Colorings beyond Fox:
the other linear Alexander quandles.
This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then so does any other diagram of the same knot. This is called p-colorability. It is also associated to systems of linear homogeneous equations over the residues mod p, by regarding the arcs of the diagram as variables and assigning the equation “twice the over-arc minus the sum of the under-arcs equals zero” to each crossing. The knot invariant is here the existence or non-existence of non-trivial solutions of these systems of equations, when working over the integers modulo p (a non-trivial solution is such that not all variables take up the same value). Another knot invariant is the minimum number of distinct colors (values) these non-trivial solutions require, should they exist. This corresponds to finding a basis, supported by a diagram, in which these solutions take up the least number of distinct values. The actual minimum is hard to calculate, in general. For the first few primes, less than 17, it depends only on the prime, p, and not on the specific knots that admit non-trivial solutions, modulo p. For primes larger than 13 this is an open problem. In this article, we begin the exploration of other generalizations of these colorings (which also involve systems of linear homogeneous equations mod p) and we give lower bounds for the number of colors.
Keywords: Linear Alexander quandle; dihedral quandle; Fox coloring; minimum number of colors; crossing number; determinant of knot or link.
Mathematics Subject Classification 2010: 57M25, 57M27
This article is about the application of linear algebra to the theory of knots and links. Knots and links are embeddings of circles into 3-dimensional space . We will use the word “knot” to mean both “knots” (one component embeddings) and “links” (multiple component embeddings); wherever necessary, we will emphasize that we mean a one component link (or a multiple component link). Knot theorists work with knots by projecting them onto a plane, thereby obtaining the so-called knot diagrams. See Figure 1 for an illustration of a knot diagram. Note that for our purposes, a knot diagram is finite in the sense that it has a finite number of arcs and crossings.
Knots obtained by deformation from a given knot are in the same equivalence class and ultimately thought of as being the same knot. At the level of diagrams, they represent the same equivalence class of knots if and only if they are related by a finite number of Reidemeister moves (this is the Reidemeister Theorem, see ). Figure 2 shows the Reidemeister moves.
One of the main issues in knot theory is to tell apart the equivalence classes referred to above. One way of doing this is by using a knot invariant i.e., a method of associating to a knot another mathematical object such as a number, a polynomial or a group. Furthermore, a knot invariant may be evaluated at knot diagrams and it remains the same as one performs Reidemeister moves. In this way, if the knot invariant evaluated at two knot diagrams returns distinct values, the knots they represent are not deformable into one another, they are not in the same equivalence class. Loosely speaking, they are not the same knot. Tri-coloring is an example of such a knot invariant. Fox defined  a tri-coloring of a knot diagram as a coloring of its arcs with three colors (say red, blue and green) such that (i) more than one color is used and (ii) at each crossing, the three arcs that meet there, all bear the same color or all bear distinct colors. If a diagram satisfies this property it is said tri-colorable. Furthermore,
Tri-colorability is invariant under Reidemeister moves i.e., it is a knot invariant.
The proof is straightforward  by analyzing what happens to the colors when Reidemeister moves are performed. ∎
Thus, the trefoil and the unknot are not deformable into one another, as shown in Figure 3. We remark that -colorability for a given odd prime, as defined in the Abstract, is a (first) generalization of tri-colorability. It is simpler to begin with tri-colorability (which is -colorability for ). However, the minimum number of colors for tri-colorabilty is (which equals the number of colors available) see Corollary 2.2. Thus we introduce -colorabilty in the Abstract in order to make sense of the minimum number of colors since it is one of the main invariants dealt with in this article.
For a given odd prime , -colorability becomes a problem in linear algebra in the following way. Given a knot diagram, we regard its arcs as algebraic variables and read equations from each crossing of the following sort: twice the over-arc minus the sum of the under-arcs equals zero - the so-called coloring condition - see Figure 4.
We then form a system of linear homogeneous equations over the integers by collecting these coloring conditions over the crossings of the diagram under study. Furthermore, we consider this system of equations over the integers modulo , . -colorability is thus the existence or non-existence of non-trivial colorings i.e., solutions of the system of linear homogeneous equations over , referred to above, which take up at least two distinct values - for the knot under study. A knot is said to be -colorable if one of its diagrams admits non-trivial -colorings.
