Quantum Enhancements via Tribracket Brackets

TL;DR

This paper introduces tribracket brackets, a new family of skein invariants for knots and links that generalize classical quantum invariants and cocycle invariants, offering new tools for knot classification.

Contribution

It develops the concept of tribracket brackets, unifying and extending existing invariants, and provides explicit examples and future research directions.

Findings

Introduced tribracket brackets as skein invariants.

Unified classical quantum and cocycle invariants within a broader framework.

Provided explicit examples demonstrating the new invariants.

Abstract

We enhance the tribracket counting invariant with \textit{tribracket brackets}, skein invariants of tribracket-colored oriented knots and links analogously to biquandle brackets. This infinite family of invariants includes the classical quantum invariants and tribracket cocycle invariants as special cases, as well as new invariants. We provide explicit examples as well as questions for future work.

Equations

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

$$[x,b,c]=d,[a,x,c]=d\mathrm{and}[a,b,x]=d$$

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

$$[a,b,c]=xb+yc-xya$$

$$\left[\left[\begin{array}[]{rrr}1&3&2\\ 2&1&3\\ 3&2&1\end{array}\right],\left[\begin{array}[]{rrr}2&1&3\\ 3&2&1\\ 1&3&2\end{array}\right],\left[\begin{array}[]{rrr}3&2&1\\ 1&3&2\\ 2&1&3\end{array}\right]\right]$$

$$\raisebox{-50.58878pt}{\includegraphics{la-sn-pr-7.pdf}}\qquad\forall b,c,d\in X\ \exists!\ a\ s.t.\ [a,b,c]=d$$

$$\raisebox{-61.42993pt}{\includegraphics{la-sn-pr-8.pdf}}\qquad\forall b,c,d\in X\ \exists!\ a\ s.t.\ [a,b,c]=d$$

$$\raisebox{-61.42993pt}{\includegraphics{la-sn-pr-9.pdf}}\qquad\forall b,c,d\in X\ \exists!\ a\ s.t.\ [a,b,c]=d$$

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

$$\begin{array}[]{rcll}A_{a,b,c}A_{c,[a,b,c],[a,c,d]}A_{a,c,d}&=&A_{b,[a,b,c],[a,b,d]}A_{a,b,d}A_{d,[a,b,d],[a,c,d]}&(ii.i)\\ A_{a,b,c}B_{c,[a,b,c],[a,c,d]}B_{a,c,d}&=&B_{b,[a,b,c],[a,b,d]}B_{a,b,d}A_{d,[a,b,d],[a,c,d]}&(ii.ii)\\ B_{a,b,c}B_{c,[a,b,c],[a,c,d]}A_{a,c,d}&=&A_{b,[a,b,c],[a,b,d]}B_{a,b,d}B_{d,[a,b,d],[a,c,d]}&(ii.iii)\\ A_{a,b,c}B_{c,[a,b,c],[a,c,d]}A_{a,c,d}&=&A_{b,[a,b,c],[a,b,d]}A_{a,b,d}B_{d,[a,b,d],[a,c,d]}\\ &&+B_{b,[a,b,c],[a,b,d]}A_{a,b,d}A_{d,[a,b,d],[a,c,d]}\\ &&+\delta B_{b,[a,b,c],[a,b,d]}A_{a,b,d}B_{d,[a,b,d],[a,c,d]}\\ &&+B_{b,[a,b,c],[a,b,d]}B_{a,b,d}B_{d,[a,b,d],[a,c,d]}&(ii.iv)\\ A_{a,b,c}A_{c,[a,b,c],[a,c,d]}B_{a,c,d}&&&\\ +B_{a,b,c}A_{c,[a,b,c],[a,c,d]}A_{a,c,d}&&&\\ +\delta B_{a,b,c}A_{c,[a,b,c],[a,c,d]}B_{a,c,d}&&&\\ +B_{a,b,c}B_{c,[a,b,c],[a,c,d]}B_{a,c,d}&=&A_{b,[a,b,c],[a,b,d]}B_{a,b,d}A_{d,[a,b,d],[a,c,d]}&(ii.v).\end{array}$$

$$\beta (C)=w^{n-p}\sum _{\mathrm{states}}\prod (\mathrm{smoothing}\mathrm{coefficients}){\delta }^k.$$

