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Thomas Kindred

Publications and source records attributed to Thomas Kindred.

17 recordsLinked to original sources
arXiv2026

Three formulas for the Ising and Potts invariants of a knot or link

Jones described how the Ising and Potts models from statistical mechanics give rise, with appropriate choices of Boltzmann weights, to invariants of an oriented link. The Boltzmann weights that Jones proposed, however, work only with a correction factor that he does not mention. We fill in the missing details in two different ways. We also show that all of these invariants, which we call the Ising and Potts invariants, are given by evaluating the Jones polynomial at $t$ equal to one of these Boltzmann weights.

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arXiv2026

Spanning solids and Murasugi sum in dimension four

We study various constructions of spanning solids for knotted surfaces in the 4-sphere. In particular, we consider a 4-dimensional analogue of Murasugi sum, which we use to define a notion of arborescent knotted surfaces. We give a variety of examples and applications to broken surface diagrams, such as deciding which resolutions of the crossings in these diagrams yield spanning solids.

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arXiv2025

Crosscap numbers of alternating links via state codes

We describe a way of encoding a Kauffman state as a set of tuples, similar to a Gauss code. Then we describe a procedure for using these state codes to determine the unoriented genus and crosscap number of any prime alternating knot or link. Finally, we compute these values for all such links through 14 crossings and all such knots through 19 crossings (this data is new for links with 10-14 crossings and knots with 14-19 crossings), and we identify several intriguing patterns in the resulting data.

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arXiv2025

Average crosscap number of a 2-bridge knot

We determine a simple condition on a particular state graph of an alternating knot or link diagram that characterizes when the unoriented genus and crosscap number coincide, extending work of Adams and Kindred. Building on this same work and using continued fraction expansions, we provide a new formula for the unoriented genus of a 2-bridge knot or link. We use recursion to obtain exact formulas for the average unoriented genus $\overline{\Gamma}(c)$ and average crosscap number $\overline{\gamma}(c)$ of all 2-bridge knots with crossing number $c$, and in particular we show that $\displaystyle{\lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\Gamma}(c)\right) = \lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\gamma}(c)\right) = 0}$.

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arXiv2024

Kink-equivalence of matrices, spanning surfaces, 4-manifolds, and quadratic forms

All checkerboard surfaces for a given knot in $S^3$ are related by isotopy and "kinking" and "unkinking" moves, which change the surfaces' Goeritz matrices like this: $G\leftrightarrow G\oplus [\pm1]=\left[\begin{smallmatrix} G&\mathbf{0}\\ \mathbf{0}^T&\pm1 \end{smallmatrix}\right]$. We call two symmetric integer matrices "kink-equivalent" if they are related by "kinking'' and "unkinking'' moves $G\leftrightarrow G\oplus [\pm1]$ and unimodular congruence. We prove constructively that every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix and to a negative-definite matrix, and we give bounds on the number of moves required. This has several implications, e.g. every knot in $S^3$ is "alternating up to fake unkinking moves" and every simply connected, closed, topological 4-manifold with nonsingular intersection pairing has a positive blow-up that is homeomorphic to a negative blow-up of a positive-definite, simply connected, closed, topological 4-manifold.

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arXiv2024

How essential is a spanning surface?

Gabai proved that any plumbing, or Murasugi sum, of $\pi_1$-essential Seifert surfaces is also $\pi_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show that the analogous statement about geometrically essential surfaces is untrue. We then introduce new numerical invariants, the algebraic and geometric essence of a spanning surface $F\subset S^3$, which measure how far $F$ is from being compressible, and we extend Ozawa's theorem by showing that plumbing respects the algebraic version of this new invariant. We also introduce a ``twisted'' generalization of plumbing and use it to compute essence for many examples, including checkerboard surfaces from reduced alternating diagrams. Finally, we extend all of these results to plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds.

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arXiv2022

The virtual flyping theorem

We extend the flyping theorem to alternating links in thickened surfaces and alternating virtual links. The proof of the former result uses work of Boden--Karimi to adapt the author's geometric proof of Tait's 1898 flyping conjecture (first proved in 1993 by Menasco--Thistlethwaite), while the proof of the latter involves a diagrammatic correspondence recently introduced by the author in a related paper. In the process, we also extend a classical result of Gordon--Litherland, establishing an isomorphism between their pairing on a spanning surface and the intersection form on a 4-manifold constructed as a double-branched cover using that surface.

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arXiv2022

End-essential spanning surfaces for links in thickened surfaces

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 curve that is $\partial$-parallel in $\Sigma\times I$. Our chief motivation comes from technical aspects of a companion paper, where we prove that Tait's flyping conjecture holds for alternating virtual links. We also describe possible applications via Turaev surfaces.