This (linear) algebraic reformulation of -colorability retains a relevant feature of the original one. In fact, any solution of the homogeneous system of equations mod , is such that, at each crossing, the colors (values) that meet there are all distinct or are all equal. To see this, just assume the existence of only two colors at a crossing and note that they have to be equal. See the proof of Proposition 2.2 below.
Note that the rule is compatible with the relations shown in Figure 5. That is , , and , so that Fox colorings are fully compatible with the Reidemeister moves. Note also that the rule is a very particular case. An algebraic structure satisfying these axioms is called a quandle - see Section 2.
We resume considering the rule where is one under-arc and is the over-arc of the crossing at issue. We remark that upon performance of a Reidemeister move on the diagram, as in Figure 5, there is a unique assignment of variables to arcs and appearance or disappearance of equations associated to the new diagram, induced by those in the original diagram . The new system of coloring conditions is such that its matrix of coefficients relates to the original one by elementary moves on matrices. Thus, the equivalence class of these systems of linear homogeneous equations over the integers modulo elementary moves on matrices, constitutes a knot invariant. In particular, the number of solutions of these systems of equations is a knot invariant. As for the solutions, there are always the trivial ones, the ones that assign the same value (henceforth color) to each arc in the diagram. The existence of this sort of solution is related to the fact that the determinant of the coefficient matrix (henceforth coloring matrix) is zero: each row is formed by exactly one , two ’s and ’s; thus, adding all columns we obtain a column of ’s.
For knots, it turns out that we can take any minor matrix of the coloring matrix and that minor matrix will have non-zero and odd determinant (for one-component knots) . The system of equations corresponding to this minor matrix can be interpreted as the coloring system of equations where we have set equal to zero one selected arc of the diagram. Since the determinant of the minor matrix is not zero, when it further is not equal to we can produce non-trivial solutions of the system of equations mod d where d is the absolute value of the aforementioned minor determinant. These results are independent of the choice of the minor matrix. Thus, is another knot invariant, called the determinant of the knot under study. This is the beginning of the topic of Fox colorings for knots. Going back to the solutions of our system of equations, we can obtain non-trivial solutions by considering the system of equations over the integers mod , , where is any factor of greater than . This is how we choose the in -colorability. It is any of the prime factors of the determinant of the knot under study.
Here is another perspective on the coloring matrix. Since it is its equivalence class that matters (it is the knot invariant) we might as well work with a preferred representative of the equivalence class. Let us choose the Smith Normal Form of the coloring matrix. It has at least one along the diagonal, because the determinant of the coloring matrix is zero. The elements of the diagonal, modulo sign, constitute also a knot invariant. (These elements can be interpreted in terms of the homology of a 2-fold branched covering along the knot of .) The absolute value of their product, except for the referred to above, is called the determinant of the knot under study (equivalent to the definition in the previous paragraph). It is a fact that the determinant for knots (one component links) is an odd integer. Going back to the solutions of our system of equations, we can obtain non-zero solutions to the system by working in an appropriate modular number system. We do that by choosing a prime factor of one of the non-zero elements, say , of this diagonal and working over the integers modulo , . In particular, if , then the knot is tricolorable. We remark that our choice of a prime factor has to do with the fact that in this way the associated ring of modular integers is a field and we can use the techniques of standard linear algebra. We can also work over for composite , if we wish, by doing linear algebra over commutative rings . We can now say that if the determinant of the knot under study is divisible by an odd prime , then the knot is -colorable or colorable mod .
1.1 Results and organization of this article.
Fox coloring is based on the rule . In fact, if is an algebraic variable, then satisfies the quandle rules, as illustrated in Figure 5. This means that we can generalize Fox colorings by working over (for a given odd prime ) and using a non-null integer , with , mod . We call such quandles Linear Alexander Quandles . If a knot diagram can be assigned integers to its arcs such that at each crossing the condition , mod is satisfied and at least two arcs are assigned distinct colors (i.e., integers mod ), then the corresponding link is said to admit non-trivial -colorings.
Let be a knot admitting non-trivial -colorings. Let be a diagram of and let be the minimum number of colors it takes to equip with a non-trivial -coloring. We let
and refer to it as the minimum number of colors for non-trivial -colorings of . When the context is clear we will say the minimum number of colors for .
