9 research outputs found
Classical results for alternating virtual links
We extend some classical results of Bankwitz, Crowell, and Murasugi to the
setting of virtual links. For instance, we show that an alternating virtual
link is split if and only if it is visibly split, and that the Alexander
polynomial of any almost classical alternating virtual link is alternating. The
first result is a consequence of an inequality relating the link determinant
and crossing number for any non-split alternating virtual link. The second is a
consequence of the matrix-tree theorem of Bott and Mayberry. We discuss the
Tait conjectures for virtual and welded links, and we note that Tait's second
conjecture is not true for alternating welded links.Comment: 18 pages, 13 figures, comments welcom
A characterization of alternating links in thickened surfaces
We use an extension of Gordon-Litherland pairing to thickened surfaces to
give a topological characterization of alternating links in thickened surfaces.
If is a closed oriented surface and is a compact unoriented
surface in , then the Gordon-Litherland pairing defines a
symmetric bilinear pairing on the first homology of . A compact surface in
is called definite if its Gordon-Litherland pairing is a
definite form. We prove that a non-split link in a thickened surface is
alternating if and only if it bounds two definite surfaces of opposite sign.Comment: 19 pages, 9 figures. Revision includes new proofs of Lemmas 6 and 18,
a slight improvement to Theorem 8, the addition of Corollary 20, and many
other minor changes. This version is to appear in Proc. Roy. Soc. Edinburgh
Sect.
The Gordon-Litherland pairing for links in thickened surfaces
We extend the Gordon-Litherland pairing to links in thickened surfaces, and
use it to define signature, determinant, and nullity invariants for links that
bound (unoriented) spanning surfaces. The invariants are seen to depend only on
the -equivalence class of the spanning surface. We prove a duality result
relating the invariants from one -equivalence class of spanning surfaces
to the restricted invariants of the other.
Using Kuperberg's theorem, these invariants give rise to well-defined
invariants of checkerboard colorable virtual links. The determinants can be
applied to determine the minimal support genus of a checkerboard colorable
virtual link. The duality result leads to a simple algorithm for computing the
invariants from the Tait graph associated to a checkerboard coloring. We show
these invariants simultaneously generalize the combinatorial invariants defined
by Im, Lee, and Lee, and those defined by Boden, Chrisman, and Gaudreau for
almost classical links.
We examine the behavior of the invariants under orientation reversal, mirror
symmetry, and crossing change. We give a 4-dimensional interpretation of the
Gordon-Litherland pairing by relating it to the intersection form on the
relative homology of certain double branched covers. This correspondence is
made explicit through the use of virtual linking matrices associated to
(virtual) spanning surfaces and their associated (virtual) Kirby diagrams.Comment: 43 pages, 17 figures, and many example
Adequate links in thickened surfaces and the generalized Tait conjectures
The Kauffman bracket of classical links extends to an invariant of links in
an arbitrary oriented 3-manifold with values in the skein module of . In
this paper, we consider the skein bracket in case is a thickened surface.
We develop a theory of adequacy for link diagrams on surfaces and show that any
alternating link diagram on a surface is skein adequate. We apply our theory to
establish the first and second Tait conjectures for adequate link diagrams on
surfaces. These are the statements that any adequate link diagram has minimal
crossing number, and any two adequate diagrams of the same link have the same
writhe.
Given a link diagram on a surface , we use to denote
its skein bracket. If has minimal genus, we show that where is the number of
connected components of , is the number of crossings, and
is the genus of This extends a classical result proved by Kauffman,
Murasugi, and Thistlethwaite. We further show that the above inequality is an
equality if and only if is weakly alternating, namely if is the
connected sum of an alternating link diagram on with one or more
alternating link diagrams on . This last statement is a generalization of
a well-known result for classical links due to Thistlethwaite, and it implies
that the skein bracket detects the crossing number for weakly alternating
links. As an application, we show that the crossing number is additive under
connected sum for adequate links in thickened surfaces.Comment: 24 pages, 13 figure
The Jones-Krushkal polynomial and minimal diagrams of surface links
We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in
thickened surfaces. It states that any reduced alternating diagram of a link in
a thickened surface has minimal crossing number, and any two reduced
alternating diagrams of the same link have the same writhe. This result is
proved more generally for link diagrams that are adequate, and the proof
involves a two-variable generalization of the Jones polynomial for surface
links defined by Krushkal. The main result is used to establish the first and
second Tait conjectures for links in thickened surfaces and for virtual links.Comment: 32 pages, 20 figures, and 1 tabl