$${\Phi }_X^{\beta ,M}(L)=\{\beta (C)|C\in {\mathcal{C}}_X(L)\}$$

$${\Phi }_X^{\beta }(L)=\sum _{C\in {\mathcal{C}}_X(L)}{u}^{\beta (C)}$$

$$w$$

$$A=\left[\left[\begin{array}[]{rrrr}A_{1,1,1}&A_{1,1,2}&\dots&A_{1,1,n}\\ A_{1,2,1}&A_{1,2,2}&\dots&A_{1,2,n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{1,n,1}&A_{1,n,2}&\dots&A_{1,n,n}\\ \end{array}\right],\dots,\left[\begin{array}[]{rrrr}A_{n,1,1}&A_{n,1,2}&\dots&A_{n,1,n}\\ A_{n,2,1}&A_{n,2,2}&\dots&A_{n,2,n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n,n,1}&A_{n,n,2}&\dots&A_{n,n,n}\\ \end{array}\right]\right]$$

$$B=\left[\left[\begin{array}[]{rrrr}B_{1,1,1}&B_{1,1,2}&\dots&B_{1,1,n}\\ B_{1,2,1}&B_{1,2,2}&\dots&B_{1,2,n}\\ \vdots&\vdots&\ddots&\vdots\\ B_{1,n,1}&B_{1,n,2}&\dots&B_{1,n,n}\\ \end{array}\right],\dots,\left[\begin{array}[]{rrrr}B_{n,1,1}&B_{n,1,2}&\dots&B_{n,1,n}\\ B_{n,2,1}&B_{n,2,2}&\dots&B_{n,2,n}\\ \vdots&\vdots&\ddots&\vdots\\ B_{n,n,1}&B_{n,n,2}&\dots&B_{n,n,n}\\ \end{array}\right]\right].$$

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

$$A=\left[\left[\begin{array}[]{rr}1&3\\ 2&1\end{array}\right],\ \left[\begin{array}[]{rr}1&2\\ 3&1\end{array}\right]\right]\ \mathrm{and}\ B=\left[\left[\begin{array}[]{rr}5&1\\ 3&5\end{array}\right],\ \left[\begin{array}[]{rr}5&3\\ 1&5\end{array}\right]\right].$$

$$\delta =-A_{111}^{-1}{B}_{111}-A_{111}{B}_{111}^{-1}=-(1)(5)-1(3)=-8=6$$

$$w=-A_{1,1,1}^2{B}_{111}^{-1}=-(1{)}^2(3)=-3=4$$

$$\text{includegraphics}.$$

$$\text{scalebox}0.85\text{includegraphics}.$$

$$\beta (C)$$

$$A=\left[\left[\begin{array}[]{rr}1&4\\ 2&1\end{array}\right],\ \left[\begin{array}[]{rr}1&1\\ 3&1\end{array}\right]\right]\ \mathrm{and}\ B=\left[\left[\begin{array}[]{rr}4&1\\ 3&4\end{array}\right],\ \left[\begin{array}[]{rr}4&4\\ 2&4\end{array}\right]\right].$$

$$\begin{array}[]{l|l}\Phi_{X}^{\beta_{1}}(L)&L\\ \hline\cr 4u^{2}&0_{1},3_{1},5_{1},5_{2},6_{1},6_{2},7_{1},7_{3},7_{4},\\ &8_{3},8_{4},8_{9},8_{12},8_{15},8_{17},8_{18},8_{19}\\ u^{3}+3u^{2}&4_{1},8_{1},8_{2},8_{5},8_{6},8_{11},8_{14}\\ u^{4}+3u^{2}&6_{3},7_{2},7_{5},7_{7},8_{7},8_{8},8_{10},8_{13}\\ u^{4}+u^{3}+2u^{2}&8_{20}\\ 2u^{4}+2u^{2}&8_{21}\\ 3u^{2}+u&7_{6},8_{16}\\ 3u^{4}+2u^{2}+3u&L7n1\\ 4u^{4}+4u&L4a1,L6a1,L7a2\\ 4u^{4}+4u^{2}&L2a1\\ 4u^{4}+4u^{3}&L6a2,L6a3\\ 4u^{4}+3u^{3}+u^{2}&L7a3,L7a4\\ 4u^{4}+u^{3}+3u^{2}&L7a6\\ 5u^{4}+2u^{2}+u&L7a1,L7a5,L7n2\\ 6u^{4}+2u^{3}&L5a1\\ 3u^{4}+5u^{3}+3u^{2}+5u&L7a7\\ 4u^{3}+12u^{2}&L6a5,L6n1\\ 12u^{3}+4u&L6a4\\ \end{array}$$