The rest of the article is devoted to these matters. In particular, the main Theorem of this article is (Section 3)
Let be a knot i.e., a -component link. Let be an odd prime. Let be an integer such that admits non-trivial -colorings. If or but , explained below then
The rest of the article is organized as follows. Section 2 introduces the algebraic structure which constitutes the underpinning of our considerations - the quandle; examples are given. Homomorphisms of quandles have as particular examples Fox colorings (which take as target quandles the so-called dihedral quandles) - this is Subsection 2.1. Instead, in this article, we use the other linear Alexander quandles - which are introduced in Subsection 2.2.
In Section 4 we present examples of non-trivial -colorings. We present results on automorphisms of colorings (Section 5), on obstructions to colorings (Section 6) and minimizations by direct calculations (Section 7). In Section 8 we reduce the number of colors in two examples. Finally, in Section 9 we collect a few questions for future work.
L. K. is pleased to thank the Simons Foundation (grant number 426075) for partial support for this research under his Collaboration Grant for Mathematicians (2016 - 2021). P.L. acknowledges partial support from FCT (Fundação para a Ciência e a Tecnologia), Portugal, through project FCT EXCL/MAT-GEO/0222/2012, “Geometry and Mathematical Physics”.
The quandles [7, 12] were conceived in order to make an algebraic version of the Reidemeister moves. We give the definition of quandles below in Definition 2.1 and illustrate the connection between Reidemeister moves and quandle axioms in Figure 5.
Definition 2.1 (Quandle).
A quandle is a set, , equipped with a binary operation, , such that
For any , ;
For any , there is a unique such that , i.e., there is a second operation, denoted , such that .
For any ,
The ordered pair denotes a quandle.
The fundamental quandle of a knot [7, 12] is defined as follows. Given any oriented diagram of , a presentation of the fundamental quandle of is obtained as follows. Arcs of the diagram stand for generators and relations are read at each crossing as illustrated in Figure 6. The fundamental quandle of the knot is also the knot quandle, in this article.
Dihedral quandle of order . The underlying set is and the operation is, for any , , mod . Note that in this case, .
Linear Alexander quandle, LAQ. Given positive integers such that , we let the underlying set be , and we let the operation be, for any , , mod . We denote such quandles .
The Alexander quandle. The underlying set is the set of the Laurent polynomials, . The quandle operation is, for any , .
Definition 2.2 (Quandle homomorphism).
Given quandles and , a quandle homomorphism is a mapping such that, for any ,
Let be a knot and a quandle. The number which may be infinite of homomorphisms from the Knot Quandle of to is a knot invariant. In this context, is called the target quandle.
If it were not, there would not be an isomorphism between the knot quandles. ∎
Let equal a dihedral quandle of order . Then the homomorphisms are the -colorings of the knot under study.
Let be a linear Alexander Quandle, other than a dihedral quandle. This article is devoted to the study of the homomorphisms with these quandles as targets.
Let be the Alexander quandle. We elaborate on the associated homomorphisms in Subsection 2.1, right below.
2.1 Alexander Colorings
We now elaborate on the quandle homomorphisms . of Example 2.2. These are the quandle homomorphisms from the knot quandle of a given knot, to the Alexander quandle. This is the quandle whose underlying set is the ring of Laurent polynomials (on the variable ), denoted , i.e.,
equipped with the operation
The coloring condition here is depicted in Figure 7. The procedure and mathematical objects that come up along the way parallel those of the preceding cases. In short, there will be a matrix (here called the Alexander matrix) whose first minor determinant (here called the (1st) Alexander polynomial) controls the existence of non-trivial solutions (colorings); should these solutions exist they belong in the appropriate quotient of (modulo the ideal generated by the Alexander polynomial, or one of its factors). More specifically, given a knot, the homomorphisms from its knot quandle to the Alexander quandle are solutions of a system of equations (the coloring conditions) read off the crossings of a given diagram of the knot. This system of equations gives rise to a matrix of coefficients (which is the Alexander matrix). The fact that along each row the non-null entries are , , and , implies this matrix has determinant; this corresponds to the existence of the trivial (i.e., monochromatic) solutions. The existence of non-trivial solutions corresponds to working on the quotient of by the ideal generated by the first minor determinant of the Alexander matrix (i.e., the Alexander polynomial). The Alexander polynomial is a Laurent polynomial on the variable and is determined up to , for any integer ; it is independent, up to , of the first minor you choose from the Alexander matrix. Further information on the Alexander matrix and the Alexander polynomial(s) can be found in , although from a different perspective.
