arXiv subjects

Micah Chrisman

Publications and source records attributed to Micah Chrisman.

At least 19 recordsLinked to original sources

arXiv2026

Thompson's Group $V$ and Virtual Link Theory

Thompson's groups $F \subset T \subset V$ were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of $F$, factoring through a surjection from $F$ to isotopy classes of links in $S^3$. The second author extended Jones' surjection to $T$, thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for $V$, defining a surjection $\mathcal{L}_{V}$ from $V$ to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup $\vec{V} \subset V$ containing Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove $\vec{V}$ realizes all oriented almost classical virtual links. We then construct unitary representations of $V$ and $\vec{V}$ from kei and operator quandle coloring invariants, respectively.

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arXiv2025

Lie superalgebra invariants and almost classical knots

A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $Σ\times [0,1]$, where $Σ$ is a closed orientable surface. AC links provide a useful window for observing the geometric topology of virtual knots. Here we take a different approach and look at AC links through the lens of quantum topology. Two adjustments are needed to the existing theory. First, it is necessary to generalize the definition of AC to include virtual tangles and, in particular, virtual braids. Secondly, to distinguish AC and non-AC tangles, the additional structure of quantum supergroups is required. For each Lie superalgebra $\mathfrak{gl}(m|n)$, we define a pair of $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors $Q^{m|n}$, $\widetilde{Q}^{m|n} \circ Zh$ on framed virtual tangles. Here $Zh$ denotes the Bar-Natan $Zh$ construction. These functors unify the Alexander polynomial (AP) of AC links and the generalized Alexander polynomial (GAP) of all virtual links into a single quantum model: $Q^{1|1}$ recovers the AP of an AC link and for any virtual link $K$, $\widetilde{Q}^{1|1}\circ Zh(K)$ is the 2-variable GAP. However, when $(m,n) \ne (1,1)$, these invariants are generally distinct from the AP and GAP. Furthermore, in contrast to the classical case, they are not determined by $m-n$. For example, there are virtual knots with trivial GAP but nontrivial $U_q(\mathfrak{gl}(2|2))$ and $U_q(\mathfrak{gl}(3|3))$ invariants. Silver and Williams proved that the GAP vanishes on all AC links. Our main result is a generalization of this theorem to almost classical tangles and the $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors. We prove that if $T$ is an almost classical tangle, then $\widetilde{Q}^{m|n}\circ Zh(T)$ is conjugate to $Q^{m|n}(T)$, with conjugation determined by an Alexander numbering of $T$.

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arXiv2025

$U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links

For links $L \subset Σ\times [0,1]$, where $Σ$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(Σ)]$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that the virtual genus of a virtual link $J$ is the smallest genus of all closed orientable surfaces $Σ$ on which $J$ can be represented by a link diagram on $Σ$. Here we generalize the CSW lower bound to all quantum supergroups $U_q(\mathfrak{gl}(m|n))$ with $m,n>0$. For $(m,n)=(1,1)$, the $U_q(\mathfrak{gl}(m|n))$ bound is the same as the CSW bound. However, changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m|n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As a first application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. As a second application, we prove that the Jaeger-Kauffman-Saleur invariant of a virtual knot is always realizable as the Alexander polynomial of an infinite cyclic cover of a knot complement in some $Σ\times [0,1]$, but is not always so on a surface of minimal genus. This is accomplished with a generalization of the $Zh$-construction, called the homotopy $Zh$-construction.

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arXiv2025

The Gordon-Litherland pairing and its many applications

Gordon and Litherland's paper $\textit{On the Signature of a link}$ introduced a bilinear form that simultaneously unifies both the quadratic forms of Trotter and Goeritz. This remarkable pairing of combinatorics and topology has had widespread application in low-dimensional topology. In this expository note, we give a picture proof (via Kirby diagrams) of their main result and discuss the numerous ways their theorem has been put to good use.

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arXiv2023

Characterization and Further Applications of the Bar-Natan Zh-Construction

Bar-Natan's Zh-construction associates to each $n$ component virtual link diagram $L$ an $(n+1)$ component virtual link diagram $Zh(L)$. If $L_0,L_1$ are equivalent virtual link diagrams, then $Zh(L_0),Zh(L_1)$ are equivalent as semi-welded links. The importance of the $Zh$-construction is that it unifies several classical knot invariants with virtual knot invariants. For example, the generalized Alexander polynomial of a virtual link diagram $L$ is identical to the usual multi-variable Alexander polynomial of $Zh(L)$. From this it follows that the generalized Alexander polynomial is a slice obstruction: it vanishes on any knot concordant to an almost classical knot. Our main result is a characterization theorem for the $Zh$-construction in terms of almost classical links. Several consequences of this characterization are explored. First, we give a purely geometric description of the $Zh$-construction. Secondly, the $Zh$-construction is used to obtain a simple derivation of the Dye-Kauffman-Miyazawa polynomial. Lastly, we show that every quandle coloring invariant and quandle 2-cocycle coloring invariant can be extended to a new invariant using the $Zh$-construction.