Full text

Keywords:

2010 MSC: 57M27, 57M25

1 Introduction

In recent decades much work has involved colorings of knot and link diagrams by algebraic structures such as quandles and biquandles. These colorings have an interpretation as elements of a homset collecting homomorphisms from an object associated to the knot or link such as the knot group, the knot quandle etc. to the finite coloring object. This interpretation gives rise to the notion of enhancements, stronger invariants obtained from the set of colorings by collecting values of an invariant of colored knots, starting with quandle cocycle invariants of [2] and explored in many subsequent works. See [5, 3] for an overview.

In [11], the second listed author and a coauthor proposed the notion of quantum enhancements, quantum invariants of biquandle-colored knots and links. In [7], the second listed author and coauthors introduced a class of quantum enhancements known as biquandle brackets, skein invariants of biquandle-colored knots and links. This infinite class of invariants includes the classical quantum invariants (Alexander-Conway, Jones, HOMFLYPT and Kauffman polynomials) and the quandle cocycle invariants as special cases, explicitly uniting these a priori unrelated invariants; moreover, examples of biquandle brackets which are apparently neither of these types were identified in [7]. In [9] biquandle brackets were extended to define biquandle virtual brackets using a skein relation including a virtual crossing, in [4] a type of biquandle bracket with skein relation involving graphs was introduced, and in [10] a method of computing biquandle bracket invariants recursively with trace diagrams as opposed to via the original state-sum definition was introduced. See [6] for a survey of biquandle bracket invariants.

In [12], colorings of knot diagrams by knot-theoretic ternary quasigroups or as we call them, Niebrzydowski tribrackets, were introduced. In this paper we extend the biquandle bracket idea to the case of tribrackets, defining quantum enhancements via tribracket brackets. The paper is organized as follows. In Section 2 we review the basics of tribracket theory. In Section 3 we introduce tribracket brackets and provide some examples, including an illustration of the computation of the new invariant, examples showing that the new invariant is proper enhancement, and the values of the tribracket bracket invariant for two brackets on all prime knots with up to eight crossings and prime links with up to 7 crossings. In Section 4 we close with some questions for future research.

2 Tribrackets

We begin with a definition. See [12, 8] etc. for more.

Definition 1**.**

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

• (i)

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

• (ii)

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

[TABLE]

The condition (i) makes $X$ a ternary quasigroup, meaning that if we think of $[a,b,c]$ as a three term product, there are well-defined operations of left, center and right division. That is, each of the equations

[TABLE]

has a unique solution in $X$. The condition (ii) makes our structure a knot-theoretic quasigroup in the terminology of [12], which we call a Niebrzydowski tribracket following [8].

Example 1**.**

Every group $G$ is a horizontal tribracket under the operation

[TABLE]

known as a Dehn tribracket.

Example 2**.**

Every $\mathbb{Z}[x^{\pm 1},y^{\pm 1}]$-module is a horizontal tribracket under the operation

[TABLE]

known as an Alexander tribracket.

Example 3**.**

Given a finite set $X=\{1,2,\dots,n\}$ we can specify a tribracket structure on $X$ with an operation 3-tensor, an ordered list of $n$ $n\times n$ matrices which we may visualize as stacked to form a 3-cube. For example,

[TABLE]

defines a 3-element tribracket where $[a,b,c]$ means the element in matrix $a$, row $b$ column $c$. For example, $[2,3,1]=1$ and $[1,1,2]=3$.

The tribracket axioms are motivated by region coloring for oriented knots and links according to the rule below:

[TABLE]

The condition (i) implies that for a given coloring of a knot or link diagram before a Reidemeister I or II move, there is a unique coloring after the move which agrees with the original coloring outside the neighborhood of the move. We illustrate with the Reidemeister II moves; the Reidemeister I moves are special cases.

[TABLE]

[TABLE]

[TABLE]

The condition (ii) ensures the same for Reidemeister III moves:

[TABLE]

Hence it follows that the number of $X$-colorings of a diagram of a knot or link $L$ is an integer-valued invariant, known as the tribracket counting invariant, denoted $\Phi_{X}^{\mathbb{Z}}(L)$.