We now illustrate the procedure with the help of the “figure eight” knot, see Figure 8. Due to the coloring of the diagram by the Alexander quandle, the matrices we associate with the system of equations in Figure 8 are , the Alexander matrix, and , one of its first minor matrices:
and the determinant of is the Alexander polynomial of the Figure eight knot. This is
(where means “equality modulo units of ”, ’s) so the reduced Alexander polynomial (defined below) of the figure eight knot, , is
The matrix in Equation (1) is the matrix of the coefficients of the associated linear homogeneous system of equations obtained from collecting the coloring conditions at each crossing of the diagram. The determinant of this matrix is zero since along each row we find exactly one , one and one along with ’s. This means that we have several solutions. We call them trivial or monochromatic because each of these solutions assign the same Laurent polynomial to the distinct arcs of the diagram. In order to obtain polychromatic (or non-trivial) solutions we have to work over a quotient of where the first minor determinant of the coloring matrix vanishes . This first minor determinant (Equation (2)) is the Alexander polynomial (modulo multiplication by units of ) of the knot at issue .
When an Alexander polynomial is not identically zero, the reduced Alexander polynomial is the one which when evaluated at is defined and positive. For instance, is the reduced Alexander polynomial of the figure eight knot.
CAVEAT: In this article we only consider knots or links whose Alexander polynomial is not identically .
We now give a formal statement and proof of the fact above.
In order to obtain polychromatic i.e., non-trivial, solutions, the system of coloring equations has to be considered over a quotient of where the first minor determinant of the coloring matrix i.e., Alexander matrix vanishes.
The proof has two parts. In the first part we prove that the sum of a polychromatic solution with a monochromatic solution is again a (polychromatic) solution, see Figure 9.
Now for the second part. Let denote the coloring matrix and assume is a polychromatic solution, so that the equation
holds (where juxtaposition on the left-hand side of the equation denotes matrix multiplication and is a column vector consisting of ’s). We now add an appropriate monochromatic solution to so that the -th component of the new solution is ; we denote it . We remark that is again a polychromatic solution. We now let denote the column vector obtained from by removing the -th component; is also polychromatic. Furthermore, Let be the matrix obtained from by removing the -th column and the -th row (although we could have removed any row). Then 
where denotes the adjoint matrix to . Since is a polychromatic solution then has to be . We bring that about by working mod i.e., by working on the quotient of by the ideal generated by (or any factor of it, should it exist).
The proof is complete. ∎
We remark that is the Alexander polynomial of the knot at issue ; it is independent of the pair row-column removed above, modulo multiplication by units. In this way, the Alexander polynomial is the means to obtain non-trivial solutions for the knot under study.
Furthermore, the Alexander polynomial is a knot invariant  so the considerations above do not depend on the knot diagram being used.
Figure 10 renders another perspective on how to retrieve the Alexander polynomial, this time for the Trefoil knot. Note that somehow we are doing a Gaussian elimination in order to calculate the relevant determinant.
Let us now look at some particular cases. Suppose we set ; then the underlying set of the target quandle is the integers, , and the quandle operation is . The Alexander polynomial evaluated at yields the so-called knot determinant and its absolute value yields the generator of the ideal associated with the nontrivial solutions i.e., the non-trivial colorings. This is the context of the so-called Fox colorings - see Figure 11 for an example with the figure eight knot.
2.2 Methodology for the colorings using the other Alexander quandles.
As in the case of Fox colorings, where we implicitly choose a dihedral quandle for target quandle, if we choose another linear Alexander quandle () for target quandle, we will obtain a system of linear homogeneous equations over the integers whose solutions yield the colorings of the knot under study with respect to the chosen linear Alexander quandle. We remark that in the case of the Fox colorings (i.e, dihedral quandles for target quandles) the parameter is already chosen; it is .
