Taycher, Benjamin_Senior Thesis 2026

Holomorphs, Yang-Baxter, and Braids: A Through-Line Benjamin Taycher Senior Thesis | 2026

Holomorphs, Yang-Baxter, and Braids: A Through-line Benjamin Taycher 2026

Abstract Mathematical braids have wide-reaching applications from fluid dynamics to cryptography and are defined the same way as real-life braids found in hair and challah, just with infinitely thin strands.

In particular, a braid is composed of several curves embedded into three dimensional space such that they do not intersect and have fixed points. These are often represented in two dimensions by projecting and then marking one strand going above another with the top strand being solid while the bottom is broken, as shown:

Braids are considered equivalent if one can be turned into the other by moving strands while leaving endpoints fixed and not letting strands intersect, so the two above braids are in fact equivalent.

One of the main problems in braid theory is telling them apart, since braids that look different can in fact be equivalent through a series 1

of strand movements. In general, telling braids apart is difficult, and so we use braid invariants to accomplish the task. These are certain values we assign braids that are the same for equivalent braids, so if two braids have a different value for an invariant, they must be different. The contribution of this project is to apply the connection between the holomorph of a group and the set-theoretic Yang-Baxter equation to produce new braid invariants. The concrete examples of this general approach that are investigated are the invariants arising from left- and right-regular subgroups of arbitrary groups and from any subgroups of abelian groups.

1

Introduction

A mathematical braid generalizes the general concept of a braid with n strands which are attached at both ends and move over and under each other between those ends. For this paper, each strand will be considered to start at the “top” and end at the “bottom”. To define this formally, let a braid be a collection of strands labeled by i, for i ∈ {1, . . . , n}, where each strand is the image of a continuous function fi : [0, 1] → R3 satisfying the following four properties: 1. fi (a) = f j (b) =⇒ i = j, a = b 2. fi (0) = (i, 0, 0) 3. fi (1) = ( j, 1, 0) with j ∈ {1, . . . n} 4. a < b =⇒ π2 ( fi (a)) > π2 ( fi (y)), where π2 is the projection function to the second element of the output Property 1 requires that none of the strands occupy the same location in R3 . Properties 2 and 3 fix the endpoints of each of the strands, fixing the starts of each of the strands on the x-axis at the x-coordinate matching the number of the strand and the ends of each strand at the line y = 1 with x-coordinates a permutation of the starting coordinates. Property 4 then implies that each strand is constantly moving down in the y direction.

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We want to consider braids the same if they can be continuously deformed into each other while keeping the ends fixed and not moving strands through each other, which is formally termed considering them under “ambient isotopy”. When a braid is mentioned for the rest of this paper, what is being referred to are equivalence classes of braids.

Braids have many applications, starting with their use in describing their sister objects, knots, and reaching as far as fluid dynamics considering the chaotic behavior of fluid with multiple stirrers [21].

For knots in particular, Alexander’s theorem states that any non-wild knot (one which can be represented with a finite number of straightedge pieces) is equivalent to a knot such that if the knot is assigned a direction, then all parts of the knot are moving in the same direction relative to some fixed axis [1]. Let this axis be the z-axis. Then take the half-plane with positive x-coordinate and y-coordinate 0 and split the knot along that plane. Then this knot can be represented as a braid where each strand is given by an intersection point of the knot with the half-plane, in addition to strands that join the tops and bottoms of certain strands. Hence studying braids gives significant insight into the world of knots.

For any of these applications, and for braid theory itself, we must be able to tell if two braids are equivalent and distinguish them if not. For this we use two key simplifications which keep all of the information of the underlying braids. First we represent braids in R2 rather than R3 . Second we reduce the ambient isotopies to consider from uncountably infinitely many to two.

Before reducing to two dimensions, note that all of the strands must lie from y = 0 to y = 1. Though the x-coordinates can be arbitrarily large for all points on the strands other than the endpoints, we can preform an ambient isotopy to bring all of the strands to have xcoordinates from 1 to n at all points by linearly contracting the braid in the x direction and then realigning the endpoints to match their original locations. The resulting braid then has all points between y = 0 and y = 1 and between x = 1 and x = n. Then to reduce the problem to two dimensions, project each braid down to R2 , specifically the xy-plane.

