NNOTE ON THE CANONICAL GENUS OF A KNOT
MARTINA AALTONEN
Abstract.
We show that every canonical Seifert surface is (up to iso-topy) given by a knot diagram in which the (open) Seifert disks arepairwise disjoint. Introduction
In this article we consider canonical Seifert surfaces and the canonicalgenus of PL knots [4]. We prove the following:
Theorem 1.
Let M ⊂ R be a Seifert surface that is canonical with respectto a knot diagram D . Then there exists a knot diagram D (cid:48) in which the (open)Seifert disks are pairwise disjoint and a Seifert surface that is canonical withrespect to D (cid:48) and isotopic to M. Recently, it was pointed out to the author that this Theorem is alreadyknown, see [1]. Theorem 1 imposes the classical result that the fundamentalgroup of R \ M is a free group for any canonical Seifert surface M of a knot.Theorem 1 also has the following corollary. Corollary 2.
The canonical genus of a knot is achieved in a subclass of knotdiagrams for which the (open) Seifert disks are pairwise disjoint.
In [2, Ch. VII] Kauffman has shown how to construct from a knot dia-gram a link diagram in which the open Seifert disks are pairwise disjoint byadding simple closed polygons on top of the knot diagram. Based on thisobservation we give an algorithm to construct from any knot diagram sucha knot diagram in which the open Seifert disks are pairwise disjoint. Weshow that this modification corresponds to an isotopy between two Seifertsurfaces that are canonical with the respective knot diagrams.
Acknowledgements.
I want to thank Sebastian Baader for his helpful-ness and his excellent suggestions, Pekka Pankka for reading the manuscriptand his suggestions for the presentation and Vadim Kulikov for discussionson the topic. 2.
Notation and preliminaries
We recall that PL knots are isotopy classes of polyhedral embeddings k : S → R . For the convenience of presentation we denote R = R × { } throughout this article. Let d : S → R be a polyhedral curve so that Date : September 25, 2018.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . G T ] J a n MARTINA AALTONEN
D D (cid:48)
Figure 1.
In the knot diagram D (cid:48) the evenly dotted PLcurve lies on top of the black PL curve coming from the knotdiagram D . The diagrams D and D (cid:48) are both knot diagramsfor the figure eight knot satisfying G ( D ) = G ( D (cid:48) ) = 1 . Thecollection of (open) Seifert disks in D is not pairwise disjoint,but the collection of (open) Seifert disks in D (cid:48) is pairwisedisjoint.(a) Im( d ) ⊂ R is a polygon with vertices v , . . . , v k ∈ R satisfying d − { v j } ) = 1 for every j ∈ { , . . . , k } , (b) d − { z } ) ≤ for all z ∈ R and(c) the set C ( d ) := { z ∈ R | d − { z } ) = 2 } is finite.Let f : d − ( C ( d )) → { , } be a function satisfying f ( d − { z } ) = { , } forall z ∈ C ( d ) . We call the pair D := ( d, f ) a knot diagram and C ( D ) := C ( d ) the crossings of D . In what follows we fix an orientation of S and note by [ x , x ] ⊂ S the segment from x to x in this orientation.Let k : S → R be a polyhedral embedding, p : R → R the projectionand D = ( d, f ) a knot diagram. We say that p induces D from k if d = p ◦ k and for x ∈ f − { } and x ∈ f − { } satisfying d ( x ) = d ( x ) the thirdcoordinate of k ( x ) is bigger than the third coordinate of k ( x ) . Let D := ( d, f ) be a knot diagram. A restriction map s := d | E is a Seifert circuit if E = (cid:83) ki =0 [ x i , y i ] for such points x , . . . , x k and y , . . . , y k in d − ( C ( D )) that(a) [ x i , y i ] ∩ d − ( C ( D )) = { x i , y i } for i ∈ { , . . . , k } and(b) d ( x i ) = d ( y i − ) and f ( x i ) (cid:54) = f ( y i − ) for all i ∈ { , . . . , k } and d ( x ) = d ( y k ) and f ( x ) (cid:54) = f ( y k ) . We denote by S ( D ) the collection of Seiferts circuits in D and for every s ∈ S ( D ) we denote C ( s ) := C ( D ) ∩ Im( s ) . Let D = ( d, f ) be a knot diagram. For a Seifert circuit s ∈ S ( D ) theimage Im( s ) ⊂ R is a simple closed polygon. We call an open boundedconnected subset b s ⊂ R satisfying ∂b s = Im( s ) the (open) Seifert disk bounded by s. Now for every point z ∈ Im( d ) there exists s ∈ S ( D ) so that z ∈ Im( s ) and for every z ∈ C ( D ) there exists a unique pair s, t ∈ S ( D ) sothat z ∈ Im( s ) ∩ Im( t ) . We denote S II ( D ) := { s ∈ S ( D ) | b t (cid:36) b s for some t ∈ S ( D ) } OTE ON THE CANONICAL GENUS OF A KNOT 3 and for every s ∈ S ( D ) we denote C II ( s ) := { z ∈ C ( s ) | z ∈ C ( t ) for t ∈ S ( D ) satisfying b t (cid:36) b s } . Lemma 3.
