Biquandle Virtual Bracket Matrix

• The biquandle virtual bracket matrix is an endpoint-refined invariant that distinguishes virtual knotoids by sorting bracket evaluations according to tail and head colors.
• It combines biquandle coloring rules with a virtual bracket state sum, transitioning from simple coloring counts to detailed polynomial invariants.
• The invariant enhances previous methods by preventing cancellation effects and capturing richer structural data through endpoint decomposition.

The biquandle virtual bracket matrix is an endpoint-refined invariant of virtual knotoids obtained by combining biquandle colorings with a biquandle virtual bracket state sum and then organizing the resulting bracket values by the colors at the tail and head. In the formulation introduced in “Biquandle Virtual Brackets and Virtual Knotoids” (Gügümcü et al., 10 Jul 2025), it is the most refined invariant in a hierarchy that begins with coloring counts and culminates in a matrix whose entries record endpoint-conditioned bracket data. Its defining feature is that it is sensitive not only to the existence of colorings and not only to the multiset of bracket evaluations, but also to how those evaluations are distributed across endpoint color pairs.

1. Position in the invariant hierarchy

A virtual knotoid is a knotoid diagram in $S^2$ with both classical and virtual crossings, considered up to the extended Reidemeister moves and isotopy. Unlike a knot diagram, it is an open immersed curve with two distinguished endpoints, the tail and the head. The endpoint structure is essential: the tail and head survive the equivalence relation, and there is no Reidemeister move that changes their colors in a biquandle coloring (Gügümcü et al., 10 Jul 2025).

Within this setting, the paper organizes its invariants in a strict refinement chain. The basic level is the total number of $X$-colorings. The next level is the biquandle counting matrix $\mathcal{M}_X(K)$, whose $(i,j)$-entry counts colorings with fixed tail color $x_i$ and head color $x_j$. Bracket enhancement then adds a state-sum value to each coloring, first as a multiset $\Phi_X^{\beta,M}(K)$, then in polynomial form $\Phi_X^\beta(K)$ when the coefficient ring is a number ring, then as endpoint-refined multisets, and finally as the biquandle virtual bracket matrix $\mathcal{M}_X^\beta(K)$, which packages those endpoint-refined evaluations into a single matrix (Gügümcü et al., 10 Jul 2025).

Invariant	What it retains	Endpoint refinement
Total number of $X$-colorings	Coloring count only	No
$X$0	Counts by tail/head colors	Yes
$X$1, $X$2	Bracket values over all colorings	No
Endpoint-refined multiset family	Bracket values by tail/head colors	Yes
$X$3	Endpoint-indexed polynomial entries	Yes

The paper explicitly states that the biquandle virtual bracket matrix is a proper enhancement of all the other invariants introduced there (Gügümcü et al., 10 Jul 2025). A plausible implication is that endpoint sensitivity is not a secondary bookkeeping device but the structural reason the invariant is stronger than the corresponding counting and bracket-polynomial data.

2. Algebraic input: biquandles and virtual brackets

The matrix invariant depends on two layers of algebraic input: a finite biquandle $X$4 and a biquandle $X$5-virtual bracket $X$6. The biquandle supplies the coloring rules at classical crossings. In the notation used in the paper, the two operations are written as $X$7 and $X$8, encoding the algebraic effect of oriented crossings on semiarc colors (Gügümcü et al., 10 Jul 2025).

Let $X$9 be a commutative, unitary ring. A biquandle $\mathcal{M}_X(K)$0-virtual bracket consists of six coefficient maps

$\mathcal{M}_X(K)$1

and two distinguished elements

$\mathcal{M}_X(K)$2

For $\mathcal{M}_X(K)$3 and $\mathcal{M}_X(K)$4, the paper writes $\mathcal{M}_X(K)$5 (Gügümcü et al., 10 Jul 2025).

These data are constrained by equations $\mathcal{M}_X(K)$6–$\mathcal{M}_X(K)$7 of Definition 4.1. The first eight relations govern Reidemeister I and II behavior: $\mathcal{M}_X(K)$8

$\mathcal{M}_X(K)$9

$(i,j)$0

$(i,j)$1

$(i,j)$2

$(i,j)$3

$(i,j)$4

$(i,j)$5

For all $(i,j)$6, the remaining fifteen equations compare the three-state expansions on the two sides of Reidemeister III. They involve the transformed color pairs $(i,j)$7, $(i,j)$8, and $(i,j)$9, and they are exactly the algebraic constraints ensuring invariance of the state sum under the generalized Reidemeister moves (Gügümcü et al., 10 Jul 2025).

