Symmetries of alternating exteriors, spatial graphs, and branched surfaces

Joint Math Meetings AMS Special Session on Developments in Spatial Graphs

Thomas Kindred, University of Nebraska-Lincoln

Wednesday, January 6, 2021 Spatial graphs (and 2-cell complexes) from link diagrams

Given a diagram D on S 2 of a link L ⊂ S 3, construct black and white chessboard surfaces B and W .

Then B ∪ W is a 2-dimensional cell complex whose 1-skeleton is a 3-valent spatial graph GD = L ∪ C, where C consists of “vertical arcs,” one at each crossing. Properties of the spatial graph GD = L ∪ C, L alternating

Properties Given a reduced alternating link diagram D:

• If K ⊂ GD is a link containing each vertex of GD , then K is alternating, and c(K) < c(L) unless K = L. Hence GD encodes C.

• One may reconstruct B ∪ W from GD by attaching a 2-cell to each cycle γ ⊂ GD for which: ◦ γ consists alternately of edges from C and from L, and ◦ C ∪ (L \ γ) is connected. 0 ◦ Hence, if D 6= D , then GD 6= GD0 . Questions about the spatial graph GD = L ∪ C Questions

• For which diagrams D does GD encode C? 0 • For which D does some D 6= D satisfy GD = GD0 ?

Given a diagram D of an , the natural edge 3-coloring of GD 4 describes a bridge trisection of an embedded surface Σ in S . Since GD is always bipartite, Σ is always orientable.

Question Is such a surface Σ always unknotted? Symmetries of exteriors

Definition: The symmetry group of a knot L ⊂ S 3 consists of homeomorphisms (S 3, L) → (S 3, L) modulo those pairwise isotopic to 1.

Facts:

• For any L ⊂ S 3:

∼ 3 ◦ ∼ 3 ◦ Sym(L) = Mapping Class Group(S \ νL) = Out(π1(S \ νL)).

• Sym(L) is finite iff L is a hyperbolic knot, a , or a cable of a torus knot. ◦ In particular, Sym(L) is finite whenever L is prime and alternating.

Note: Many have interesting symmetries which are pairwise isotopic to the identity. Extending GD to GL when L is alternating Flyping theorem (Menasco-Thistlethwaite, 1993): All reduced

alternating diagrams of any nonsplit link are related by flype moves:

T 2 T1 T2 T1

Extend GD = L ∪ C to a spatial graph L ∪ C ∪ A by attaching the set A of all “flyping arcs” for D. Then any flype on D is an isotopy move on L ∪ C ∪ A, exchanging an arc in C with one in A. By the flyping theorem, L ∪ C ∪ A depends only on L. Denote it GL. Symmetries of GL

M-T’s stronger flyping theorem: For any alternating link L, Sym(L) comes from flypes and “visible symmetries” of alternating diagrams.

In GL = L ∪ A ∪ C, think of A ∪ C as “the set of all possible crossings” in reduced alternating diagrams of L.

By M-T’s theorem, any symmetry of L comes from a symmetry of GL. Extending GL to a pair of branched surfaces

Just as one may extend the spatial graph GD to the 2-complex B ∪ W by attaching 2-cells, one may extend GL to a 2-complex comprised of the chessboard surfaces from all reduced alternating diagrams of L.

The 2-complex is a union of two surfaces branched along a subset of GL \ L.

The construction comes from the way flyping affects chessboard surfaces. Plumbing and re-plumbing

Let F be a spanning surface, V ⊂ S 3 \\F a properly embedded disk s.t.

• ∂V bounds a disk U ⊂ F . 3 • Denoting S \\(U ∪ V ) = Y1 t Y2, neither Fi = F ∩ Yi is a disk. Then V is a plumbing cap for F , and U is its shadow.

= along

Say that F is obtained by (generalized) plumbing F1 and F2 along U, denoted F1 ∗ F2 = F . This operation is also called Murasugi sum. The operation F → F 0 = (F \ U) ∪ V is called re-plumbing, and can also be realized via proper isotopy through the 4-ball: on D re-plumb and isotop D’s chessboards F± Proposition (K)

0 Any flype D → D follows a plumbing cap V for F±. Re-plumbing F± 0 0 along V gives a surface isotopic to F±; also, F∓ is isotopic to F∓.

Proof.

T T 2 2 2 T

T T 1 1

Flyping re-plumbing theorem (K) 0 Reduced alternating diagrams D and D are related by flypes iff F+ and 0 0 F+ are related by re-plumbing and isotopy moves, as are F− and F−. Classical problems in Tait’s conjectures (1898) Given reduced alternating diagrams D, D0 of a prime nonsplit link L: 0 (1) D and D minimize crossings: | |D = | |D0 = c(L). 0 0 (2) D and D have the same : w(D) = w(D ) = | |D0 − | |D0 .