The linear Alexander quandles constitute another family of quandles that contains the dihedral quandles. Each linear Alexander quandle is indexed on the modulus, say (which is its order) and a second integer parameter, say , such that . We recover the dihedral quandles by setting , for each modulus . The operation here is
The requirement guarantees the right-invertibility of the operation i.e, the second axiom for quandles. We warn the reader that different choices of for the same modulus may give rise to isomorphic linear Alexander quandles. Notwithstanding, if we choose a prime modulus , then each of the integers comply with . Moreover, these are the only linear Alexander quandles with the indicated prime order. Thus, for each odd prime , each such that corresponds to a non-isomorphic quandle . In the current article we will always choose a prime modulus, unless otherwise stated. Furthermore, once is chosen, the prime modulus, say , should be a factor of the Alexander polynomial of the knot at stake and evaluated at , leaning on the results of Subsection 2.1. In this article we will prefer this prime factor to be the reduced Alexander polynomial evaluated at the given . Table 2.1 shows the values of the reduced Alexander polynomial of the figure eight knot at the first few positive integers.
Given a knot , we call -determinant of , notation , the value of the reduced Alexander polynomial of evaluated at . We will write often for when it is clear from context which we are referring to.
For positive integer and integer such that , mod is a quandle over , and knot , we call -colorings of , the homomorphisms from the knot quandle of to the linear Alexander quandle of order and parameter . We note that, at each crossing, there is a preferred under-arc which receives the product, in the case of -colorings for mod - see Figure 13.
We leave it to the reader to show that for no orientation of the diagram is needed. We conclude this Subsection with preliminary results on the minimum number of colors, Proposition 2.2 and Corollary 2.2, below.
Let be a link multiple component knot. is said to be a split link if there exist disjoint balls in -space and a deformation of such that some components of deformed are in one of the balls while the other components are in the other ball.
If is not a split link, is said to be a non-split link.
We regard a one-component knot as a non-split link.
If a knot or non-split link admits a non-trivial -coloring with odd prime and , then at least three colors are needed to assemble such a coloring.
If there is a non-trivial coloring, there has to be at least one crossing where two distinct colors meet (call these colors and ). Otherwise the link would be split. We will show that there has to be a third color in order for the -coloring condition to be satisfied at this crossing. We prove this by showing that the existence of only two colors in this crossing implies they are equal.
Assume the over-arc is assigned the same color, say , as one of the under-arcs.
Assume that under-arc is the one that does not receive the product. Then the coloring condition yields the color on the arc that receives the product. This is which implies the three colors meeting at this crossing are the same. This does not comply with the hypothesis;
Assume now the over-arc and the under-arc which receives the product are both colored . Then where is the yet unknown color of the other under-arc. The equation simplifies to and since is prime and , then is invertible so the equation further simplifies to which again is a contradiction.
Then the color on the over-arc has to be distinct from the colors on the under-arcs. Forcing the existence of two colors, this implies that the over-arc is colored and the under-arcs are colored . Applying the coloring condition we obtain which simplifies to mod which implies that , since is prime and . This is again a contradiction.
The proof is complete. ∎
Keeping the conditions of Proposition 2.2 and setting , then it takes exactly distinct colors to assemble such a non-trivial coloring.
In this case, the least number of colors expected (Proposition 2.2) and the number of colors available both equal . The proof is complete. ∎
This article is devoted mainly to generalizing the results on minimum number of colors from dihedral quandles to linear Alexander quandles and to trying to expose differences that may occur. Proposition 2.2 is an example of a phenomenon already known for Fox colorings. An instance which does no occur for Fox colorings is illustrated in Subsection 4.1.
Remark We will assume that the knot under study will admit non-trivial -colorings for a given prime standing for the value of the reduced Alexander polynomial of the knot. The parameter will eventually turn out to depend on . For instance for the trefoil knot, , which is realized for and .
3 The palette graph
In this Section we study the coloring structure where we use for target quandles linear Alexander quandles of type - is an odd prime and is an integer such that, preferably, but not necessarily, . We estimate the determinants of the coloring matrices for these target quandles and we eventually arrive at a lower bound for the number of colors (Theorem 1.2). This is useful because when removing colors from a coloring, if we arrive at the corresponding lower bound, as dictated by Theorem 1.2, we will know we have reached the minimum number of colors. Figure 26 provides such an example although the lower bound attained is not provided by Theorem 1.2 but by Proposition 2.2. The work in this Section is a generalization of .
We begin by examining the matrices that are of the sort that arise as coloring matrices for a knot, as we have described them in the previous Section. For instance, the coloring matrix associated with Figure 12 is
Let be an integer and a positive integer. Let be the set of matrices over the integers and such that
each row contains at most one , at most one , and at most one , all other entries being , should there be any more entries in the row.