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arXiv2023

A geometric foundation of virtual knot theory

Virtual knots are defined diagrammatically as a collection of figures, called virtual knot diagrams, that are considered equivalent up to finite sequences of extended Reidemeister moves. By contrast, knots in $\mathbb{R}^3$ can be defined geometrically. They are the points of a space $\mathbb{K}$ of knots. The knot space has a topology so that equivalent knots lie in the same path component. The aim of this paper is to use sheaf theory to obtain a fully geometric model for virtual knots. The geometric model formalizes the intuitive notion that a virtual knot is an actual knot residing in a variable ambient space; the usual diagrammatic theory follows as in the classical case. To do this, it is shown that there exists a site $(\textbf{VK}, J_{\textbf{VK}})$ so that its category $\text{Sh}(\textbf{VK},J_{\textbf{VK}})$ of sheaves can be naturally interpreted as the ``space of virtual knots''. A point of this Grothendieck topos, that is a geometric morphism $\textbf{Sets} \to \text{Sh}(\textbf{VK})$, is a virtual knot. The virtual isotopy relation is generated by paths in this space, or more precisely, geometric morphisms $\text{Sh}([0,1]) \to \text{Sh}(\textbf{VK},J_{\textbf{VK}})$. Virtual knot invariants valued in a discrete topological space $\mathbb{G}$ are geometric morphisms $\text{Sh}(\textbf{VK},J_{\textbf{VK}}) \to \text{Sh}(\mathbb{G})$, just as classical knot invariants valued in $\mathbb{G}$ are continuous functions $\mathbb{K} \to \mathbb{G}$. The embedding of classical knots into virtual knots is also realized as a geometric morphism.

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arXiv2022

Algebraic concordance order of almost classical knots

Torsion in the concordance group $\mathscr{C}$ of knots in $S^3$ can be studied with the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$. Here $\mathbb{F}$ is a field of characteristic $χ(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine, who also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. While the concordance group $\mathscr{C}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathscr{VC}$. It is unknown if $\mathscr{VC}$ admits non-classical finite torsion. Here we define the virtual algebraic concordance group $\mathscr{VG}^{\mathbb{F}}$ for almost classical knots . This is an analogue of $\mathscr{G}^{\mathbb{F}}$ for homologically trivial knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. The main result is an algebraic classification of $\mathscr{VG}^{\mathbb{F}}$. A consequence of the classification is that $\mathscr{G}^{\mathbb{Q}}$ embeds into $\mathscr{VG}^{\mathbb{Q}}$ and $\mathscr{VG}^{\mathbb{Q}}$ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$, we give a generalization of the Arf invariant.

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arXiv2022

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 $S^*$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^*$-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.

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arXiv2022

Hyperbolic Knots and Torsion in Khovanov Homology

In this note, we show that if there is a knot in $S^3$ having $\mathbb{Z}_m$ torsion in its Khovanov homology, then there are infinitely many hyperbolic knots and infinitely many prime satellite knots having $\mathbb{Z}_m$ torsion in their Khovanov homology. As an application, we give the first known examples of hyperbolic knots and links with odd (and other non-$\mathbb{Z}_2$ torsion) in their Khovanov homology.

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arXiv2021

Milnor's concordance invariants for knots on surfaces

Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\barμ$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

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arXiv2020

Concordances to prime hyperbolic virtual knots

Let $Σ_0,Σ_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset Σ_0 \times [0,1]$ and $K_1 \subset Σ_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=Σ_1 \sqcup -Σ_0$ and $\partial A=K_1 \sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \subset Σ\times [0,1]$ admits a nontrivial decomposition along a separating vertical annulus that intersects $K$ in two points. Here we prove that every knot $K \subset Σ\times [0,1]$ is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in $Σ\times [0,1]$, we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby-Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in $Σ\times [0,1]$ is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in $3$-balls we call complementary tangles. Properties of complementary tangles are studied in detail.

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arXiv2020

Virtual concordance and the generalized Alexander polynomial

We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots.

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arXiv2018

Relating virtual knot invariants to links in $\mathbb{S}^{3}$

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J \sqcup K_1 \sqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot $\upsilon$ to a link $J \sqcup K$, where $J$ is fibered and $\text{lk}(J,K)=0$. Here we extend virtual covers to all multicomponent links $L=J \sqcup K$, with $K$ a knot. It is shown that an unknotted component $J_0$ can be added to $L$ so that $J_0 \sqcup J$ is fibered and $K$ has algebraic intersection number zero with a fiber of $J_0 \sqcup J$. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.

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arXiv2018

Signature and concordance of virtual knots

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and $ω$-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots. The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial $Δ(t)$ satisfying $Δ(1)=1$ occurs as the Alexander polynomial of an almost classical knot. In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, $ω$-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four. Table 2 at the end summarizes our findings.

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arXiv2017

Virtual knot cobordism and bounding the slice genus

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with 6 or fewer crossings.

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arXiv2017

Virtual Seifert Surfaces

A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $Σ\times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset Σ\times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in $Σ\times [0,1]$ are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing $F \subset Σ\times [0,1]$. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow-Tchernov-Vdovina.

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arXiv2017

Band-Passes and Long Virtual Knot Concordance

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group $\mathscr{VC}$. It is shown that for every concordance class $[K] \in \mathscr{VC}$, there is a $J \in [K]$ that is not band-pass equivalent to $K$ and an $L \in [K]$ that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that $v_{2,1}+v_{2,2} \pmod 2$ is a band-pass invariant but not a concordance invariant of long virtual knots, where $v_{2,1}$ and $v_{2,2}$ generate the degree two Polyak group for long virtual knots.

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arXiv2016

Virtual Covers of Links

We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in $\mathbb{S}^3$ come from taking $K$ to be in the complement $N$ of a fibered link $J$. If $J \sqcup K$ is invertible and $K$ is "close to" a fiber of $J$, then $\upsilon$ satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual covers of links $J \sqcup K$ where $J$ is not fibered but is virtually fibered (in the sense of W. Thurston).

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