Example 4**.**

Consider colorings of the trefoil knot $K$ and the unknot $U$ by the tribracket $X$ in Example 3.

[TABLE]

It is easy to see that there are nine $X$-colorings of $U$ since we can choose any color $x\in X$ for outside the circle and any $y\in X$ for inside; on the other hand, for the trefoil $K$ a check of which possible solutions $(a,b,c,d,e)$ satisfy the conditions

[TABLE]

reveals 27 $X$-colorings. Then since $\Phi_{X}^{\mathbb{Z}}(K)=27\neq 9=\Phi_{X}^{\mathbb{Z}}(U)$, the invariant distinguishes the knots.

We will denote the set of $X$-colorings of an oriented link diagram $L$ by $\mathcal{C}_{X}(L)$; then the counting invariant is the cardinality of this set, i.e. $\Phi_{X}^{\mathbb{Z}}(L)=|\mathcal{C}_{X}(L)|$.

3 Tribracket Brackets

We will now define a family of quantum enhancements of the tribracket counting invariant using structures we call tribracket brackets.

Definition 2**.**

Let $X$ be a tribracket and $R$ a commutative ring with identity. Then a tribracket bracket $\beta$ consists of a pair of maps $A,B:X^{3}\to R^{\times}$ such that

• (i)

for all $a,b,c\in X$, the elements $-A_{a,b,c}B_{a,b,c}^{-1}-A_{a,b,c}^{-1}B_{a,b,c}$ are all equal, with their common value denoted as $\delta$, and

• (ii)

for all $a,b,c,d\in X$ we have

[TABLE]

We also define a distinguished element $w=-A_{a,b,b}^{2}B_{a,b,b}^{-1}$ which is required to be the same for all $a,b\in X$.

The tribracket bracket axioms are motivated by the skein relations

[TABLE]

[TABLE]

with the elements $\delta$ and $w$ representing the values of a smoothed component and a positive kink:

[TABLE]

Then the tribracket bracket axioms are the conditions required to make the following proposition true:

Proposition 1**.**

Given a tribracket $X$, and $X$-colored link diagram $C$ and tribracket bracket $\beta$, we compute an element $\beta(C)$ by first computing the set of Kauffman states of $C$ and for each state, sum a contribution consisting of the product of the smoothing coefficients times $\delta^{k}$ where $k$ is the number of smoothed components in the state. Finally, we multiply this sum by the writhe correction factor $w^{n-p}$ where $p$ is the number of positive crossings and $n$ is the number of negative crossings in $L$. More symbolically, that is

[TABLE]

Then $\beta(C)$is unchanged by $X$-colored Reidemeister moves.

This implies our main result:

Corollary 2**.**

Given a tribracket $X$ and tribracket bracket $\beta$, for any oriented link $L$ the multiset

[TABLE]

and the “polynomial”

[TABLE]

are invariants of oriented links.

Proof.

(of Proposition 1) For each of the moves in our generating set of oriented Reidemeister moves, we compare the contributions to $\beta(C)$ on both sides of the move and see that they agree.

First, applying the skein relation at either positive $X$-colored Reidemeister I move

[TABLE]

we obtain the requirement that

[TABLE]

and a similar computation at either negative Reidemeister move yields $w^{-1}=-A_{a,b,b}^{-2}B_{a,b,b}$.

Next, comparing coefficients of the smoothed tangles on both sides of each Reidemeister II move

[TABLE]

yields the requirement that $\delta=-A_{a,b,b}B_{a,b,b}^{-1}-A_{a,b,b}^{-1}B_{a,b,b}$. Here we depict one of the four oriented Reidemeister II moves; the others yield the same result.

Finally, in the Reidemeister III in our generating set in Section 2, on the left side we have

[TABLE]

and on the right side, we have

[TABLE]

Comparing coefficients then yields the equations (ii.i)-(ii.v).

For a finite tribracket $X=\{1,2,\dots,n\}$ we can specify a tribracket bracket $\beta$ with a pair of 3-tensors $A$ and $B$ whose entries are the $A_{x,y,z}$ and $B_{x,y,z}$ coefficients:

[TABLE]

and

[TABLE]

Example 5**.**

If $X=\{1\}$ then setting $A_{1,1,1}=A$ and $B_{1,1,1}=A^{-1}$ gives us $\Phi_{X}^{\beta}(L)$ equal to a normalization of the Kauffman bracket/Jones polynomial.