It is possible to move the strands such that at most two share the 3

same xy-coordinate and that shared coordinates are points and not entire segments. These points in the plane to which two strands are projected are called intersections, since that is what they look like, even though the strands themselves do not intersect, as on the left of Figure 1. At each such intersection, one of the strands must have been going over the other before projection with the other going under, which is distinguished by drawing the over strand as a solid line while adding a break to the under strand. The resulting projection is called a braid diagram, as is shown on the right of Figure 1

Figure 1: Braids without and with over-under distinction.

The second simplification is ascribed to Kurt Reidemeister, who in 1927 proved that any ambient isotopy of knot diagrams (defined the same way as braid diagrams, but without endpoints) can be described as a sequence of the three Reidemeister moves, as shown in Figure 1 [22].

Figure 2: Reidemeister moves, from left to right, numbers 1, 2, and 3.

The first of these moves can never happen in a braid diagram since each strand has monotonically increasing y-coordinate, and so can never cross over itself, but the second and third moves are still possible. Hence the theorem can be transferred to apply to braids simply by removing the first move and only considering the second and third.

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These two simplifications reduce the problem of telling whether two braids are equivalent to finding a sequence of Reidemeister moves that transform one to the other or proving that such a sequence does not exist. Still, this leaves an infinite number of sequences to consider, and to prove that two braids are not equivalent we must prove that none of the possible sequences take one to the other. To avoid comparing all infinite permutations of moves, mathematicians have resorted to braid invariants, properties of braids that are unchanged by ambient isotopy, since then if two braids have different values for an invariant property, they must be different. Since all ambient isotopies can be expressed as sequences of Reidemeister moves, though, the task is already significantly simplified, since not all ambient isotopies must be considered, but all that needs to be done is to prove that both Reidemeister moves applicable to braids do in fact leave the property unchanged. A simple invariant is just considering the final positions of each strand, since both Reidemeister moves leave the endpoints fixed, but there are still many braids that are different and yet have the same endpoints, so it is important to look for more invariants to be able to split braids into as many equivalence classes as possible. The rest of this paper will thus be dedicated to generating invariants through a process that starts at a seemingly completely unrelated construct of groups and their holomorphs but ends with the generation of a family of powerful braid invariants.

2

Goal

The purpose of this paper is to give a method for generating braid invariants. We will proceed by starting with a group, extracting information about its relation with its automorphisms to generate a solution to the set-theoretic Yang-Baxter equation, and then using that solution to generate a braid invariant. While the individual steps have been described in previous papers, the contribution of this manuscript is to combine all of the steps to create a route from groups all the way to braid invariants.

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3

Holomorph of a Group

3.1

Definition of the Holomorph

The first step of the process is to take the holomorph of a group, but what is the holomorph? It is, in essence, a way of generalizing the concept of affine transformations from vector spaces to groups and giving these transformations a group structure. Recall that an affine transformation is a linear transformation followed by a translation by a vector. The translation transforms into a group context simply by adding an element of the group. Then the linear transformations are those that alter the vector space but keep it fundamentally the same since they preserve vector sums (the transformed sum of two vectors before the transformation and the sum of the two transformed vectors are always equal). When moving into the context of groups, preserving sums translates to preserving the group operation. Hence the equivalent of a linear transformation is a function A : G → G, with G a group, such that A(gh) = A(g)A(h), which is simply a group homomorphism from G to itself. If the linear transformations are then restricted to be invertible, then they must be bijective, so the group homomorphism then becomes an isomorphism.

Definition 1. Let G be a group. Then an automorphism of G is an isomorphism from G to itself.

Furthermore, since the goal is to turn these affine transformations into a group, let us first check that the invertible linear transformations, the automorphisms, form a group themselves.

Proposition 1. Let G be a group and let Aut(G) be the set of automorphisms of G. Then Aut(G) under composition forms a group, also denoted Aut(G).

Proof. Aut(G) is closed since composing two isomorphisms gives an isomorphism, and the domain and codomain are still both G as they are for every automorphism in Aut(G). The identity I ∈ Aut(G) is then simply given by the identity isomorphism of G, which always exists, and inverses of isomorphisms are given by their inverses as functions, which must exist and be isomorphisms by the definition 6

of an isomorphism. Lastly, Aut(G) is associative by the associativity of function composition, and so Aut(G) forms a group.