If then , where .
The proof is by induction on . If then the only entry in is either , , or and the inequality holds. Now for the induction step. For positive , we split the proof into three instances.
If has a row without a , then the minor expansion along this row yields
If has a row with exactly one , apart from the ’s, then
Otherwise, any row of has a and at least one of or . If need be by swapping columns, we let its entry be , and still call this matrix . We let be the -th column of . Then the following matrix satisfies . above, by way of its first row:
For negative the argument is analogous to the one above but takes up the role of . In this way, the splitting into the three instances is now the following.
If has a row without an , then the minor expansion along this row yields
If has a row with exactly one , apart from the ’s, then
Otherwise, any row of has an and at least one of or . If need be by swapping columns, we let its entry be , and still call this matrix . We let be the -th column of . Then the following matrix satisfies . above, by way of its first row:
The proof is complete. ∎
-palette graphs and -palette graphs of diagrams.
Let be an integer greater than , and an integer such that defines a quandle over . Let be a subset of . The -palette graph of is a directed graph whose vertices are the elements of and whose directed edges are defined as follows. For each , there is an edge from to if there exists an such that , mod . This edge is labeled . A sequence from as above is called a local -coloring of a potential crossing mod - or simply a local coloring, when the context is clear.
Since a diagram equipped with a non-trivial -coloring and as above uses a subset of colors mod , we call the -palette graph of the coloring, the -palette graph whose vertices correspond to the colors used in this coloring and whose directed edges together with the source vertex and the target vertex correspond to the distinct solutions of the coloring condition present in the coloring under study. A sequence from such that and are the colors of the under-arcs at a crossing whose over-arc is colored , and , mod is called a local -coloring of a crossing mod - or simply a local coloring, when the context is clear. If need be, we will add broken lines to the graph for those edges that belong to the palette graph of but do not correspond to colors on over-arcs in the coloring under study i.e., do not belong to the -palette graph of the coloring.
In general, we consider the palette graph of a coloring, usually drawn next to the diagram, without the broken lines. Also, “palette graph” will mean the palette graph of a coloring, unless explicitly stated.
Finally, for finite and for any diagram, we remark that both notions palette graph and palette graph of coloring make sense for i.e., upon replacement of for , above - see Figure 17, for example. The corresponding colorings are called integral colorings. This instance will also be considered in the sequel, namely in Lemma 3.4 and Corollary 3.2.
In Figure 14 we give an illustrative example for this definition. We remark that the graphs are not directed in Figure 14 since the quandle operation does not have a preferred product for dihedral quandles. On the other hand, in Figure 15 the graphs are directed because here the quandle at stake is a linear Alexander quandle other than a dihedral quandle.
The following remark is in order here. In the sequel, the figures pertaining to colorings of knots by linear Alexander quandles display a boxed pair of integers in the top left. This pair of integers stands for the pair which characterizes the linear Alexander quandle which is being used for target quandle. will be an odd prime, or , and is the value of the reduced Alexander polynomial of the knot under study, evaluated at the corresponding . For Fox colorings dihedral quandle for target quandle, ; for all other linear Alexander quandles for target quandles, will satisfy .
For example, in Figure 14, the boxed “”, stands for .
For knots 1-component links, the palette graph is connected.
Walk along the oriented diagram (equipped with a non-trivial (p, m)-coloring) starting at a given under-arc. Mirror that walk in the palette graph of the coloring, starting at the vertex whose color is the color of the under-arc where you started in the diagram. If you are going under a monochromatic crossing in the diagram, you stay in the palette graph, at the vertex whose color is the color of the monochromatic crossing at stake. Otherwise move, in the palette graph, to the vertex whose color is the color of the under-arc on the other side of the crossing you just went under in the diagram. Since there is only one component in the diagram, you must visit all colors in the diagram, thus visiting all colors in the palette graph. Hence there is a path in the palette graph connecting any two colors (i.e., vertices). ∎
For each circuit in the palette graph of a coloring there is an equation, , (represented by one of the edges of the circuit) which is a consequence of the other equations (represented by the other edges in the circuit). Thus, the coloring at stake is still a solution of the system of equations obtained by removal of equation from the system of coloring conditions associated with the colored diagram at stake.