(3) D and D0 are related by flype moves:

T 2 T1 T2 T1

Question (Fox, ∼ 1960) What is an alternating link?

Conjecture (open)

If L = L1#L2, then c(L) = c(L1) + c(L2). Proofs of Tait’s conjectures, answers to Fox’s question Tait’s conjectures (1898) Given reduced alternating diagrams D, D0 of a prime nonsplit link L: 0 (1) D and D minimize crossings: | |D = | |D0 = c(L). 0 0 (2) D and D have the same writhe: w(D) = w(D ) = | |D0 − | |D0 . (3) D and D0 are related by flype moves.

Question (Fox, ∼ 1960) What is an alternating link?

• 1987: Kauffman, Murasugi, Thistlethwaite independently used the to prove (1), which implies (2). • 1993: Menasco-Thistlethwaite used geometric techniques and the Jones polynomial to prove (3), which implies (2) and part of (1). • 2017: Greene, Howie independently answered Fox’s question; Greene gave the first purely geometric proof of (2) and part of (1). • 2020: K gave the first purely geometric proof of (3). Greene’s characterization of alternating links

Definition: A spanning surface F is +-definite iff for every s.c.c. γ ⊂ F : • The framing of γ in F is positive, or • γ bounds an orientable subsurface of F .

Theorems (Greene, 2017) • A conn’d diagram D is alternating iff its chessboards are ±-definite. • A non-split link L ⊂ S 3 is alternating iff it has spanning surfaces F± which respectively are ±-definite. • Any reduced alternating diagrams D, D0 of the same link satisfy 0 | |D = | |D0 and w(D) = w(D ).

Yet, it does not follow that any reduced alternating diagram minimizes crossings. All existing proofs of this fact use the Jones polynomial. Geometric proof of the flyping theorem

Let D, D0 be reduced alternating diagrams of a prime, nonsplit link L 0 with chessboards F±, F±. Two intermediate results, Theorems (K) 0 0 • D and D are related by flypes if and only if F+ and F+ are related 0 by isotopy and re-plumbing moves, as are F− and F−. • Any essential positive- (resp. negative-) definite surface spanning L is related to F+ (resp. F−) by isotopy and re-plumbing moves.

together with Greene’s characterization, give a geometric proof of:

Flyping theorem (Menasco-Thistlethwaite; K) All reduced alternating diagrams of any prime nonsplit link are related by flypes. Geometric proofs: open problems

Open problems • Prove geometrically that any reduced alternating link diagram realizes the underlying link’s crossing number. • Prove geometrically that any reduced alternating diagram realizes the underlying tangle’s crossing number. • Prove geometrically that any adequate link diagram realizes the underlying link’s crossing number. • Prove that crossing number is additive under connected sum.

Approach: Translate statements about diagrams to statements about chessboard surfaces (a la Greene, Howie) or spatial graphs (e.g. GD or GL). J. Greene, Alternating links and definite surfaces, w/appendix by A. Juhasz, M Lackenby, Duke Math. J. 166 (2017), no. 11, 2133-2151. J. Howie, A characterisation of exteriors, Geom. Topol. 21 (2017), no. 4, 2353-2371. A. Issa, H. Turner, Links all of whose cyclic branched covers are L-spaces, preprint, arXiv:2008.06127. T. Kindred, A geometric proof of the flyping theorem, preprint, arXiv:2008.06490. L.H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407. A. Kawauchi, A survey of knot theory, Birkh¨auserVerlag, Basel, 1996. xxii+420 pp. J. Meier, A. Zupan, Bridge trisections of knotted surfaces in S 4, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7343-7386. J. Meier, A. Thompson, A. Zupan, Cubic graphs induced by bridge trisections, preprint, arXiv:2007.07280. W. Menasco, M. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 403-412. W. Menasco, M. Thistlethwaite, The classification of alternating links, Ann. of Math. (2) 138 (1993), no. 1, 113-171. K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187-194. L. Paoluzzi, Cyclic branched covers of alternating knots, preprint, arXiv:2006.12922 P.G. Tait, On Knots I, II, and III, Scientific papers 1 (1898), 273-347. M.B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297-309. M.B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93 (1988), no. 2, 285-296. M.B. Thistlethwaite, On the algebraic part of an alternating link, Pacific J. Math. 151 (1991), no. 2, 317-333. Thank you!