The holomorph combines these automorphisms with the addition of an element of G, corresponding to vector addition. Then each element of the holomorph, denoted Hol(G), can be represented as a pair (g, A), where g ∈ G and A ∈ Aut(G). Then to compose these elements, follow the composition of affine transformations: (h, B)(g, A) means that A is applied first, then g is added, then B is applied, and finally h is added. Note that since B is applied after A, the combined transformation composes, and since B is applied after g, the resulting added element is B(g), which gives the following definition for composition: (h, B)(g, A) = (hB(g), BA) This notion can be formalized with the semidirect product of groups: Definition 2. Let G and H be groups and let φ : H → Aut(G) be a homomorphism. Then the (external) semidirect product of G and H (with the homomorphism φ ), G ⋊φ H is defined to be the set {(g, h)|g ∈ G, h ∈ H}, with multiplication defined by (g1 , h1 )(g2 , h2 ) = (g1 φ (h1 )(g2 ), h1 h2 ), where φ (h1 ) may be written as φh1 to avoid confusion, giving (g1 φh1 (g2 ), h1 h2 ).

Proposition 2. For any groups G and H with homomorphism φ : H → Aut(G), the group G ⋊φ H is a group under the above defined multiplication rule.

The holomorph follows the same multiplication rule as the semidirect product of G and Aut(G), and the homomorphism φ is simply the identity since Aut(G) is already the group of automorphisms of G, so we have the following: Definition 3. Let G be a group. Then the holomorph of G is Hol(G) := G ⋊I Aut(G), where I : Aut(G) → Aut(G) is the identity homomorphism and is accordingly often omitted.

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Example 1. Consider D3 = {e, r, r2 , s, sr, sr2 } to be the group of symmetries of a triangle under composition, where r represents a rotation and s represents a reflection over a set line of symmetry, which can be presented as {r, s|r3 = s2 = e, rs = sr−1 }. D3 has two generators, namely r and s, and any automorphism must map them to elements with matching order, so r is mapped either to r or r2 and s is mapped to s, sr, or sr2 . Then all automorphisms are obtained by extending these mappings by using the composition preservation of automorphisms. Hence Aut(D3 ) = {I, (r 7→ r2 ), (s 7→ sr), (r 7→ r2 , s 7→ sr), (s 7→ sr2 ), (r 7→ r2 , s 7→ sr2 )}, and so Hol(D3 ) contains all pairs of elements of D3 and Aut(D3 ), and so is a 36 element group.

3.2

Regular Subgroups

The holomorph is an object used in its own right, but many of its applications, including using it as a solution to the set-theoretic YangBaxter equation require what is called a regular subgroup of the holomorph. If a group is taken to act on G, permuting its elements, then a regular subgroup is one where each mapping p(x) = y corresponds to a unique permutation p, but to consider such subgroups, we must first rephrase the notion of the holomorph in the language of permutations.

Let us first begin by defining a permutation group: Definition 4. A permutation group on a set X, denoted Perm(X) is the set of permutations of elements of X (bijective functions from X to itself), which is a group under composition. When X = {1, . . . , n} for some n ∈ N, this group is called the symmetric group on n elements and is denoted Sn .

Note that the structure of the permutation group is only dependent on |X|, not on the elements themselves, so all permutations groups on the same number of elements are isomorphic to each other, and in particular isomorphic to Sn for |X| finite. Then |Sn | = n!, simply because for each of n elements a distinct choice from n elements must be made.

Let H ≤ Perm(X), so the following definitions describe how H acts on X by permutation.

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Definition 5. H is transitive if for all x, y ∈ X, there exists h ∈ H such that h(x) = y, so for every pair x, y ∈ X, one can be brought to the other by a permutation in H. Definition 6. H is fixed-point free if for all h ∈ H, x ∈ X, h(x) = x =⇒ h = I, where I is the identity permutation, so distinct elements of H must act differently on every element of X. These two properties, as well as the easy-to-check property that |H| = |X| are related to each other by the following theorem.

Proposition 3. [5] Let H ≤ Perm(X). Then any two of the following imply the third: • H is transitive • H is fixed-point free • |H| = |X| Definition 7. Let H ≤ Perm(X). Then H is regular if it is transitive and fixed-point free.

