0, but some powe of ϕ fixes a vetex, then ϕ is only ealizable by an oientation-peseving homeomophism if it falls into one of cases (4) (9), by the Oientation Peseving Lemma. In these cases, ϕ is ealizable by a glide otation. If fix(ϕ) = 0 and no powe of ϕ fixes a vetex, then ϕ is ealizable by eithe a otation o a glide otation, depending on whethe ϕ(v) = V. If fix(ϕ) 3, then ϕ is only ealizable by an oientation-evesing homeomophism if it falls into case (11) by the Fixed Vetices Lemma. In this case, ϕ is ealizable by a eflection. If 1 fix(ϕ), then ϕ is only ealizable by an oientation-evesing homeomophism if it falls into case (1), by the Oientation Peseving Lemma. In this case, ϕ is ealizable by an impope otation. If fix(ϕ) = 0, then ϕ is only ealizable by an oientation-evesing homeomophism if it falls into case (10), (1) o (13), by the Oientation Peseving Lemma. In each of these cases, ϕ is ealizable by an impope otation. This accounts fo all possibilities, and completes the poof. Now that we have detemined which automophisms of K n,m can be induced by a homeomophism of an embedding in S 3, the next step is to exploe which goups of automophisms can be ealized as the goup of symmeties of an embedding in S 3. It is known that fo evey finite subgoup G of Diff + (S 3 ), thee is an embedding Γ of some complete bipatite gaph K n,n such that the goup of all automophisms of Γ which ae induced by oientation peseving homeomophisms of S 3 is isomophic to G [FNPT]. Howeve, this esult does not tell us which goups ae possible fo embeddings of a paticula bipatite gaph. Question. Given n and m, which subgoups of the automophism goup of K n,m ae induced by the oientation peseving homeomophisms of the pai (S 3,Γ) fo some embedding Γ of K n,m? The analogous question fo complete gaphs has been completely answeed [FMNY]. Fo bipatite gaphs, the question has been studied fo K n,n when the subgoup is isomophic to A 4, S 4 o A 5 [M] and when it is isomophic to Z n, D n o Z n Z m [HMP], but thee is substantial wok still to be done. Refeences [BLP] M. Boileau, B. Leeb, and J. Poti, Geometization of 3-dimensional obifolds, Ann. Math. 16 (005),

18 BIPARTITE SYMMETRIES 17 [D] P. Du Val, Homogaphies, Quatenions and Rotations, Oxfod Univesity Pess, [HMP] K. Hake, B. Mello and M. Pittluck, Topological symmety goups of complete bipatite gaphs, in pepaation. [F1] E. Flapan, Symmeties of Möbius laddes, Math. Ann. 83 (1989), [F] E. Flapan, Rigidity of gaph symmeties in the 3-sphee, J. Knot Theoy Ramifications 4 (1995), [FNPT] E. Flapan, R. Naimi, J. Pommesheim and H. Tamvakis, Topological symmety goups of gaphs embedded in the 3-sphee, Comment. Math. Helv. 80 (005), [FNT] E. Flapan, R. Naimi and H. Tamvakis, Topological symmety goups of complete gaphs in the 3-sphee, J. London Math. Soc. 73 (006), [FMNY] E. Flapan, B. Mello, R. Naimi and M. Yoshizawa, Classification of topological symmety goups of K n, Top. Poc. 43 (014), [M] B. Mello, Complete bipatite gaphs whose topological symmety goups ae polyhedal, Tokyo J. Math. 37 (014), to appea. [Si] J. Simon, Topological chiality of cetain molecules, Topology 5 (1986), [Sm] P.A. Smith, Tansfomations of finite peiod, ii, Ann. Math. 40 (1939), Depatment of Mathematics, Pomona College, Claemont, CA 91711, USA addess: eflapan@pomona.edu Depatment of Mathematics, Pomona College, Claemont, CA 91711, USA Depatment of Mathematics, Loyola Maymount Univesity, Los Angeles, CA 90045, USA addess: blake.mello@lmu.edu Depatment of Mathematics, Loyola Maymount Univesity, Los Angeles, CA 90045, USA Depatment of Mathematics, Pomona College, Claemont, CA 91711, USA