arXiv subjects

Katherine Vance

Publications and source records attributed to Katherine Vance.

4 recordsLinked to original sources

arXiv2026

Gordian distance and clasper surgery for links

In 2000, Habiro introduced the notion of $C_k$-equivalence of knots and links. This geometric filtration is closely connected to finite type invariants, a class of invariants including Milnor's invariants. Shortly thereafter, Ohyama, Taniyama, and Yamada proved that $C_k$-equivalence, and by extension finite type invariants, say very little about the unknotting number by showing that any knot is at most one crossing change away from being $C_k$-trivial for any $k\in \mathbb{N}$. The same is not true for links, since the pairwise linking number gives a lower bound on unlinking and is an invariant of $C_2$-equivalence. We prove that, aside from the linking number, the result of Ohyama, Taniyama, and Yamada extends to links: any $n$-component link with linking number zero can be reduced to a $C_k$-trivial link in at most $n^2$ crossing changes. As a consequence, Milnor's invariants carry only limited information about the unlinking number. To establish a lower bound, we produce a sequence of $n$-component links for which the crossing change distance to a $C_k$-trivial link grows quadratically in $n$. Notably, these bounds are independent of the choice of $k\in \mathbb{N}$. Finally, we determine the exact number of crossing changes to a $C_k$-trivial link for links with nonzero linking numbers and where no component is $C_k$-trivial.

math.GT↗

arXiv2024

How many crossing changes or Delta-moves does it take to get to a homotopy trivial link?

The homotopy trivializing number, $n_h(L)$, and the Delta homotopy trivializing number, $n_\Delta(L)$, are invariants of the link homotopy class of $L$ which count how many crossing changes or Delta moves are needed to reduce that link to a homotopy trivial link. In 2022, Davis, Orson, and Park proved that the homotopy trivializing number of $L$ is bounded above by the sum of the absolute values of the pairwise linking numbers and some quantity $C_n$ which depends only on $n$, the number of components. In this paper we improve on this result by using the classification of link homotopy due to Habegger-Lin to give a quadratic upper bound on $C_n$. We employ ideas from extremal graph theory to demonstrate that this bound is close to sharp, by exhibiting links with vanishing pairwise linking numbers and whose homotopy trivializing numbers grows quadratically. In the process, we determine the homotopy trivializing number of every 4-component link. We also prove a cubic upper bound on the difference between the Delta homotopy trivializing number of $L$ and the sum of the absolute values of the triple linking numbers of $L$.

math.GT↗

arXiv2024

Diagonal knots and the tau invariant

In 2003, Ozsv\'ath, Szab\'o, and Rasmussen introduced the $\tau$ invariant for knots, and in 2011, Sarkar published a computational shortcut for the $\tau$ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots, and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot.

math.GT↗

arXiv2018

Tau invariants for balanced spatial graphs

In 2003, Ozsváth and Szabó defined the concordance invariant $τ$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $τ$ for knots in $S^3$ and a combinatorial proof that $τ$ gives a lower bound for the slice genus of a knot. Recently, Harvey and O'Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in $S^3$ which extends knot Floer homology. We define a $\mathbb{Z}$-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O'Donnol's graph Floer homology. We use this to show that there is a well-defined $τ$ invariant for balanced spatial graphs generalizing the $τ$ knot concordance invariant. In particular, this defines a $τ$ invariant for links in $S^3$. Using techniques similar to those of Sarkar, we show that our $τ$ invariant gives an obstruction to a link being slice.

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