Regular subgroups give a natural way to embed a group in a permutation group, which gives the basis for another definition of the holomorph.

Definition 8. Let G be a group and Perm(G) be the group of permutations of G as a set. Then λ : G → Perm(G) is defined with g 7→ λg , where λg is the permutation with h 7→ gh. Essentially, each λg multiplies each element of G by the given element g, which is always bijective and so always a permutation, and this is called the left-regular representation.

Furthermore, λ is an injective homomorphism from G to Perm(G).

We have that |λ (G)| = |G| which is the size of the permuted set, G, and that element multiplication in groups is transitive so by 3, λ (G) is in fact a regular subgroup of Perm(G) fitting its name as the leftregular representation.

Then recall that the holomorph essentially extended the concept of an affine transformation to a group context. While defining the 9

holomorph through the semi-direct product, an affine transformation was thought of as an automorphism followed by a translation.

Alternatively, one may think of affine transformations as those that transform element addition to other element addition. This follows from the semi-direct product definition as (g, A)(x) = gA(x) and (g, A)(x + y) = gA(xy) = gA(x)A(y), since A is a homomorphism, so the addition of y to x simply shifts the output by A(y), which is a fixed element. Since all affine transformations we are concerned with are bijective, they can all be viewed elements of Perm(G) acting on elements of G embedded in that same permutation group through λ . Then the elements of the holomorph must be precisely those elements σ ∈ Perm(G) such that σ λg = λh σ : adding g and then applying σ is the same as applying σ and then adding h, for some g, h ∈ G. Then multiplying by σ −1 on the right, we have σ λg σ −1 = λh σ σ −1 = λh , which reduces the requirement to be that for any g ∈ G, σ λg σ −1 ∈ λ (G), or more succinctly, σ λ (G)σ −1 ⊆ λ (G).

One may notice that this states that the holomorph contains all elements of Perm(G) such that λ (G) is a normal subgroup of it.

Definition 9. The normalizer of a subgroup H ≤ G, denoted NormG (H) is the largest subgroup of G such that H remains fixed under conjugation of all of its elements, and so it is the largest subgroup of G that has H as a normal subgroup. This subgroup can be expressed as set {g ∈ G|∀h ∈ H, ghg−1 ∈ H} Then a second definition of the holomorph follows: Definition 10. The permutation definition of the holomorph, also denoted Hol(G), is given by NormPerm(G) (λ (G)).

Theorem 4. [13] Let G be a group. Then Norm(λ (G)) ∼ = Hol(G) = G ⋊ Aut(G) and so the two definitions of the holomorph coincide.

Proof. Take some n ∈ Norm(λ (G)). Then by the definition of the normalizer, we have that nλg n−1 = λh and so nλg = λh n. Then let us first consider the case where n stabilizes the identity of G, so then n(g) = nλg (eG ) = λh n(eG ) = λh (eG ) = h. Hence nλg = λh n = λn(g) n 10

and similarly, λn(gh) n = nλgh = nλg λh = λn(g) nλh = λn(g) λn(h) n = λn(g)n(h) n, so we have n(gh) = n(g)n(h), so n is a permutation of G that preserves the preserves the group operation, so n must be an automorphism. Furthermore, all automorphisms of G normalize λ (G), so the subgroup of Hol(G) that fixes the identity is in fact equal (not just isomorphic) to Aut(G). Then since λ (G) is a transitive subgroup of Hol(G) and the only one of its elements that fixes the identity is λe , we have λ (G) ◁ Norm(λ (G)) = Hol(G) by definition, λ (G) transitive, and λ (G) ∩ Aut(G) = {e}, so Hol(G) ∼ = λ (G) ⋊ Aut(G).

4

The Yang-Baxter Equation

In this section, we will first define skew left braces and create them from the regular subgroups of holomorphs described in the previous section. Then these skew left braces will be applied to give a solution to the set-theoretic Yang-Baxter equation (which will be referred to simply as the Yang-Baxter equation).