At the level of local skein data, the virtual bracket differs from the ordinary two-smoothing bracket by admitting three local replacements at each classical crossing: vertical smoothing, horizontal smoothing, and virtual smoothing. The coefficient used is color-dependent. At a positive crossing the three weights are $x_i$0, $x_i$1, and $x_i$2; at a negative crossing they are $x_i$3, $x_i$4, and $x_i$5 (Gügümcü et al., 10 Jul 2025).

3. Construction of the biquandle virtual bracket matrix

Fix a finite biquandle

$x_i$6

and an oriented virtual knotoid diagram $x_i$7. An $x_i$8-coloring labels each semiarc by an element of $x_i$9 subject to the biquandle relations at each classical crossing; virtual crossings do not change labels. Equivalently, colorings are homomorphisms

$x_j$0

where $x_j$1 is the fundamental biquandle of $x_j$2. For fixed endpoint colors $x_j$3, the subset

$x_j$4

consists of those homomorphisms sending the tail semiarc generator to $x_j$5 and the head semiarc generator to $x_j$6 (Gügümcü et al., 10 Jul 2025).

For a fixed coloring $x_j$7, the colored diagram $x_j$8 is expanded into states by resolving every classical crossing in one of three ways. If $x_j$9 has $\Phi_X^{\beta,M}(K)$0 classical crossings, then there are $\Phi_X^{\beta,M}(K)$1 states. Each state is a disjoint union of one open component and possibly some closed components, possibly with virtual crossings remaining (Gügümcü et al., 10 Jul 2025).

If $\Phi_X^{\beta,M}(K)$2 is such a state and $\Phi_X^{\beta,M}(K)$3 is the number of components of $\Phi_X^{\beta,M}(K)$4, its contribution is

$\Phi_X^{\beta,M}(K)$5

where $\Phi_X^{\beta,M}(K)$6 is the writhe of $\Phi_X^{\beta,M}(K)$7. Summing over all states gives the bracket value of the coloring: $\Phi_X^{\beta,M}(K)$8 Collecting these values over all colorings yields the multiset

$\Phi_X^{\beta,M}(K)$9

If $\Phi_X^\beta(K)$0 is a number ring, this multiset is encoded as

$\Phi_X^\beta(K)$1

The endpoint-refined multiset family is defined by restricting to $\Phi_X^\beta(K)$2, and its polynomial representative is

$\Phi_X^\beta(K)$3

The biquandle virtual bracket matrix is then

$\Phi_X^\beta(K)$4

Thus each matrix entry is not a count but an aggregate of bracket evaluations attached to those colorings having a specified tail color and head color (Gügümcü et al., 10 Jul 2025).

The distinction between this matrix and the underlying multiset or polynomial is structural. The multiset forgets endpoints; the matrix remembers where each bracket value came from in the endpoint-color decomposition. Since knotoids are open objects, that additional stratification is mathematically natural rather than merely notational.

4. Two meanings of “bracket matrix” in the literature

The phrase “bracket matrix” has two distinct uses in the biquandle literature. One use is coefficient storage: a matrix records the values of coefficient functions such as $\Phi_X^\beta(K)$5 and $\Phi_X^\beta(K)$6. The other use is matrix-valued invariant: a matrix records endpoint-indexed invariant data. The biquandle virtual bracket matrix belongs to the second class.

Earlier work on biquandle brackets for knots and links encoded finite bracket coefficients by block matrices such as $\Phi_X^\beta(K)$7 (Nelson et al., 2015), and the virtual-bracket extension encoded six coefficient families by a block matrix

$\Phi_X^\beta(K)$8

for a finite biquandle (Nelson et al., 2017). In classical knotoid theory, “Biquandle Brackets and Knotoids” introduced a matrix-valued invariant $\Phi_X^\beta(K)$9 whose $\mathcal{M}_X^\beta(K)$0-entry collects bracket contributions from colorings with fixed tail and head colors (Gügümcü et al., 2019). The 2025 virtual-knotoid construction extends that endpoint-refined philosophy to the virtual setting and uses virtual smoothings in the state sum (Gügümcü et al., 10 Jul 2025).