Let D be a knot diagram. Then the open Seifert disks are pairwisedisjoint if and only if S II ( D ) = ∅ . Proof.
Suppose that S II ( D ) = ∅ , but that there exists Seifert circuits s, t ∈S ( D ) , s (cid:54) = t, so that b s ∩ b t (cid:54) = ∅ . Then there exists a point in
Im( t ) ∩ b s . By property (a) of a knot diagram and property (b) of a Seifert circuitthe simple closed polygons
Im( t ) and Im( s ) do not cross each other. Thus Im( t ) ⊂ Im( s ) ∪ b s . In particular, b t ⊂ b s and b s ∩ b t = b t . Thus s ∈ S II ( D ) , which is a contradiction. Thus the open Seifert disks are pairwise disjoint.Suppose S II ( D ) (cid:54) = ∅ and s ∈ S II ( D ) . Then there exists trivially a pair ofSeifert disks with non-empty intersection. (cid:3)
For the definition of a canonical Seifert surface, we give first the definitionof a crossing area. Let D = ( d, f ) be a knot diagram and z ∈ C ( D ) . Let z , z ∈ Im( s ) be points so that [ z, z ] , [ z, z ] ⊂ Im( s ) and [ z , z ] ∩ Im( d ) = { z , z } and let ∆ zi ⊂ R be the closed set that has the polygon [ z, z ] ∪ [ z, z ] ∪ [ z , z ] as its boundary for every s ∈ S ( D ) satisfying z ∈ C ( s ) . Thenwe call w z := (cid:83) i ∈{ s,t } ∆ zi , where s, t ∈ S ( D ) are distinct Seiferts circlessatisfying z ∈ C ( s ) ∩ C ( t ) , a crossing area of z. Remark 4.
Let D be a knot diagram s ∈ S ( D ) and ( w i ) i ∈C ( s ) be a collectionof pairwise disjoint crossing areas. Let b s ⊂ R be the open Seifert diskbounded by s. Then the boundary of b s \ (cid:0) (cid:83) i ∈C ( s ) ∆ is (cid:1) ⊂ R is a simpleclosed polygon. Further, b s \ (cid:0) (cid:83) i ∈C ( s ) ∆ is (cid:1) = b s \ (cid:0) (cid:83) i ∈C ( s ) \C II ( s ) ∆ is (cid:1) = b s \ (cid:0) (cid:83) i ∈C ( s ) \C II ( s ) w i (cid:1) , since b s ∩ ∆ is = ∅ for i ∈ C II ( s ) and b s ∩ w i = b s ∩ ∆ is for i ∈ C ( s ) \ C II ( s ) . Let M ⊂ R be a Seifert surface and D = ( d, f ) a knot diagram. Then M is canonical with respect to D , if D is induced from Bd( M ) by the projection p : R → R and there exists a triple (( u i ) i ∈S ( D ) ∪C ( D ) , ( w i ) i ∈C ( D ) , ( r i ) i ∈S ( D ) ) , where ( u i ) i ∈S ( D ) ∪C ( D ) is a collection of PL -manifolds in R with one bound-ary component and genus , ( w i ) i ∈C ( D ) is a collection of pairwise disjointcrossing areas of D and ( r i ) i ∈S ( D ) is a collection of real numbers, so that(a) M = (cid:83) i ∈S ( D ) ∪C ( D ) u i , (b) if i, j ∈ S ( D ) ∪ C ( D ) satisfy i ∈ S ( D ) and j ∈ C ( i ) or j ∈ S ( D ) and i ∈ C ( j ) , then u i ∩ u j = Bd( u i ) ∩ Bd( u j ) ≈ [0 , , otherwise u i ∩ u j = ∅ , (c) u i = (cid:8) z + (0 , , r i ) | z ∈ Cl (cid:0) b i \ (cid:83) j ∈C ( i ) \C II ( i ) w j (cid:1)(cid:9) for i ∈ S ( D ) and b i ⊂ R the open Seifert disk bounded by i, (d) u i ⊂ p − ( w i ) for i ∈ C ( D ) and(e) u j ∩ ( R + (0 , , r i )) = u j ∩ u i for i ∈ S ( D ) and j ∈ C II ( i ) . We say that the triple (( u i ) i ∈S ( D ) ∪C ( D ) , ( w i ) i ∈C ( D ) , ( r i ) i ∈S ( D ) ) is the data ofa canonical Seifert surface M with respect to D . We say that a Seifert surface M of a knot is a canonical Seifert surface ,if M is isotopic to a Seifert surface that is canonical with respect to a knot MARTINA AALTONEN diagram D of that knot. Note that, if a Seifert surface M is canonical withrespect to D , then the genus G ( D ) of M is ( C ( D ) − S ( D ) + 1) / . We recall that using Seiferts algorithm [3] one can construct for every knotdiagram D a Seifert surface M ⊂ R that is canonical with respect to D . Indeed, let ( w i ) i ∈C ( D ) be a collection of pairwise disjoint crossing areas of D and ( r i ) i ∈S ( D ) a collection of real numbers so that by defining u i := (cid:8) z + (0 , , r i ) | z ∈ Cl (cid:0) b i \ (cid:83) j ∈C ( i ) \C II ( i ) w j (cid:1)(cid:9) for every i ∈ S ( D ) the collection ( u i ) i ∈S ( D ) is pairwise disjoint and { z + (0 , , r ) | z ∈ w k , r ∈ [ r i , r j ] } ∩ { u l | l ∈ S ( D ) } ⊂ u i ∩ u j for every i, j ∈ S ( D ) and k ∈ C ( i ) ∩ C ( j ) . Then there exists a collection ( u i ) i ∈C ( D ) of bands corresponding to crossings, see Figures 2 and 3, so that M := (cid:83) i ∈S ( D ) ∪C ( D ) u i is a Seifert surface and the data of M with respect to D is the triple (( u i ) i ∈S ( D ) ∪C ( D ) , ( w i ) i ∈C ( D ) , ( r i ) i ∈S ( D ) ) . u t u s u z z b t b s Figure 2.
For D = ( d, f ) , s, t ∈ S ( D ) and z ∈ ( C ( s ) \C II ( s )) ∩ C ( t ) the band u z connecting u s and u t on the leftand Im( d ) ⊂ R around the crossing z on the right. u t u s u z b t b s b s z Figure 3.
For D = ( d, f ) , s, t ∈ S ( D ) and z ∈ C II ( s ) ∩ C ( t ) the band u z connecting u s and u t on the left and Im( d ) ⊂ R around the crossing z on the right. OTE ON THE CANONICAL GENUS OF A KNOT 5 Proof of Theorem 1 s s s s s s Figure 4.
On the left an illustration of s ∈ S II ( D ) . In themiddle is illustrated the changes in the induced knot diagramobtained by the first isotopy that takes β to β . On the rightis illustrated the changes in the induced knot diagram by thesecond isotopy that takes β to such β that p ( β ) ∩ Im( d ) ⊂ Im( s ) and the Seifert circuits s , . . . , s ∈ S ( D ) \ S II ( D ) . Proposition 5.
Let D = ( d, f ) be a knot diagram and M ⊂ R a Seifertsurface that is canonical with respect to D . Suppose S II ( D ) (cid:54) = ∅ and s ∈S II ( D ) . Then there exists a knot diagram D (cid:48) = ( d (cid:48) , f (cid:48) ) and a Seifert surface M (cid:48) so that (i) M (cid:48) is canonical with respect to D (cid:48) , (ii) M (cid:48) is isotopic to M, (iii) S II ( D (cid:48) ) = S II ( D ) \ { s } and (iv) C ( D (cid:48) ) − C ( D ) = 2 C II ( s ) . Proof.
