Biquasiles and Dual Graph Diagrams
Abstract
We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.
2010 MSC: 57M27, 57M25
1 Introduction
The checkerboard colorings of a planar knot complement have long been used in knot theory, going back to papers such as [6]. From the undecorated checkerboard graph, one can reconstruct an unoriented alternating knot or link up to mirror image. In [5], signs are added to edges, enabling reconstruction of not necessarily alternating unoriented knots and links. In this paper, we introduce dual graph diagrams for oriented knots and links, a type of diagram using both of the (mutually dual) checkerboard graphs decorated with some edges having signs and others having directions, enabling recovery of arbitrary oriented knots and links. A similar graph without decorations was used in the study of the dimer model of the Alexander and twisted Alexander polynomials in [1].
Analogously to the construction of quandles and biquandles from a coloring scheme for arcs and semiarcs in oriented knot and link diagrams [3], we introduce a coloring scheme for vertices in a dual graph diagram. This coloring scheme motivates a new algebraic structure known as a biquasile with axioms determined by the dual graph Reidemeister moves. More precisely, the biquasile axioms are chosen so that biquasile colorings of dual graph diagrams are preserved faithfully by Reidemeister moves. This enables us to define biquasile counting invariants of knots and links and allows the introduction of enhancements of these invariants.
The paper is organized as follows. In Section 3 we introduce dual graph diagrams and the dual graph Reidemeister moves, as well as the reconstruction algorithm; a generic dual graph diagram presents a type of directed bivalent spatial graph, sometimes known as a magnetic graph [4, 7, 8]. We introduce a geometricstyle oriented link invariant defined from the dual graph representation, the dual graph component number, and show that this invariant is bounded above by the braid index. In Section 4 we introduce biquasiles, deriving the biquasile axioms from the dual graph diagram Reidemeister moves and introduce biquasile counting invariants for oriented knots and links. We provide several examples illustrating the computation of the invariant and collecting results. In Section 5 we turn our focus to the case of Alexander biquasiles, a type of biquasile structure defined on modules over the threevariable Laurent polynomial ring . We conclude in Section 6 with some questions for future research.
2 Acknowledgements
The authors would like to thank the referee for helpful comments and Jieon Kim and Philipp Korablev for catching mistakes in the first version of this paper.
3 Dual graph diagrams
We begin with a definition.
Definition 1.
Let be an oriented knot or link diagram. The dual graph diagram associated to has a vertex associated to each region of the planar knot complement and edges joining vertices whose regions are opposite at crossings. The edges are given directions or signs as pictured:
Conversely, given a pair of dual planar graphs (i.e., such that each region of contains a unique vertex of with adjacent regions in corresponding to adjacent vertices in ), becomes a dual graph diagram when we assign either a direction or or sign to each edge such that each pair of crossed edges has one signed edge and one directed edge.
Example 1.
The oriented knot diagram below has the corresponding dual graph diagram below.
We can understand the dual graph diagram as the result of superimposing the two checkerboard graphs associated to the knot or link diagram and decorating the edges to indicate orientation and crossing information.
A natural question is which dual graph diagrams present oriented knots or links. Let us consider the reconstruction algorithm for obtaining the original oriented knot or link diagram from its dual graph diagram. First, we note that each crossing of edges in the dual graph represents a crossing in the original link diagram, so we can start by putting a crossing at each edgecrossing with crossing information as determined by the direction and sign decorations carried by the edges:
We then observe that a dual graph diagram tiles the sphere with quadrilaterals with two corners given by vertices and two corners given by edge crossings as depicted:
Some such quadrilaterals may be degenerate, with boundary formed by a leaf and a loop:
Within each quadrilateral there is a unique path (up to planar isotopy) connecting the ends of the crossings; drawing these paths completes the diagram:
We note that for some dual graph diagrams, the orientations at the ends of a strand may disagree; in these cases, we can include bivalent vertices in the interior of such an arc, obtaining a bivalent spatial graph with sourcesink orientations; such diagrams are known as magnetic graphs in [4]:
We have the following:
Proposition 1.
Every oriented knot or link has a dual graph diagram with directed edges consisting of disjoint cycles.
Proof.
Simply put the knot or link diagram in closed braid form; the dual graph diagram then consists of disjoint directed cycles (where the braid being closed to form has strands) running vertically between the strands of the braid overlaid by locally horizontal signed edges. ∎
Example 2.
The figure eight knot has closed braid presentation with corresponding dual graph diagram as depicted:
Definition 2.
The dual graph component number of a knot or link is the minimal number of components in the directed subgraph of a dual graph diagram representing , i.e. the graph obtained from by deleting the signed edges, taken over the set of all dual graph diagrams representing .