4.1

Skew Left Braces

Definition 11. A skew left brace is an algebraic structure combining two group structures on a set. A skew left brace (G, ·, ◦) has operations · and ◦, both of which give a group structure on the underlying set and interact via the following equation: a ◦ (bc) = (a ◦ b)a−1 (a ◦ c)

(1)

Let G be a group and H ≤ Hol(G) be a regular subgroup of the holomorph with elements written in the form (g, A), where g ∈ G and A ∈ Aut(G). Since H is regular, it must be transitive, so for every x, y ∈ G, there must be an element (g, A) ∈ Hol(G) such that (g, A)(x) = y. Setting x = e, we have (g, A)(e) = gA(e) = g, so for every y ∈ G, there must be an element of H of the form (y, A). This gives at least |G| elements already since there are |G| unique first entries, and since a regular subgroup must have exactly |G| elements, these are all the elements of H, so H has exactly one element with 11

each first entry in G. As such, every element g ∈ G can be associated uniquely with an automorphism fg by taking the automorphism in its element of H. Theorem 5. Let · be the group operation of G and define ◦ by a◦b = a fa (b). Then (G, ·, ◦) is a skew left brace with (G, ◦) ∼ = H, which is well-defined since fa is unique as each a appears exactly once in the first entry of the semi-direct product due to regularity.

Proof. Clearly, · gives a group structure on G since · is the binary operation of G as a group. Then since fa is unique as proven earlier, fa (b) is well-defined, so a fa (b) ∈ G, so G is closed under ◦.

Then take (a, fa )(b, fb ) as a product in H. Then (a, fa )(b, fb ) = (a fa (b) fa fb ), so taking the first entry of both the input and output gives a ◦ b = a fa (b) to be a group isomorphic to H, since the association of a and fa is unique by regularity.

Lastly, a ◦ (bc) = a fa (bc), which, since fa is an automorphism and so a homomorphism, is equal to a fa (b) fa (c) = a fa (b)a−1 a fa (c) = (a ◦ b)a−1 (a ◦ c), proving that · and ◦ interact as needed to form a brace, completing the proof.

Example 2. Following the earlier example of Hol(D3 ), take the regular subgroup H = {(e, I), (r, s 7→ sr2 ), (r2 , s 7→ sr), (s, r 7→ r2 ), (sr, [r 7→ r2 , s 7→ sr2 ]), (sr2 , [r 7→ r2 , s 7→ sr])}, which gives the following multiplication table for ◦: ◦ e r r2 s sr sr2

e e r r2 s sr sr2

r r r2 e sr2 s sr

r2 r2 e r sr sr2 s

s s sr sr2 e r r2

sr sr sr2 s r2 e r

sr2 sr2 s sr r r2 e

As seen, (G, ◦) is also isomorphic to D3 , though it is not equal to (G, ·). In this case specifically, a ◦ b = ba, though this is not always the case.

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4.2

The Set Theoretic Yang-Baxter Equation

The Yang-Baxter equation originates from statistical mechanics, describing how particles change states as they switch positions along a line, but it has since been generalized to a variety of applications, notably including knot and braid theory. The (set theoretic) equation is as such: (S × I)(I × S)(S × I) = (I × S)(S × I)(I × S)

(2)

Here, S is a function S : X × X → X × X for some set X, I is the identity function on X, and the × operation between them signifies that S or I acts on the first one or two elements of the input, and then the other acts on the last one or two elements of the input.

Hence each function S × I or I × S is a function from X 3 to itself.

The set-theoretic aspect of the equation refers to the fact that X is a set and the functions all act on X and its direct products, whereas the non-set-theoretic version typically uses vector spaces and their tensor products as the domains and codomains. In this context, S is a solution to the Yang-Baxter equation and we call it a switch.

This equation can be imagined pictorially with braids, where strand i going over strand i + 1 corresponds to applying S to the values assigned to those strands, where the Yang-Baxter equation specifies the third Reidemeister move as shown in Figure 3.

Figure 3: The Yang-Baxter equation pictured as the third Reidemeister move for a braid.

At each level where there is a crossing, S is applied to the two crossing strands and I is applied to the untouched strand. Hence the 13

Yang-Baxter equation requires that the results of either of the braidequivalent pathways are equal, giving the third Reidemeister move.

For example, one can let S(a, b) = (b, a), simply switching the order of the two inputs, which gives (S × I)(I × S)(S × I)(a, b, c) = (S × I)(I × S)(b, a, c) = (S × I)(b, c, a) = (c, b, a) = (I × S)(c, a, b) = (I × S)(S × I)(a, c, b) = (I × S)(S × I)(I × S)(a, b, c). This switch corresponds to keeping track of the order of the elements, since the switch permutes the two elements it acts on, and so if it is used as a braid invariant, it only keeps track of the final positions of the strands. Skew left braces and switches derived from them can be used to generate other invariants.

