Multi-tribrackets

TL;DR

This paper introduces multi-tribrackets, a new algebraic framework for coloring knot and link diagrams with different operations at various crossings, enhancing the ability to distinguish complex links.

Contribution

It develops the concept of multi-tribrackets, including component multi-tribrackets, and demonstrates their effectiveness in distinguishing links beyond standard tribracket invariants.

Findings

Multi-tribrackets can distinguish links not separable by standard tribracket invariants.

Examples show the effectiveness of multi-tribracket counting invariants.

Reinterpretation of previous results in terms of multi-tribrackets.

Abstract

We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-component crossings. We provide examples to show that the resulting counting invariants can distinguish links which are not distinguished by the counting invariants associated to the standard tribracket coloring. We reinterpret the results of [11] in terms of multi-tribrackets and consider futuredirections for multi-tribracket theory.

Equations

$$[c,[a,b,c],[a,c,d]]=[b,[a,b,c],[a,b,d]]=[d,[a,b,d],[a,c,d]].$$

$$[x,y,a]=z,[x,b,y]=z\mathrm{and}[c,x,y]=z.$$

$${\alpha }_{x,y}(z)=[x,y,z],{\beta }_{x,y}(z)=[x,z,y]\mathrm{and}{\gamma }_{x,y}(z)=[z,x,y]$$

$$[c,[a,b,c],[a,c,d]]=[c,b{a}^{-1}c,c{a}^{-1}d]=b{a}^{-1}c{c}^{-1}c{a}^{-1}d=b{a}^{-1}c{a}^{-1}d$$

$$[b,[a,b,c],[a,b,d]]=[b,b{a}^{-1}c,b{a}^{-1}d]=b{a}^{-1}c{b}^{-1}b{a}^{-1}d=b{a}^{-1}c{a}^{-1}d$$

$$[d,[a,b,d],[a,c,d]]=[d,b{a}^{-1}d,c{a}^{-1}d]=b{a}^{-1}d{d}^{-1}c{a}^{-1}d=b{a}^{-1}c{a}^{-1}d$$

$$\left[\left[\begin{array}[]{rr}1&2\\ 2&1\end{array}\right],\left[\begin{array}[]{rr}2&1\\ 1&2\end{array}\right]\right]\quad\mathrm{and}\quad\left[\left[\begin{array}[]{rr}2&1\\ 1&2\end{array}\right],\left[\begin{array}[]{rr}1&2\\ 2&1\end{array}\right]\right].$$

$$⟨a,b,⟨b,c,d⟩⟩$$

$$⟨⟨a,b,c⟩,c,d⟩$$

$$[a,b,⟨a,b,c⟩]=c=⟨a,b,[a,b,c]⟩.$$

$$\begin{array}[]{rcl}3x_{1}+x_{2}+2x_{4}&=&x_{3}\\ 3x_{1}+2x_{2}+x_{4}&=&x_{3}\\ \end{array}$$

$$\left[\begin{array}[]{rrrr}3&1&4&2\\ 3&2&4&1\\ \end{array}\right]\leftrightarrow\left[\begin{array}[]{rrrr}1&0&3&1\\ 0&1&0&4\\ \end{array}\right]$$

$$\text{includegraphics}\mathrm{and}\text{includegraphics}$$

$$[,,{]}_x:X\times X\times X\to X$$

$$\forall a,b\in X\exists !c\in X\mathrm{such}\mathrm{that}[a,b,b{]}_x=c,$$

$$\forall a,b\in X\exists !c\in X\mathrm{such}\mathrm{that}[c,a,a{]}_x=b,$$

$$\forall a,b,d\in X\exists !c\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$\forall a,b,c\in X\exists !d\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$\forall b,c,d\in X\exists !a\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$[b,[a,b,c{]}_x,[a,b,d{]}_y{]}_z=[c,[a,b,c{]}_x,[a,c,d{]}_z{]}_y=[d,[a,b,d{]}_y,[a,c,d{]}_z{]}_x.$$

$$[b,[a,b,c{]}_x,[a,d,b{]}_z{]}_y=[c,[a,b,c{]}_x,[a,c,d{]}_y{]}_z=[d,[a,d,b{]}_z,[a,c,d{]}_y{]}_x.$$