Pick a circuit in the palette graph with say n vertices, , where is succeeded by as you go along the circuit and is . Let be the color of the edge connecting to . Then
where the ’s mean either or , according to the coloring. ∎
In this way, we are interested in the graph obtained from the palette graph by taking the spanning tree of each of its components; we call this graph the spanning forest of the palette graph. If we are working with a knot (i.e., a 1-component link), the spanning forest of its palette graph is a spanning tree, thanks to Lemma 3.2.
Keeping the notation and terminology from Definition 3.2, let be an -palette graph with colors. Let be a spanning forest of . In the sequel, the relationships “vertex-variable-column” and “edge-equation-row” should be born in mind. We now refer the reader to Figure 16. We call the -adjacency matrix of , the matrix, , whose -entry is
, if edge ends at vertex ,
, if edge begins at vertex ,
, if edge begins at vertex and ends at vertex such that , mod i.e., if is labeled with color ,
Let be an odd prime and be an integer. Let be a knot i.e., a -component link, admitting non-trivial -colorings.
Let be the -palette graph of such a coloring, let be its -adjacency matrix, and be the square matrix, obtained by deletion of the column from . Then,
either or is divisible by ; and
, mod .
We remark that knot is an example where , for , see Figure 17.
Assume over the integers. There are two independent solutions for the system of homogeneous linear equations (mod ) whose coefficient matrix is , (the ubiquitous trivial coloring) and (the non-trivial -coloring the palette graph of the statement stems from). Then the rank of is at most . Thus, mod , since rank of is at most , mod .
First note that the expression is trivially true for , since any two integers are equivalent, mod . Assume now and consider the matrix with the entries read mod ; call it . The entries of are ’s, ’s, and ’s. Specifically, the entry is if edge is directed to vertex , it is if edge stems from vertex , and otherwise. Also note that the entries in correspond to a ”subgraph”, , obtained from the spanning tree of , by removing the vertex corresponding to column . There is then a unique bijection, call it , mapping each edge in to a vertex in incident to that edge. Note that “incident” here means that the vertex may either play the role of source or of sink for the edge at stake. The existence and uniqueness of is stated and proved below in Claim 3.1. Then, the only non-null summand of is the one associated with . Thus,
The proof of the following Claim completes the proof of this Lemma.
Let be an integer greater than and let denote a tree on vertices and consequently edges. Then, upon removal of one vertex from , say , there is a unique bijection from the edges of to such that each edge is mapped to the vertex incident to it.
The proof is by induction. It is clearly true for . Assuming it is true for all positive integers up to a given , consider the instance. If we remove a vertex, call it , from which has more than one edge incident to it, then we obtain a finite number of disconnected trees, each one satisfying the induction hypothesis. Putting together the bijections for each of these trees, we obtain the bijection for . If we remove a vertex, call it , which has only one edge incident to it, call it , let us then remove and let us also remove the other vertex incident to it, call it . The resulting tree now satisfies the induction hypothesis and so there is a unique bijection from edges to vertices such that each edge is mapped to a vertex incident to it. We now augment this bijection by sending to , thus obtaining the desired bijection for . The proof of Claim 3.1 is complete. ∎
The proof of Lemma 3.4 is complete. ∎
If mod , then for some integer . If mod , then may be zero. This occurs for the knot .
The only integer root of any reduced Alexander polynomial of a knot i.e., -component link is .
Suppose is an integer root of the reduced Alexander polynomial of i.e., . Note that is the minor determinant of the coloring matrix of , with coloring condition . This coloring matrix is an integer matrix and along each row there is exactly one , one and one , and otherwise ’s. Hence the determinant of the coloring matrix is . Its Smith Normal Form has therefore at least two ’s along the diagonal; one because the coloring matrix it stems from has determinant, and the other one because its first minor determinant is , ( is a root). There is thus at least one non-trivial -coloring of the diagram, for any odd prime i.e., a non-trivial integral coloring. Consider the palette graph of this non-trivial coloring, consider a spanning forest of this palette graph and obtain its -adjacency matrix, . Then, with the notation of Lemma 3.4,
But this is true if and only if . The proof is complete. ∎
We note that the reduced Alexander polynomial of the Hopf link equals at .