4.3

Switches from Skew Left Braces

Theorem 6. [12] Given (G, ·, ◦), let λa (b) = a−1 (a ◦ b). Then S(a, b) = (λa (b), λλ−1 ((a ◦ b)−1 a(a ◦ b))) is a switch where the set a (b) X is the underlying set of G. Furthermore, S is involutive (its own inverse) if and only if ab = ba for all a, b ∈ G. This central theorem allows us to translate the knowledge of holomorphs and their regular subgroups into switches to shift into the perspective of braids and knots.

Example 3. Let us continue the example of D3 with H ≤ Hol(D3 ) a regular subgroup of the holomorph as in example 2. Then we have the following table for S(a, b), which shall be called the dihedral switch: e r r2 s sr sr2 S(a, b) e (e, e) (r, e) (r2 , e) (s, e) (sr, e) (sr2 , e) r (e, r) (r, r) (r2 , r) (sr2 , r) (s, r) (sr, r) 2 2 2 r (e, r ) (r, r ) (r2 , r2 ) (sr, r2 ) (sr2 , r2 ) (s, r2 ) s (e, s) (r2 , s) (r, s) (s, s) (sr2 , s) (sr, s) 2 2 sr (e, sr) (r , sr) (r, sr) (sr , sr) (sr, sr) (s, sr) sr2 (e, sr2 ) (r2 , sr2 ) (r, sr2 ) (sr, sr2 ) (s, sr2 ) (sr2 , sr2 )

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5

Braids

Moving on to braids, the goal is to apply the above generated switches to create invariants. To do so, first start by representing braids in the context of groups to turn braids, fundamentally geometric objects, into ones that can be classified by an equation concerning functions on sets.

One way to to do so is by considering the braid group, written as a presentation generated by the over crossings. Let σi represent strand i crossing over strand i + 1. σi−1 naturally represents strand i crossing under strand i + 1 since crossing over and then under are inverse operations in terms of braids, as shown:

The braid group is generated by {σ1 , . . . , σn−1 } as all braids are composed of a sequence of strands crossing over and under each other. This presentation naturally covers the second Reidemeister move, since σi σi−1 or σi−1 σi are the only ways that the second Reidemeister move position can be formed. Note also that since strands can only exchange places with the strands next to them, the first Reidemeister move, which requires a strand to pass over itself, is impossible. Hence the only remaining move to be concerned with is the third:

This move can be represented in the braid group presentation as σi σi+1 σi = σi+1 σi σi+1 , corresponding to the switches seen above.

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Lastly, if two crossings are made far from each other, then they cannot affect each other and so must commute. Specifically, the only crossings that can interact with each other are those σi and σ j with |i − j| = 1 (also when |i − j| = 0, but in that case σi = σ j so they still commute). Hence we have the following presentation for the braid group on n strands: Bn = ⟨σ1 , . . . , σn−1 |σi σi+1 σi = σi+1 σi σi+1 , |i− j| = ̸ 1 ⇐⇒ σi σ j = σ j σi ⟩ (3) On one hand, this gives a succinct definition for all braids with the operation of concatenation of braids: taking one and then putting another right below it using the strands from above. The problem is that the problem of telling whether or not two words in such a presentation are equivalent is called the group word problem and is unsolvable, as proven by Boone in 1957 [2]. Hence we turn to invariants of braids, now armed with the knowledge that all that needs to be checked is that σi σi+1 σi = σi+1 σi σi+1 and |i − j| = ̸ 1 ⇐⇒ σi σ j = σ j σi .

5.1

Assigning Switches

The method we use to generate invariants is to assign some value to each strand and then give rules for how those values change when strands interact. Formally, let the state of n strands before strands i and i + 1 cross be represented by the n-tuple (x1 , . . . , xn ). Then after the switch, let all strand values but i and i + 1 remain unchanged, since those strands do not interact with any others, and let xi and xi+1 be changed according to some function S : X × X → X × X, where X is the set of possible values for each xi .