$$\forall a,b\in X\exists !c\in X\mathrm{such}\mathrm{that}[a,b,b{]}_x=c,$$

$$\forall a,b\in X\exists !c\in X\mathrm{such}\mathrm{that}[c,a,a{]}_x=b,$$

$$\forall a,b,d\in X\exists !c\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$\forall a,b,c\in X\exists !d\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$\forall b,c,d\in X\exists !a\in X\mathrm{such}\mathrm{that}[a,b,c{]}_x=[a,c,b{]}_y=d,$$

$$[b,[a,b,c{]}_x,[a,b,d{]}_y{]}_z=[c,[a,b,c{]}_x,[a,c,d{]}_z{]}_y=[d,[a,b,d{]}_y,[a,c,d{]}_z{]}_x,$$

$$[b,[a,b,c{]}_x,[a,d,b{]}_z{]}_y=[c,[a,b,c{]}_x,[a,c,d{]}_y{]}_z=[d,[a,d,b{]}_z,[a,c,d{]}_y{]}_x.$$

$$M=\{{\mathrm{I}}_{\mathrm{CP}},{\mathrm{I}}_{\mathrm{CN}},{\mathrm{I}}_{\mathrm{CP}}^{\prime },{\mathrm{I}}_{\mathrm{CN}}^{\prime },{\mathrm{II}}_{\mathrm{CP},\mathrm{CN}},{\mathrm{II}}_{\mathrm{CN},\mathrm{CP}},{\mathrm{II}}_{\mathrm{CP},\mathrm{CN}}^{\prime },{\mathrm{II}}_{\mathrm{CN},\mathrm{CP}}^{\prime },{\mathrm{III}}_{\mathrm{CP},\mathrm{CP},\mathrm{CP}}\}$$

$$[a,b,c{]}_{\mathrm{CP}}=[a,b,c]=[a,c,b{]}_{\mathrm{CN}}.$$

$$M=\left\{\begin{array}[]{l}\mathrm{I}_{\mathrm{SP}},\ \mathrm{I}_{\mathrm{SN}},\ \mathrm{I}_{\mathrm{SP}}^{\prime},\ \mathrm{I}_{\mathrm{SN}}^{\prime},\\ \mathrm{II}_{\mathrm{SP},\mathrm{SN}},\ \mathrm{II}_{\mathrm{SN},\mathrm{SP}},\ \mathrm{II}_{\mathrm{SP},\mathrm{SN}}^{\prime},\ \mathrm{II}_{\mathrm{SN},\mathrm{SP}}^{\prime},\ \mathrm{II}_{\mathrm{MP},\mathrm{MN}},\ \mathrm{II}_{\mathrm{MN},\mathrm{MP}},\ \mathrm{II}_{\mathrm{MP},\mathrm{MN}}^{\prime},\ \mathrm{II}_{\mathrm{MN},\mathrm{MP}}^{\prime},\\ \mathrm{III}_{\mathrm{SP},\mathrm{SP},\mathrm{SP}},\ \mathrm{III}_{\mathrm{SP},\mathrm{MP},\mathrm{MP}},\ \mathrm{III}_{\mathrm{MP},\mathrm{SP},\mathrm{MP}},\ \mathrm{III}_{\mathrm{MP},\mathrm{MP},\mathrm{SP}},\ \mathrm{III}_{\mathrm{MP},\mathrm{MP},\mathrm{MP}}\end{array}\right\}$$

Full text

Keywords: Niebrzydowski tribrackets, enhancements, oriented knot and link invariants, link tribrackets

2010 MSC: 57M27, 57M25

1 Introduction

Niebrzydowski Tribrackets, also known as knot-theoretic ternary quasigroups, are ternary operations on sets which satisfy conditions motivated by the Reidemeister moves in knot theory. A set $X$ with a ternary operation in which the actions of all three variables are invertible is called a ternary quasigroup, and a ternary quasigroup is knot-theoretic if it satisfies a condition related to the Reidemeister III move.

In [12], Niebrzydowski considered region colorings of knot diagrams by ternary quasigroups. In [13] he introduced a (co)homology theory for these objects analogous to quandle (co)homology and further developed this theory in [15] and [14]. An isomorphism between this (co)homology theory and a cohomology theory of objects called local biquandles was established in [10] by the first author and collaborators.