In light of the proof of Proposition 1, we have the following easy observation:
Proposition 2.
The dual graph component number of an oriented link is bounded above by the braid index.
Recall that two oriented knot or link diagrams represent ambient isotopic knots or links if and only if they are related by a sequence of oriented Reidemeister moves:
Translating these Reidemeister moves into dualgraph format, we obtain the following moves:
Note that while dual graph Reidemeister moves do allow local vertexintroducing and vertexremoving moves, they do not allow strands of the knot or link to move past bivalent vertices; thus, the larger category of dual graph diagrams modulo dual graph Reidemeister moves is a slightly different category from the usual case of directed bivalent spatial graphs. In particular, classical knots and links form Reidemeister equivalence classes of the subset of dual graph diagrams whose reconstructions do not require bivalent vertices, with moves restricted to forbid local orientationreversing moves.
4 Biquasiles
Let be a set. We would like to define an algebraic structure on with operations and axioms motivated by the Reidemeister moves on dual graph diagrams in order to define knot and link invariants. Let us impose on two binary operations, and each defining a quasigroup structure on , i.e., such that each operation has both a right and left inverse operation (not necessarily equal), which we will denote respectively by and . More precisely, we have the following definition:
Definition 3.
Let be a set with binary operations satisfying
Then we say is a biquasile if for all we have
We will interpret these operations as the following vertex coloring rules at crossings in a dual graph diagram:
Definition 4.
Let be a biquasile and a dual graph diagram. Then any assignment of elements of to the vertices of satisfying the conditions above is an coloring of .
We want to establish axioms for our algebraic structure to ensure that for each valid coloring on one side of the move, there is a unique valid coloring on the other side.
Consider the two Reidemeister I moves involving positively signed crossings:
The condition we need is that for all there exist unique elements such that
That is, we must be able to solve the equations and for and respectively; this requires that has a left inverse operation and that has both a left and right inverse operation and ; provided that is a quasigroup under both and , these conditions are satisfied, with
Since the coloring rule at negative crossings is equivalent to
the conditions arising from the Reidemeister I moves involving negatively signed crossings are the same as those arising from the moves involving positively signed crossings.
Taking the direct Reidemeister II moves (i.e., the type II moves with both strands oriented in the same direction)
we get the requirement that for every , there exists a unique such that
then we have
as required by definition of the operations and . The other direct II move is similar.
The reverse Reidemeister II moves, i.e., the type II moves in which the strands are oriented in opposite directions,
require that for every there exists a unique such that
The existence of the right and left inverse operations for and means we can solve both equations for , obtaining in both cases:
as required.
The third Reidemeister move
yields the conditions
so we have
whence
must be satisfied for all . We can consider these to be somewhat complicated analogues of the distributive law. We can reformulate these slightly by defining functions
(1) 
then (i) and (ii) are the requirements that
(2) 
Example 3.
(Dehn Biquasile of an abelian group) Let be any abelian group; then is a biquasile under the operations
as we can easily verify:
and
Since the Dehn presentation relation abelianizes to , this type of biquasile can be understood as a generalization of the Dehn presentation of the knot group.
As with other algebraic structures, we can specify a biquasile structure on a finite set with a pair of matrices encoding the operation tables of the and operations. More precisely, the biquasile matrix of a biquasile of cardinality is the block matrix with entry where
Example 4.
Our python computations reveal biquasile structures on the set of three elements, including for instance
or more compactly
As with other algebraic categories, we have the following standard definitions.
Definition 5.
A biquasile homomorphism is a map between biquasiles such that and for all . A bijective homomorphism is an isomorphism, and an isomorphism is an automorphism.
Example 5.
There are four biquasile structures on the set , given by
Of these, there are two isomorphism classes, and . The 72 biquasiles of order three break down into 19 isomorphism classes, and our computations reveal 2880 biquasiles of order four comprising 177 isomorphism classes.
Definition 6.
A subbiquasile of is a subset which is itself a biquasile under the operations of ; for to be a subbiquasile we need closure of under and the right and left inverse operations of both. Say a biquasile is simple if it has no nonempty subbiquasiles.
Example 6.
The biquasile structure on with operation matrix
in Example 4 has one nontrivial subbiquasile, the singleton set . The biquasile with operation matrix
is simple since we have , and , so the closure of every nonempty subset of under the biquasile operations is all of .
Definition 7.
Let be a set and the set defined recursively by the following rules:

implies , and

implies the formal expressions and .
We call the elements of biquasile words in the generators . Then we define the free biquasile on to be the set of equivalence classes of biquasile words in modulo the relations determined by the biquasile axioms, e.g. , , etc. More generally, given a set of generators and a set of relations , i.e., equations of biquasile words, the biquasile presented by is the set of equivalence classes of biquasile words in modulo the equivalence relation generated by the biquasile axioms together with the relations in .
As in other universal algebraic systems, biquasiles presented by presentations related by the following Tietze moves are isomorphic:

Adding or removing a generator and relation of the form where is a word in the other generators not involving , and

Adding or removing a relation which is a consequence of the other relations.
An important example is the fundamental biquasile of an oriented knot or link, defined as follows:
Definition 8.
Let be a dual graph diagram and its set of vertices. Then the fundamental biquasile of is the biquasile with presentation where at each edge crossing in the dual graph diagram we have a relation as pictured.
Example 7.
The dual graph diagram below has the fundamental biquasile presentation listed.
By construction, we have the following:
Proposition 3.
The isomorphism class of the fundamental biquasile of an oriented link is a link invariant.
Definition 9.
In light of proposition 3, we define the fundamental biquasile of an oriented knot or link to be the fundamental biquasile of any dual graph diagram representing .
Definition 10.
Let be a biquasile and a dual graph diagram. The biquasile counting invariant of , denoted , is the cardinality of the set of colorings of , i.e., assignments of elements of to vertices in satisfying the conditions
Colorings of by can be interpreted as homomorphisms from the fundamental biquasile of to .
By construction, we have the following:
Theorem 4.
If is finite, then where is the set of vertices in . If and are related by Reidemeister moves, then and hence is an oriented link invariant.
Example 8.
Example 9.
Using our Python code, we selected three biquasiles of order 5,
and computed the counting invariant for all prime knots with up to eight crossings and prime links with up to seven crossings; the results are collected below. Nontrivial values (i.e., values differing from that of the unlink of the same number of components) are listed in bold.
5 Alexander Biquasiles
As with previous knotcoloring structures, we can consider the case of biquasile structures with linear operations, which we call Alexander biquasiles. We can think of Alexander biquasiles as generalizations of Dehn biquasiles.
Proposition 5.
Let . An module is a linear biquasile, also called an Alexander biquasile, under the operations
(3) 
Proof.
Example 10.
(Alexander biquasile structures on ) As a special case, one can consider groups and allow . For example, in there are seven unique (nonisomorphic) possibilities for assigning that satisfy (2):
d  n  s  

1  1  2  1 
2  1  1  1 
2  2  1  1 
1  1  1  2 
1  2  1  2 
2  1  2  2 
2  2  2  2 
We may also be interested in the number of configurations of that satisfy the conditions (2) (as well as forming a quasigroup), and how many of those configurations create nonisomorphic structures:
m  # configurations  # nonisomorphic 
2  1  1 
3  8  7 
4  8  7 
5  64  34 
6  8  7 
7  216  137 
8  64  33 
9  216  152 
10  64  34 
As an extension to this example, we consider a dual graph diagram and compute the number of colorings, , of by the biquasile with the operations
which correspond to the selection of and . Consider the following knot and corresponding dual graph diagram , where we have labeled the nodes for reference.
This dual graph diagram yields the following four equations (along with their equivalent forms):
. 
Rewriting as homogeneous equations and putting in matrix form (with respect to the vector ), these give:
The solution space is thus twodimensional, and so we have .
Example 11.
As in the case of quandles and biquandles [3], we can define a modulevalued Alexander invariant of oriented knots and links by considering the Alexandrization of the fundamental biquasile of an oriented knot or link, i.e. the fundamental biquasile written as an Alexander biquasile. More precisely, the kernel of the coefficient matrix of the homogeneous system of linear equations over is an module valued invariant of oriented knots and links analogous to the classical Alexander invariant; from it, we can derive polynomialvalued invariants via the Gröbner basis construction described in [2].
For instance, the knot in example 10 has Alexander biquasile given by the kernel of the matrix below with entries in :
These invariants will be the subject of another paper.
6 Questions
We end with a few collected questions for future research.

What, if anything, is the relationship between biquasiles and biquandles?

What enhancements of the biquasile counting invariants can be defined?

What kinds of categorifications of biquasiles and their invariants are possible?
References
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