Example 6**.**

If $A_{x,y,z}=B_{x,y,z}$ for all $x,y,z\in X$ then $\beta$ is a tribracket 2-cocycle and $\Phi_{X}^{\beta}(L)$ equals the tribracket 2-cocycle invariant times the Kauffman bracket polynomial evaluated at $A=-1$.

Example 7**.**

Let $X$ be the tribracket specified by the 3-tensor

[TABLE]

Then our python computations reveal tribracket bracket structures with $\mathbb{Z}_{7}$ coefficients including

[TABLE]

For instance here we have $A_{1,2,2}=1$ and $B_{2,1,2}=3$. This tribracket bracket has

[TABLE]

and

[TABLE]

and satisfies sixteen skein relations including

[TABLE]

and

[TABLE]

Example 8**.**

The tribracket bracket polynomial invariant $\Phi_{X}^{\beta}$ associated to the tribracket bracket in example 7 distinguishes the Hopf link from the $(4,2)$-torus link with invariant values $4u^{6}+4u$ and $8u$ respectively, demonstrating that the enhancement is proper. Let us illustrate the computation of the invariant with the Hopf link. First, we find the set of all $X$-colorings of the link:

[TABLE]

Next, for each coloring $C$, we compute the state-sum value $\beta(C)$. We illustrate the process with a choice of coloring:

[TABLE]

Thus for this coloring we obtain state-sum value

[TABLE]

and this coloring contributes $u^{6}$ to $\Phi_{X}^{\beta}(\mathrm{Hopf\ link})$. Repeating for the other colorings and for the $(4,2)$-torus link, we obtain the stated result.

Example 9**.**

Using python code, we computed the tribracket bracket polynomial for all of the prime knots with up to 8 crossings and prime links with up to 7 crossings as found in [1] (with a choice of orientation for each component) with respect to the tribracket $X$ from example 7 and the $\mathbb{Z}_{5}$ tribracket bracket $\beta$ given by

[TABLE]

The results are collected in the table. We note that this invariant distinguishes the square knot $SK$ from the granny knot $GK$ with $\Phi_{X}^{\beta_{1}}(SK)=u^{4}+3u^{2}\neq 4u^{2}=\Phi_{X}^{\beta_{1}}(GK)$.

[TABLE]

Remark 1**.**

We have defined tribracket brackets analogously to biquandle brackets as in [7], using the state-sum formulation in which we do all smoothings simultaneously to obtain a completed set of states. With classical skein invariants one often wants to compute the invariant recursively by doing Reidemeister moves on partially smoothed diagrams. In [10] a method for such computations is defined for biquandle brackets using trace diagrams, decorated trivalent graphs which can be given biquandle colorings for partially smoothed states. We can make an analogous construction with tribracket brackets by defining tribracket colorings of trace diagrams in the following way:

[TABLE]

We then have skein relations

[TABLE]

[TABLE]

with the value of a crossingless diagram given by $\delta^{k}w^{n-p}$ where $n$ is the number of negative signed traces, $p$ is the number of positive signed traces, and $k$ is the number of components of the bivalent graph obtained by deleting all traces.

As observed in [10] for the biquandle bracket case, for any biquandle bracket there is some subset of possible trace moves, i.e. moving a strand over, under or through a trace, which are allowed and some which are not depending on whether certain conditions are satisfied by the bracket; an analogous result holds in the tribracket bracket case.

4 Questions

We conclude with a few questions for future research.

In the case of biquandle brackets over a ring $R$, some brackets represent cocycles in the second cohomology of the coloring biquandle with coefficients in $R$, and cohomologous cocycles define the same invariant. To what extent is the same thing true in the tribracket bracket case? Indeed, what is the relationship between tribracket brackets and biquandle brackets?

As with biquandle brackets, what other quantum enhancements of the tribracket counting invariant are there beyond those which can be represented as tribracket brackets?

For ease of computation, we have considered only examples of tribracket brackets over finite rings; what are some good examples of tribracket brackets over infinite rings such as $\mathbb{Z}$ or $\mathbb{Z}[t]$?

What is the geometric meaning of a tribracket bracket? How can we categorify tribracket brackets?