Let (( u i ) i ∈S ( D ) ∪C ( D ) , ( w i ) i ∈C ( D ) , ( r i ) i ∈S ( D ) ) be the data of M with re-spect to D . Let s ∈ S II ( D ) . Then
Bd( u s ) ⊂ ( R + (0 , , r s )) is a sim-ple closed polygon and there exists β = [ z , z ] ⊂ Bd( M ) so that β ⊂ Bd( u s ) \ p − (cid:0) (cid:83) i ∈C ( D ) w i (cid:1) . Now α = (Bd( u s ) \ [ z , z ]) ∪ { z , z } is a simplePL curve and there exists an embedding ι : α → u s \ Bd( u s ) so that the PLcurve β = [ z , ι ( z )] ∪ ι ( α ) ∪ [ ι ( z ) , z ] is such a simple curve in ( R +(0 , , r s )) that F = (cid:83) z ∈ α [ z, ι ( z )] is a PL disk having α ∪ β as its boundary. In par-ticular, there exists an isotopy that takes M to M := ( M \ u s ) ∪ F andkeeps M \ u s fixed. Now the projection p : R → R induces a knot diagram D := ( d , f ) of Bd( M ) so that Im( d ) = (cid:0) Im( d ) \ p ( β ) (cid:1) ∪ p ( β ) , see Figures 4 and 5.We note that M is not in general canonical with respect to D . However,by contracting F linearly along the segments [ z, ι ( z )] for each z ∈ α towards α and rotating it for every i ∈ C II ( s ) locally around α in a neighbourhoodof p − ( w i ) ∩ α either upwards or downwards to avoid u i and then stretchingit away from α in ( R + (0 , , r s )) , we have an isotopy that fixes M \ F andtakes F to a PL 2-manifold G ⊂ R and β to β so that M := ( M \ F ) ∪ G satisfies:(a) ∂ R ( p ( G )) = p ( α ) ∪ p ( β ) and p ( G ) ∩ Im( d ) = p ( α ) , MARTINA AALTONEN u t F u z b t p ( F ) z Figure 5.
For z ∈ C II ( s ) ∩ C ( t ) , t ∈ S ( D ) , the band u z connecting F and u t on the left and Im( d ) ⊂ R around thecrossing z on the right.(b) the projection p : R → R induces a knot diagram D := ( d , f ) of Bd( M ) so that Im( d ) = (cid:0) Im( d ) \ p ( β ) (cid:1) ∪ p ( β ) , C ( D ) \ C ( D ) = p ( α \ { z , z } ) ∩ p ( β ) and C ( D ) \ C ( D )) = 2 C II ( s ) (c) w i ∩ p ( β ) = ∅ for every i ∈ C ( D ) and(d) there exists a collection ( w i ) i ∈C ( D ) \C ( D ) of pairwise disjoint crossingareas in R \ (cid:16)(cid:83) i ∈C ( D ) w i (cid:17) so that G \ p − { z ∈ w i | i ∈ C ( D ) \ C ( D ) } ⊂ ( R + (0 , , r s )) , see Figures 4 and 6. u t GGG u z b s i b s i − b s i +1 b t zz (cid:48) z (cid:48)(cid:48) Figure 6.
For z ∈ C II ( s ) ∩ C ( t ) , t ∈ S ( D ) , and the band u z connecting G and u t on the left and Im( d ) ⊂ R and z (cid:48) , z (cid:48)(cid:48) ∈ C ( D ) \ C ( D ) around the crossing z on the right. OTE ON THE CANONICAL GENUS OF A KNOT 7
Now M is isotopic to M and thus satisfies (ii). By property (d) M satisfies (iv). Towards proving (iii), let Ω , Ω , . . . , Ω k ⊂ R , where k =2 C II ( s ) , be the collection of components of p ( G ) \ p ( α ∪ β ) and s i := ∂ R (Ω i ) for every i ∈ { , . . . , k } . Since p ( β ) ∩ Im( d ) ⊂ p ( α ) and p ( α ) ⊂ Im( s ) , the collection S ( D ) of Seiferts circles of D satisfies S ( D ) = ( S ( D ) \ { s } ) ∪ { s i : i ∈ { , . . . , k }} and b s i = Ω i . Now s i / ∈ S II ( D ) for every i ∈ { , . . . , k } , since by property(a) we have b s i ∩ Im( d ) ⊂ ( p ( G ) \ p ( α )) ∩ Im( d ) = ∅ . Since s ∈ S II ( D ) and s / ∈ S ( D ) , we have S II ( D ) = S II ( D ) \ { s } . Thus M satisfies (iii).It remains to show that M satisfies condition (i). By property (b) theprojection p : R → R induces D from Bd( M ) and by property (c) thecrossing area w i of D is a crossing area for D for every i ∈ C ( D ) . Thus ( w i ) i ∈C ( D ) is a pairwise disjoint collection of crossing areas and there existsa collection ( u (cid:48) i ) i ∈ ( C ( D ) \C ( D )) ∪ ( S ( D ) \S ( D )) of closed PL 2-manifolds with oneboundary component and genus in Cl (cid:0) G \ (cid:0) (cid:83) i ∈ C II ( s ) w i (cid:1)(cid:1) so that whenwe set u (cid:48) i = u i ∪ ( p − ( w i ) ∩ G ) for every i ∈ C II ( s ) and u (cid:48) i = u i for every i ∈ ( S ( D ) ∩ S ( D )) ∪ ( C ( D ) \ C II ( s )) and r i = r s for every i ∈ S ( D ) \ S ( D ) the collection (( u (cid:48) i ) i ∈S ( D ) ∪C ( D ) , ( w i ) i ∈C ( D ) , ( r i ) i ∈S ( D ) ) is a data of M with respect to D . Thus M is canonical with respect to D . Thus M satisfies condition (i). (cid:3) Corollary 6.