In [9] biquasiles, algebraic structures defined by pair of quasigroups satisfying certain conditions related to Reidemeister III moves interpreted in terms of dual graph diagrams, were introduced. In [3] the biquasile coloring invariant was enhanced with Boltzmann weights, which can be understood as special cases of tribracket cocycle invariants defined in [13]. In [8] biquasile colorings and Boltzmann enhancements were used to study orientable surface-links.

In [11], tribracket colorings were extended to virtual knots and links and in [4], tribrackets were enhanced with partial products to define Niebrzydowski algebras, algebraic structures for coloring $Y$-oriented spatial graphs and handlebody-links.

In this paper we define multi-tribrackets, a generalization of the tribracket structure which includes virtual tribrackets from [11] as a special case and which defines stronger counting invariants for virtual knots and links with multiple components. The paper is organized as follows. In Section 2 we review tribrackets. In Section 3 we introduce multi-tribrackets and identify special cases for use with virtual knots and with multicomponent links. We compute examples of counting invariants associated to these structures and show that they are stronger than the single tribracket counting invariants. We end in Section 4 with questions for future research.

2 Tribrackets

We begin with a definition; see [12, 10] etc. for more.

Definition 1**.**

Let $X$ be a set. A horizontal tribracket operation on $X$ is a ternary operation $[,,]:X\times X\times X\to X$ satisfying

• (i)

In the equations $[a,b,c]=d$, any three of the variables determines the fourth, and

• (ii)

For all $a,b,c,d\in X$

[TABLE]

Remark 1**.**

Axiom (i) can be rephrased in a few ways. One is that for every $x,y,z\in X$ there exist unique $a,b,c\in X$ such that

[TABLE]

Equivalently, for all $x,y\in X$, the maps $\alpha_{x,y},\beta_{x,y},\gamma_{x,y}:X\to X$ defined by

[TABLE]

are invertible.

Example 1**.**

Let $G$ be a group. Then the map $[,,]:G\times G\times G\to G$ defined by $[x,y,z]=yx^{-1}z$ is a horizontal tribracket known as a Dehn tribracket. Let us illustrate axiom (ii):

[TABLE]

while

[TABLE]

and

[TABLE]

as required.

Example 2**.**

Let $X$ be a module over the ring $\mathbb{Z}[x^{\pm 1},y^{\pm 1}]$ of two-variable Laurent polynomials. Then the map $[,,]:X\times X\times X\to X$ defined by $[a,b,c]=xb+yc-xya$ is a horizontal tribracket known as an Alexander tribracket.

Example 3**.**

For a finite set $X=\{1,2,\dots,n\}$ we can specify a tribracket operation with an operation 3-tensor, i.e., an ordered list of $n$ $n\times n$ matrices, where the entry in matrix $i$ row $j$ column $k$ is $[i,j,k]$. For example, there are two horizontal tribracket maps on $X=\{1,2\}$, which are specified by the operation 3-tensors

[TABLE]

The horizontal tribracket axioms are motivated by the following region coloring rules:

[TABLE]

and

An assignment of elements of $X$ to the regions in an oriented link diagram $L$ satisfying these rules will be called a tribracket coloring or an $X$-coloring of $L$.

Moreover, we have the following:

Theorem 1**.**

[12]** Let $X$ be a set with a tribracket and $L$ an oriented link diagram. The number $\Phi_{X}^{\mathbb{Z}}(L)$ of $X$-colorings of $L$ does not change under Reidemeister moves and hence is an integer-valued invariant of oriented links.

Remark 2**.**

We can also define vertical tribrackets, motivated by the coloring rule

[TABLE]

as follows: A ternary operation $\langle,,\rangle:X\times X\times X\to X$ is a vertical tribracket if it satisfies

• (i)

In the equations $\langle a,b,c\rangle=d$, any three of the variable determines the fourth, and

• (ii)

For all $a,b,c,d\in X$

[TABLE]

From these coloring rules, we can observe that given a horizontal tribracket, there is an induced vertical tribracket (and vice-versa) satisfying

[TABLE]

In this paper we will work in terms of horizontal tribrackets, but all of our results can be rephrased in terms of vertical tribrackets. See also [10].

