Related papers: Alternating links and definite surfaces
We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams,…
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…
A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…
It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due…
In this paper, we study alternating links in thickened surfaces in terms of the lattices of integer flows on their Tait graphs. We use this approach to give a short proof of the first two generalised Tait conjectures. We also prove that the…
In this paper, we apply Kauffman bracket skein algebras to develop a theory of skein adequate links in thickened surfaces. We show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first…
Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all…
We show that among alternating knots, those which have diagrams whose Seifert and Tait graphs are isomorphic are dominant.
The Tait conjecture states that alternating reduced diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. Kauffman and K. Murasugi studying the Jones polynomial. The author proved…
We use an extension of Gordon-Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in…
We give a complete proof of results announced by Hirasawa and Sakuma describing explicitly the Kakimizu complex of a non-split, prime, special, alternating link.
Let $D$ be a cellular alternating link diagram on a closed orientable surface $\Sigma$. We prove that if $D$ has no removable nugatory crossings then each checkerboard surface from $D$ is $\pi_1$-essential and contains no essential closed…
The Tait conjecture states that reduced alternating diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L.H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper…
We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so,…
Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare.…
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement…
In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled…
Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties.…
Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram…
Dasbach and Lin proved a "volumish theorem" for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of…