Let D be a knot diagram and M ⊂ R a Seifert surface thatis canonical with respect to D . Then there exists a knot diagram D (cid:48) and aSeifert surface M (cid:48) so that the open Seifert disks in D (cid:48) are pairwise disjoint, M (cid:48) is canonical with respect to D (cid:48) and isotopic to M and C ( D (cid:48) ) − C ( D ) = (cid:88) i ∈S ( D ) C II ( i ) . Proof.
By Lemma 3 the collection of Seifert disks in a knot diagram D (cid:48) ispairwise disjoint if S II ( D (cid:48) ) = ∅ . Thus the statement follows from Proposition5 by induction on S II ( D )) . (cid:3) This concludes the proof of Theorem 1.4.
Algorithm
Let D = ( d, f ) be a knot diagram of a knot [ k ] . The integer G ( D ) =( C ( D ) − S ( D ) + 1) / does not depend on the function f. Thus the proofof Proposition 5 yields an algorithm to construct a knot diagram D (cid:48) of [ k ] so that S II ( D (cid:48) ) = ∅ and G ( D (cid:48) ) = G ( D ) . Suppose S II ( D ) (cid:54) = ∅ . Let s ∈ S II ( D ) and [ x , x ] ⊂ d − (Im( s ) \ C ( D )) . By tracing
Im( s ) \ s ([ x , x ]) from s ( x ) and crossing Im( s ) twice close toevery i ∈ C II ( s ) on top of Im( s ) to avoid crossing Im( d ) \ Im( s ) we get aknot digram D (cid:48) := ( d (cid:48) , f (cid:48) ) so that MARTINA AALTONEN (a) d | ( S \ [ x , x ]) = d (cid:48) | ( S \ [ x , x ]) and f (cid:48) ( x ) = f ( x ) for x ∈ d − ( C ( D )) ,f (cid:48) ( x ) = 1 for x ∈ d (cid:48)− (( C ( D (cid:48) ) \ C ( D )) ∩ [ x , x ] f (cid:48) ( x ) = 0 for x ∈ d (cid:48)− (( C ( D (cid:48) ) \ C ( D )) \ [ x , x ] , (b) S ( D ) \ S ( D (cid:48) ) = { s } and S ( D (cid:48) ) \ S ( D )) = C ( D (cid:48) ) \ C ( D )) + 1 and(c) S II ( D (cid:48) ) ⊂ S II ( D ) . Now D (cid:48) := ( d (cid:48) , f (cid:48) ) is a knot diagram of [ k ] satisfying S II ( D (cid:48) ) < S II ( D ) and G ( D (cid:48) ) = G ( D ) . Thus the Seifert circuits S II ( D ) can be removed from D one by one. References [1] Mikami Hirasawa,
The flat genus of links , Kobe J. Math. (1995), no. 2, 155–159.MR 1391192 (97g:57006)[2] Louis H. Kauffman, On knots , Annals of Mathematics Studies, vol. 115, PrincetonUniversity Press, Princeton, NJ, 1987.[3] Dale Rolfsen,
Knots and links , Publish or Perish, Inc., Berkeley, Calif., 1976, Mathe-matics Lecture Series, No. 7.[4] H. Seifert,
Über das Geschlecht von Knoten , Mathematische Annalen (1935), no. 1,571–592 (German).
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