We are now ready to prove Theorem 1.2. We state it here again for the reader’s convenience. We remark again that we exclude from our considerations knots or links whose Alexander polynomial is identically .
Let be a knot i.e., a -component link and be an odd prime. Let be an integer such that admits non-trivial -colorings mod . If or but then
Remark Care should be taken when choosing the representative for the formula . Let us consider the case of Fox colorings while keeping the notation from Theorem 3.1. Here either or . Since represents the number of colors in the diagram, and we know that it takes at least colors to obtain a non-trivial coloring (Proposition 2.2), then the relation is a trivial relation (for each , which are the relevant ’s here). On the other hand, the relation is not a trivial relation.
4 Illustrative examples.
In this Section we present other examples and the associated calculations. Each figure portrays a knot diagram equipped with a non-trivial -coloring along with the corresponding palette graph and spanning tree. We recall that the boxed pair “prime, integer” at the top left of each figure, stands for the parameters of the linear Alexander quandle being used as target quandle in the figure. Later (Section 8) we will try to reduce the number of the colors in these colorings. If the number of colors we are left with at the end equals the lowest bound dictated by Theorem 1.2 or Proposition 2.2 or Proposition 7.1- we know we reached the corresponding , for the knot at stake.
In general, each figure intends to represent a different feature within the non-trivial -colorings. For instance, Figure 17, for knot , with , the determinant of the coloring matrix is , which means that the diagram in the Figure is colored integrally (the coloring conditions at each crossing are satisfied over the integers). This phenomenon does not occur for knots (one component) and Fox colorings, since, for knots (one component) the Alexander polynomial evaluated at is always an odd integer . Also, Theorem 1.2 does not apply here.
In Figure 18 a non-trivial -coloring of the knot is found. In this case, Theorem 1.2 provides an estimate for the lower bound of the number of colors: . Since we are using eight colors in this coloring we may state .
In Figure 19 there is a non-trivial -coloring of knot . Here again Theorem 1.2 provides an estimate for the lower bound of the number of colors: . Since we are using six colors in this coloring we may state .
We remark that in Figures 18 and 19 we have alternating reduced diagrams equipped with colorings associated to a prime determinant. Moreover, these colorings are such that distinct arcs receive distinct colors. In Subsection 4.1 we present examples of alternating reduced diagrams equipped with colorings associated to a prime determinant but such that different arcs receive the same color - we call this the anti KH behavior; we elaborate further below. This does not occur for Fox colorings.
4.1 Further examples: anti-KH behavior.
Let be an alternating knot one component of prime determinant, , i.e., . Any non-trivial -coloring Fox coloring on a reduced alternating diagram of assigns different colors to different arcs.
We say a knot, , displays KH behavior mod , if is alternating, and if there is a prime and an integer such that with , and for some reduced alternating diagram of equipped with a non-trivial -coloring, different arcs receive different colors. Otherwise we say displays anti-KH behavior.
5 Equivalence classes of colorings by linear Alexander quandles
We recall here the definition of Linear Alexander Quandle also to introduce notation which will shorten the statements in the sequel.
Definition 5.1 (Linear Alexander Quandles).
Let be an odd prime and a positive integer such that . We let stand for the linear Alexander quandle of order and parameter i.e., the quandle whose underlying set is the integers modulo , , and whose quandle operation is, for any ,
We remark that for we obtain the dihedral quandle of order .
Definition 5.2 (Automorphism groups of Linear Alexander Quandles).
Let be an odd prime and a positive integer such that . We let stand for the group of automorphisms of the linear Alexander quandle of order and parameter , . This is the set of bijections such that, for any , ; the group operation is composition of functions.
Let be an odd prime and a positive integer such that . Then
where is the affine group over , . In particular, only depends on .
We start by proving that . For and , consider
For any ,
We now prove . We pick i.e. such that for any ,
Then, for any , set
Then . Moreover,
In particular, setting , we obtain
whereas setting , we obtain
It follows that for any positive integer , and . In particular, there are positive integers such that . Thus, for any ,
Let . For any ,
The proof is clear for . We now prove for positive , by induction.
For we already know . Now for the inductive step.
The proof is complete for non-negative . For negative we write and invoke the first part of the proof. The proof is complete. ∎
Let , let . Then,
The proof is complete. ∎
So for any , pick positive integers such that and . Then
so that is linear i.e., there is such that