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arXiv2022

Primeness of alternating virtual links

Using a new tool called lassos, we establish a new correspondence between cellular link {diagrams} on closed surfaces and equivalence classes of virtual link {diagrams}. This is analogous to a well-known correspondence among the links represented by these diagrams, but with a crucial subtlety. We explain how, under these correspondences, the traditional notion of primeness for virtual links is stricter than the one for links in thickened surfaces. We extend a classical result of Menasco by proving that an alternating link in a thickened surface is prime in the stricter sense unless it is ``obviously" composite. (Adams et al and Howie--Purcell previously extended Menasco's result for the other notion of primeness.) We describe, given an alternating virtual link diagram, how to determine by inspection whether the virtual link it represents is prime in either sense.

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arXiv2022

A simple proof of the Crowell-Murasugi theorem

We give an elementary, self-contained proof of the theorem, proven independently in 1958-9 by Crowell and Murasugi, that the genus of an alternating knot equals half the breadth of its Alexander polynomial, and that applying Seifert's algorithm to any alternating knot diagram gives a surface of minimal genus.

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arXiv2020

Efficient multisections of odd-dimensional tori

Rubinstein--Tillmann generalized the notions of Heegaard splittings of 3-manifolds and trisections of 4-manifolds by defining {\it multisections} of PL $n$-manifolds, which are decompositions into $k=\lfloor n/2\rfloor+1$ $n$-dimensional 1-handlebodies with nice intersection properties. For each odd-dimensional torus $T^n$, we construct a multisection which is {\it efficient} in the sense that each 1-handlebody has genus $n$, which we prove is optimal; each multisection is {\it symmetric} with respect to both the permutation action of $S_n$ on the indices and the $\Z_k$ translation action along the main diagonal. We also construct such a trisection of $T^4$, lift all symmetric multisections of tori to certain cubulated manifolds, and obtain combinatorial identities as corollaries.

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arXiv2020

A geometric proof of the flyping theorem

In 1898, Tait asserted several properties of alternating knot diagrams. These assertions became known as Tait's conjectures and remained open until the discovery of the Jones polynomial in 1985. The new polynomial invariants soon led to proofs of all of Tait's conjectures, culminating in 1993 with Menasco--Thistlethwaite's proof of Tait's flyping conjecture. In 2017, Greene (and independently Howie) answered a longstanding question of Fox by characterizing alternating links geometrically. Greene then used his characterization to give the first {\it geometric} proof of part of Tait's conjectures. We use Greene's characterization, Menasco's crossing ball structures, and a hierarchy of isotopy and {\it re-plumbing} moves to give the first entirely geometric proof of Menasco--Thistlethwaite's flyping theorem.

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arXiv2019

Crosscap numbers of alternating knots via unknotting splices

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

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arXiv2017

Plumbing is a natural operation in Khovanov homology

Given a connect sum of link diagrams, there is an isomorphism which decomposes unnormalized Khovanov chain groups for the product in terms of normalized chain groups for the factors; this isomorphism is straightforward to see on the level of chains. Similarly, any plumbing $x*y$ of Kauffman states carries an isomorphism of the chain subgroups generated by the enhancements of $x*y$, $x$, $y$: \[ \mathcal{C}_R(x*y)\to \left(\mathcal{C}_{R,p\to1}(x)\otimes \mathcal{C}_{R,p\to1}(y)\right)\oplus\left(\mathcal{C}_{R,p\to0}(x)\otimes \mathcal{C}_{R,p\to0}(y)\right). \] We apply this plumbing of chains to to prove that every homogeneously adequate state has enhancements $X^\pm$ in distinct $j$-gradings whose $A$-traces (which we define) represent nonzero Khovanov homology classes over $\mathbb{F}_2$, and that this is also true over $\mathbb{Z}$ when all $A$-blocks' state surfaces are two-sided. We construct $X^\pm$ explicitly.

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arXiv2017

Alternating links have representativity 2

We prove that if $L$ is a non-trivial alternating link embedded (without crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$-incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.

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arXiv2012

A classification of spanning surfaces for alternating links

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, orientability, and aggregate slope that a surface spanning that link can have. To this end, we describe a straightforward algorithm, much like Seifert's Algorithm, through which to construct certain spanning surfaces called layered surfaces. A particularly important subset of these will be what we call basic layered surfaces. We can alter these surface by performing the entirely local operations of adding handles and/or crosscaps, each of which increases genus. The main result then shows that if we are given an alternating projection P(L) and a surface S spanning L, we can construct a surface T spanning L with the same genus, orientability, and aggregate slope as S that is a basic layered surface with respect to P, except perhaps at a collection of added crosscaps and/or handles. Furthermore, S must be connected if L is non-splittable. This result has several useful corollaries. In particular, it allows for the determination of nonorientable genus for alternating links. It also can be used to show that mutancy of alternating links preserves nonorientable genus. And it allows one to prove that there are knots that have a pair of minimal nonorientable genus spanning surfaces, one boundary-incompressible and one boundary-compressible.

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