Example 4**.**

Let us compute the number of colorings $\Phi_{X}^{\mathbb{Z}}(L2a1)$ of the Hopf link $L2a1$ by the Alexander tribracket structure on $X=\mathbb{Z}_{5}$ with $x=1$ and $y=2$, i.e. the horizontal tribracket $[a,b,c]=1b+2c-(1)(2)a=3a+b+2c$.

[TABLE]

We have coloring equations

[TABLE]

or in matrix form

[TABLE]

so there are $5^{2}=25$ $X$-colorings of $L2a1$, including

[TABLE]

etc. This invaraint thus distinguishes the Hopf link from the unlink of two components, which has $5^{3}=125$ $X$-colorings.

3 Multi-tribrackets

Let us now generalize tribrackets to the case of multi-tribrackets. The motivation is to have distinct tribracket operations at different kinds of crossings with interaction laws determined by the Reidemeister move analogues with specified crossing types we wish to allow. We begin with a definition inspired by conversations of the first author with Allison Henrich and Aaron Kaestner [5]:

Definition 2**.**

An explicitly typed oriented knot or link diagram is a planar 4-valent directed graph with every vertex having two adjacent inputs and two adjacent outputs and labeled with a crossing type chosen from a set $T$ of defined crossing types.

Examples of crossing types include but are not limited to:

• •

Positive and negative classical crossings

[TABLE]

• •

Virtual crossings

[TABLE]

• •

Single-component and multi-component positive and negative crossings

[TABLE]

[TABLE]

• •

Even and odd parity positive and negative single-component crossings where we have an even or odd number of crossing points between the over and under instances of the crossing

[TABLE]

[TABLE]

etc.

Then given a set of explicit crossing types, we can select versions of Reidemeister moves of types I, II and III with various combinations of explicitly typed crossings to define an explicitly typed knot theory. The resulting equivalence classes of knot diagrams will be known as explicitly typed knots.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Next, we make our main definition.

Definition 3**.**

Let $X$ be a set and suppose we have an explicitly typed knot theory with set of types $T$ and typed moves $M$. A horizontal multi-tribracket structure on $X$ of type $(T,M)$ is a set of ternary operations

[TABLE]

indexed by elements $x\in T$ satisfying

• (i)

For every move I${}_{x}\in M$, we have the condition that

[TABLE]

• (i*′*)

For every move I${}_{x}^{\prime}\in M$, we have the condition that

[TABLE]

• (ii)

For every move II${}_{x,y}\in M$, we have the condition that

[TABLE]

• (ii*′*)

For every move II${}_{x,y}^{\prime\prime}\in M$, we have the condition that

[TABLE]

• (ii*′′*)

For every move II${}_{x,y}^{\prime\prime}\in M$, we have the condition that

[TABLE]

• (iii)

For every move III${}_{x,y,z}\in M$, we have for all $a,b,c,d\in X$

[TABLE]

• (iii*′*)

For every move III${}^{\prime}_{x,y,z}\in M$, we have for all $a,b,c,d\in X$

[TABLE]

Definition 3 is motivated by the the coloring rule

[TABLE]

applied to the explicitly typed Reidemeister moves. Specifically, the axioms are chosen to ensure that given a valid colorings of an explicitly diagram on one side of a move, there is a unqiue valid coloring of the diagram on the other side of the move which agrees with the original outside the neighborhood of the move. Specifically, we have:

Theorem 2**.**

The number of colorings of an explicitly typed link diagram by a multi-tribracket of type $(T,M)$ is unchanged by Reidemeister moves in $M$.

Proof.

We consider the explicitly typed moves systematically.

(i) We note that this one is implied by the fact that $[,,]_{x}$ is an operation, but we include it for completeness:

[TABLE]

[TABLE]

(i*′*)

[TABLE]

[TABLE]

(ii)

[TABLE]

[TABLE]

(ii*′*)

[TABLE]

[TABLE]

(iii)

[TABLE]

[TABLE]

(iii*′*)

[TABLE]

[TABLE]

and (iii*′*)

[TABLE]

[TABLE]

∎

Definition 4**.**

Let $X$ be a multi-tribracket of type $(T,M)$. Then the number of $X$-colorings of an oriented link diagram is an integer-valued invariant of explicitly typed knots and links of type $(T,M)$. We denote this invariant as $\Phi_{X}^{\mathbb{Z}}(L)$.