Matrix notion	Data stored	Representative source
$\mathcal{M}_X^\beta(K)$1	Classical bracket coefficients	(Nelson et al., 2015)
$\mathcal{M}_X^\beta(K)$2	Virtual bracket coefficients	(Nelson et al., 2017)
$\mathcal{M}_X^\beta(K)$3	Knotoid endpoint-refined bracket invariant	(Gügümcü et al., 2019)
$\mathcal{M}_X^\beta(K)$4	Virtual-knotoid endpoint-refined bracket invariant	(Gügümcü et al., 10 Jul 2025)

This distinction matters because the matrix in $\mathcal{M}_X^\beta(K)$5 is not part of the input data. It is the invariant itself. A plausible consequence is that confusion can arise if one identifies the six-block coefficient matrix of a virtual bracket with the endpoint-refined invariant matrix; the former parameterizes a bracket theory, while the latter packages the values of that theory on a virtual knotoid.

5. Endpoint sensitivity and enhancement strength

The defining reason the biquandle virtual bracket matrix is stronger than the preceding invariants is that it refines bracket information by endpoint colors. The paper emphasizes that it does not merely count colorings and does not merely collect all bracket values into one polynomial; it sorts bracket evaluations according to the colors at the tail and head. Because knotoids have endpoints, this endpoint-sensitive packaging is stronger than the more familiar counting and polynomial invariants (Gügümcü et al., 10 Jul 2025).

This endpoint-refined logic already appeared in the classical knotoid setting. “Biquandle Brackets and Knotoids” showed that the matrix-valued bracket enhancement there is stronger than both the coloring matrix and the ordinary biquandle bracket polynomial, with examples in which knotoids sharing the same counting matrix and the same bracket polynomial are separated by their bracket matrices (Gügümcü et al., 2019). The 2025 virtual-knotoid paper makes the corresponding claim at the virtual level in a sharper form: the biquandle virtual bracket matrix is a proper enhancement of all the other invariants introduced in that paper (Gügümcü et al., 10 Jul 2025).

The computational mechanism behind this strength is straightforward. A coloring contributes to exactly one endpoint sector $\mathcal{M}_X^\beta(K)$6, and the state-sum value associated to that coloring is recorded only in the corresponding entry of the matrix. Any cancellation or coincidence that occurs after forgetting endpoints may therefore disappear once the data are stratified by tail-head color pair. This suggests that endpoint decomposition is not only a refinement of information but also a way of preventing accidental identifications produced by global aggregation.

The paper also notes that examples encode the coefficient maps $\mathcal{M}_X^\beta(K)$7 as a single block matrix. For $\mathcal{M}_X^\beta(K)$8, one Section 4 example is presented as a $\mathcal{M}_X^\beta(K)$9 block matrix together with

$X$0

illustrating the finite-data format used in explicit computations (Gügümcü et al., 10 Jul 2025). The invariant matrix $X$1 is then built from state sums computed with such coefficient data.

6. Related extensions and adjacent theories

The biquandle virtual bracket matrix sits within a broader family of color-dependent skein constructions. “Trace Diagrams and Biquandle Brackets” developed a recursive calculus for ordinary biquandle brackets, using trace diagrams and extra adequacy conditions to simplify computations; that work is directly relevant to the classical part of any bracket-state method, but it does not define a virtual biquandle bracket (Nelson et al., 2017). “Biquandle Bracket Quivers” extended the bracket formalism to quiver-valued enhancements and stated that the construction applies to virtual knots and links by ignoring virtual crossings in determining colorings (Falkenburg et al., 2021).

Several later theories modify the local coefficient package rather than the endpoint organization. “Kaestner Brackets” introduced parity-sensitive coefficient families $X$2 on parity biquandles for oriented virtual knots and links (Kobayashi et al., 2019). “Picture-valued biquandle bracket” enlarged the coefficient system to six tables $X$3 and produced picture-valued invariants for classical and virtual knots (Ilyutko et al., 2017). “Biquandle Power Brackets” replaced the constant loop value by a subset-dependent function $X$4, but explicitly stated that these invariants do not extend to virtual knots by merely ignoring virtual crossings (Gügümcü et al., 2024). “Psyquandle Brackets” carried the matrix-style bracket philosophy to singular knots and pseudoknots, using four coefficient families $X$5 (Nelson et al., 18 Aug 2025).

These developments indicate that the phrase “biquandle virtual bracket matrix” should be interpreted narrowly. In the specific virtual-knotoid sense of (Gügümcü et al., 10 Jul 2025), it denotes the endpoint-refined invariant matrix $X$6. In earlier literature, closely related phrases often referred instead to coefficient tables such as $X$7 or $X$8. A plausible historical reading is that the 2025 construction synthesizes two earlier strands: virtual biquandle brackets on the one hand and matrix-valued endpoint refinements for knotoids on the other.