Not all functions S give braid invariants, since some of them do not follow the third Reidemeister move, such as S(a, b) = (a2 , b).

With that function S, take the input tuple (1, 2, 3) ∈ (Z34 , +), which gives the following two braids:

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These two braids are equivalent via the third Reidemeister move, and so for S to give a valid invariant, the bottom values should be the same for both cases, but with S(a, b) = (a2 , b), and input (1, 2, 3), they are not, providing a counterexample and showing that this S does not in fact produce a braid invariant.

What functions S do work then? We have that if S is applied to two pairs of strands switching places that have an index differing by more than 1, then the strand switches commute since neither function is applied to the outputs of the other but instead to completely separate strand entries. Then let σi give the switch S(xi , xi+1 ). Then since σi−1 σi = e, there must be a way to return from S(xi , xi+1 ) to (xi , xi+1 ), so S must be invertible and so bijective. Lastly, we have the relation σi σi+1 σi = σi+1 σi σi+1 . Consider the triple (xi , xi+1 , xi+2 ).

Applying the left hand side to these values gives equation 4, while applying the right hand side gives equation 5.

(S × I)(I × S)(S × I)(xi , xi+1 , xi+2 )

(4)

(I × S)(S × I)(I × S)(xi , xi+1 , xi+2 )

(5)

Then since xi , xi+1 , xi+2 can be any elements of X, these are precisely the switches arising from the Yang-Baxter equation.

Example 4. Following example 3, take the dihedral switch S to apply for σi and S−1 to apply for σi−1 , which gives the following values for the endpoints of the strands of this braid, noting that S(a, b) = (a−1 ba, b) and S−1 (a, b) = (b, bab−1 ):

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6

Specific Results

6.1

Abelian Groups

Abelian groups are typical groups to consider as a general example of a group for the method described, since they are well understood and their holomorphs have been the centers of numerous papers [18, 19]. From [12] we have the following theorem about the switches resulting from abelian groups: Theorem 7. The switch S(a, b) = (λa (b), λλ−1 ((a ◦ b)−1 a(a ◦ b))), a (b) given by a skew left brace (G, ·, ◦) is involutive (its own inverse) if and only if ab = ba for all a, b ∈ G. Proof. Here we will prove only one direction: that if ab = ba for all a, b ∈ G, that the switch is involutive. This can simply be checked algebraically, as we have the following: S(a, b) = (λa (b), λλ−1 ((a ◦ b)−1 a(a ◦ b))) = (λa (b), λλ−1 (a)) (6) a (b) a (b) (a)) S(S(a, b)) = S(λa (b), λλ−1 a (b) = (λλa (b) (λλ−1 (a)), λ −1 a (b)

λλa (b) (λλ−1(b) (a))

= (a, λa−1 (λa (b))) = (a, b)

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a

(λa (b)))

(7)

Then since S2 is the identity function, we have S = S−1 , which gives the following two braid diagrams:

Because of this, regardless of the specific abelian group that S is derived from, the resulting invariant cannot differentiate between σi and σi−1 , and so the maximum amount of information it can hold is that of the braid group with the additional relation that σi = σi−1 or alternatively σi2 = e for all i from 1 to n − 1. This presentation simply gives the symmetric group, Sn as in [23]. Though this result follows from the view of the presentation as an algebraic object alone, it can also be obtained intuitively from the braid invariants themselves, since if switching two strands over and under are considered identical, then the path of any given strand is forgotten, which only leaves its final position, which is simply a permutation, since any strand can end up in any final position. Unfortunately, this means that this class of invariants are of little use when distinguishing between braids.

6.2

Left- and Right-Regular Representation Subgroups

Hol(G) can have many different regular subgroups, and counting and classifying them is an active area of research. The common approach to is by analyzing a construct called the multiple holomorph, as in [16, 17]. Every holomorph has at least the subgroups given by the left- and right-regular representation of G, and for non-abelian G, these two subgroups are non-isomorphic.