Example 5**.**

A tribracket is a multi-tribracket with $T=\{\mathrm{CP},\mathrm{CN}\}$ and

[TABLE]

where we define

[TABLE]

We note that other generating sets of classical Reidemeister moves are possible, and that this particular set is not minimal.

Example 6**.**

The case that originally motivated this paper was the idea of having different tribracket operations at single-component crossings and at multicomponent crossings. In this case we need $T=\{SP,SN,MP,MN\}$ and

[TABLE]

since (1) Reidemeister I moves are only single-component, (2) in a Reidemeister II move both crossings are single-component or both are multi-component, and (3) in a Reidemeister III move we can have all three crossings on the same component, one crossing with both strands on the same component and two on different components, or all three crossings on different components.

[TABLE]

As in the classical case, we can simplify this somewhat by setting

[TABLE]

[TABLE]

We call such an $X$ a multicomponent multi-tribracket. For such an $X$, $\Phi_{X}^{\mathbb{Z}}(L)$ is an invariant of oriented links which agrees with the usual counting invariant with respect to the $X_{0}$ 3-tensor for knots but can be different for links.

For instance, the pair of operation 3-tensors

[TABLE]

[TABLE]

defines a multi-tribracket whose counting invariant distinguishes the links $L6a2$ and $L6a4$ while the single tribracket counting invariant with respect to the single-component tribracket $X_{0}$ does not.

[TABLE]

Example 7**.**

We computed the counting invariant $\Phi_{X}^{\mathbb{Z}}(L)$ for all prime links with up to 7 crossings with respect to the multicomponent multi-tribracket

[TABLE]

[TABLE]

using our python code. The results are collected in the table.

[TABLE]

We also computed the invariant using the 4-element tribracket

[TABLE]

[TABLE]

using our python code. The results are collected in the table.

[TABLE]

Example 8**.**

A virtual tribracket, originally defined in [11] (where vertical tribracket notation was used), is a multi-tribracket with $T=\{\mathrm{CP},\mathrm{CN},\mathrm{V}\}$ and

[TABLE]

Then for example, the pair of 3-tensors

[TABLE]

[TABLE]

defines a virtual tribracket by

[TABLE]

and

[TABLE]

whose counting invariant distinguishes the link

[TABLE]

from the unlink of two components, since while the latter has $3^{3}=27$ colorings, a coloring of the former must satisfy

[TABLE]

and it is easy to see that no such triple exists; hence this virtual link has no $X$-colorings.

Example 9**.**

For our final example let us consider the case of welded links. Welded links can be understood as virtual links in which we are allowed to move a classical strand over but not under a virtual crossing, thinking of the virtual crossing as “welded” to the paper (see e.g. [1].)

This extra move means we can obtain invariants of welded isotopy by taking counting invariants using welded tribrackets, multi-tribrackets with $T=\{CP,CN,V\}$ and

[TABLE]

Our python code reveals welded tribrackets including for example

[TABLE]

[TABLE]

It then follows that the number of colorings of a welded link diagram by a welded tribracket is an invariant of welded links.

4 Questions

We conclude in this section with several possibilities for future directions of research.

• •

Parity multi-tribrackets. For any crossing in a knot or link diagram, the parity (even or odd) of the number of crossing points one encounters on the journey from the overcrossing point to the undercrossing point is unchanged by Reidemeister moves. This has led to the notion of parity biquandles in [6], which are effectively just usual biquandles when applied to classical knots but yield stronger invariants for virtual links. Parity multi-tribrackets are more cumbersome because of the necessity of including an operation at virtual crossings, but a parity virtual tribracket can be descrbed with a triple of compatible 3-tensors defining operations at even, odd and virtual crossings.

• •

Enhancements. As with quandles, biquandles and other coloring structures, we can define enhancements of the multi-tribracket counting invariant, e.g. extending the tribracket modules in [7] to the case of multi-tribrackets.

• •

Homology. How can we modify the tribracket homology found in [14, 10] to adapt to the case of multi-tribrackets?

• •

Multi-cocycle enhancements. Following [2, 7], define a general theory of multi-tribracket cocycle enhancements.