Recall from Definition 8 that the left-regular representation of G, denoted λ (G) is an embedding of G into Perm(G) with g 7→ λg , with λg (h) = gh. The right-regular representation is defined analogously, denoted ρ(G), with g 7→ ρg and ρg (h) = hg−1 . Since this is a right-regular representation, the g is put on the right, but notably the element that is multiplied is g−1 and not g itself, since we 19

−1 −1 −1 = ρ need ρg1 (ρg2 (h)) = ρg1 (hg−1 g1 g2 (h) 2 ) = hg2 g1 = h(g1 g2 ) for composition to hold for the right-regular representation to give a group. Similar to λ (G), ρ(G) also gives a regular subgroup since multiplying on the right by the inverse of every element gives a bijective function from G to itself.

Example 5. Consider λ (G). Then every element of Hol(G) must act as a left-multiplication by some element, which is represented in the semidirect product definition as (g, I). Hence when building the skew left brace from λ (G), we have fa = I for all a ∈ G, and so a ◦ b = a fa (b) = ab. Therefore, we have λa (b) = a−1 (a ◦ b) = a−1 ab = b, so λa = I = λa−1 for all a ∈ G. Then we have S(a, b) = (λa (b), λλ−1 ((a ◦ b)−1 a(a ◦ b))) = (b, (a ◦ b)−1 a(a ◦ b)) = a (b) (b, (ab)−1 aab) = (b, b−1 a−1aab) = (b, b−1 ab), so regardless of the group G, the switch S returns the second element followed by the conjugation of the first element by the second. This still produces different invariants for different groups, and in fact gives an infinite family of invariants, since conjugation acts differently in different groups.

Example 6. Similarly, consider the case of ρ(G). The elements of the corresponding subgroup of the semidirect product version of Hol(G) are varied, since they must have an automorphism that, composed with multiplication by an element gives the equivalent of multiplying by the inverse of a different element on the right. We have that for an element a, there must be some element (a′ , fa′ ) such that a′ fa′ (b) = ba−1 for all b ∈ G. Then let b = e, so then a′ = a′ fa′ (e) = ea−1 = a−1 , so a′ = a−1 . Hence we have a−1 fa−1 (b) = ba−1 , and since every element has a unique inverse, we can replace all instances of a with a−1 and vice versa, giving a fa (b) = ba, so a ◦ b = a fa (b) = ba. Then λa (b) = a−1 (a◦b) = a−1 ba and λa−1 (b) = aba−1 , and so S(a, b) = (λa (b), λλ−1 ((a ◦ b)−1 a(a ◦ b))) a (b) −1 −1 = (a−1 ba, λa−1 b aba)) −1 ba (a

= (a−1 ba, a−1 ba(a−1 b−1 aba)a−1 b−1 a) = (a−1 ba, a) 20

(8)

As expected, this is similar to the left-regular representation version but reversed, since both the left- and right-regular representations are just shifts of the group by its elements and their inverses with no other automorphisms.

Note also that, following the states of λ (G) and ρ(G) as “inverses” in a way, subgroups acting in opposite ways, the switches that they generate are inverses of each other.

7

Conclusion and Future Work

In this paper, several existing results have been combined to produce a pathway that takes any group and a regular subgroup of its holomorph and turns that into an invariant on braids. Some specific choices of groups and regular subgroups are then discussed to provide general results for some specific cases.

There are two main areas for future work relating to these results. First, this pathway requires that a regular subgroup of the holomorph be found, and though there are many existing families of examples, more research into finding regular subgroups of the holomorph would be useful to broaden the invariants obtainable via this method. As the holomorph’s regular subgroups are also useful for many other applications such as Hopf-Galois theory, there has already been some research in this area, such as Kohl’s papers on the case for groups of order 4p for p prime and on the dihedral and quaternionic groups and Tsang’s paper on centerless groups [16,17,24]. One specific construction useful in this area is the multiple holomorph, which is defined NHol(G) := NormPerm(G) (Hol(G)), where the quotient NHol(G)/ Hol(G) gives insight into those regular subgroups of the holomorph of G which are isomorphic to G itself, though this restriction is not required for the Yang-Baxter equation.

The second is the application of these braid invariants to the construction of knot invariants. Although Alexander’s Theorem allows one to turn a knot into a braid, this removes some information about the knot and so the invariants do not convert completely.

Still, Fenn in [7] uses the switches as present in this paper to construct biquandles which naturally turn into knot invariants. Only 21

some switches give valid biquandles, though, and the restrictions on the switches needed to guarantee a biquandle are unspecified, but solving that problem would generalize the braid invariants generated here to broadly applicable knot invariants.

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