Jones slopes and coarse volume of near-alternating links
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of essential surfaces and the hyperbolic volume of their complements: we show that the Strong Slope Conjecture is true for near-alternating knots with spanning Jones surfaces, their colored Jones polynomials admit stable coefficients, and the stable coefficients provide two-sided bounds on the volume of the knot complement. We also discuss extensions of these results to their Murasugi sums and a class of highly twisted links.
- 1 Introduction
- 2 Graphical skein theory
- 3 Jones slopes
- 4 Boundary slope and Euler characteristic
- 5 Near-alternating knots are not adequate
- 6 Stable coefficients and volume bounds
- 7 Highly twisted links
Since the discovery of the Jones polynomial and related quantum knot invariants, a central problem in quantum topology has been to understand the connection between those invariants and the geometry of the knot complement. An important example of these quantum invariants is the colored Jones polynomial, which assigns a sequence of Laurent polynomials from the representation theory of to a link , and contains the Jones polynomial as the first term of the sequence, see Definition 2.2. Conjectures such as the Volume Conjecture [Kas97, MM01, MMO02] and the Strong Slope Conjecture [Gar11, KT15] predict that the colored Jones polynomial is closely related to the hyperbolic geometry and the topology of surfaces of the knot complement.
Much evidence for this relationship comes from the class of semi-adequate links. These are a class of links satisfying a diagrammatic condition, see Definition 1.7. An adequate knot satisfies the Strong Slope Conjecture, see Conjecture 1.5, and certain stable coefficients of their colored Jones polynomial give volume bounds on the complement of an adequate knot [DL07, FKP08, FKP13]. For these results, a key component is the existence of essential spanning surfaces, see Deifnition 1.4, along which the complement may be decomposed into simpler geometric components. Such surfaces have also been shown to be fundamental to the characterization of alternating knots [Gre17, How17] and adequate knots [Kal17].
In this paper, we are motivated by the question of when we can expect the Slope Conjecture and coarse volume bounds to be realized by spanning surfaces from state surfaces of the knot diagram beyond adequate links. Our answer to this question in this paper is the introduction of the class of near-alternating links, to be defined below in Definition 1.1. For a near-alternating knot, we compute its Jones slopes, show that there exist essential spanning surfaces in its exterior realizing the Strong Slope Conjecture, and we prove that the first, second, penultimate, and the last coefficient of its colored Jones polynomial are stable. If the near-alternating diagram is prime, twist-reduced, and highly twisted with more than 7 crossings in each twist region, then the link is hyperbolic by [FKP08], and we show that these stable coefficients provide coarse volume bounds for the link exterior. These results closely mirror those for adequate links, and we show that near-alternating links are not adequate, thus they form a strictly new class.
We also consider extensions of these results to more general classes of links in this paper. The first direction for extension is motivated by Murasugi sums of knots, which is a method for producing link diagrams that can produce all link diagrams. A near -adequate link is obtained by taking a certain Murasugi sum of a near-alternating link diagram with an -adequate link diagram. We compute a Jones slope for a near- adequate knot and find a spanning Jones surface for the slope. For the second direction, we consider the class of highly twisted knots with multiple () negative twist regions. We show that with certain restrictions on the negative twist regions, a highly twisted knot that is sufficiently positively-twisted, which means that a sufficient number of positive crossings are added to every positive twist region, satisfies the Strong Slope Conjecture with stable first, second, penultimate, and last coefficient giving coarse volume bounds on the knot complement.
We give the necessary definitions in order to state the main results below. We shall always consider a link . The indices should be considered independently in each instance unless explicitly stated otherwise.
1.1. Near-alternating link
Let be a finite, weighted planar graph in . For each edge of let be the weight. We may replace each vertex of with a disk and each edge with a twisted band consisting of right-handed (positive) or left-handed (negative) half twists if , or if , respectively. We denote the resulting surface by and consider the link diagram . All link diagrams may be represented as for some finite, weighted planar graph .
A path in a weighted graph with vertex set and a weighted edge set is a finite sequence of distinct vertices such that for . We define the length of a path as
where is the weight of the edge in .
A graph is said to be -connected if it does not have a vertex whose removal results in a disconnected graph, such a vertex is called a cut vertex.
We say that a non-split link diagram is near-alternating if , where is a 2-connected, finite, weighted planar graph without one-edged loops with a single negative edge of weight , such that . In addition, the graph obtained by deleting the edge from satisfies the following conditions.
Let be the minimum of taken over all paths in starting at and ending at and let be the total number of such paths. Then , and
The graph remains 2-connected, and is prime. We also require that the diagram , where is the graph obtained from by contracting the edge , be adequate.
A link is said to be near-alternating if it admits a near-alternating diagram, see Figure 1 for an example and the conventions for a negative or a positive twist region.
A pretzel link is near-alternating if , for all , and
1.2. Strong Slope Conjectures
Let be a link diagram. A Kauffman state is a choice of replacing every crossing of by the - or -resolution as in Figure 2, with the (dashed) segment recording the location of the crossing before the replacement.
Applying a Kauffman state results in a set of disjoint circles called state circles. We form a -state graph for each Kauffman state by letting the resulting state circles be vertices and the segments be edges. The all- state graph comes from the Kauffman state which chooses the -resolution at every crossing of . Similarly, the all- state graph comes from the Kauffman state which chooses the resolution at every crossing of .
is the number of crossings of , is the writhe of with an orientation, and is the number of vertices in the all- state. We can now state the main result of this paper.
Let be the minimum degree of , the th colored Jones polynomial of .
Let be a link admitting a near-alternating diagram with a single negative twist region of weight and let be defined by (1), then
Note that the case for many 3-tangle pretzel knots with a near-alternating diagram was already shown in [LvdV], and the degree of the Jones polynomial of pretzel knots was computed in [HTY00] for certain pretzel knots which are mostly not near-alternating.
Theorem 1 proves the Strong Slope Conjecture for near-alternating knots which we now describe. An orientable and properly embedded surface is essential if it is incompressible, boundary-incompressible, and non boundary-parallel. If is non-orientable, then is essential if its orientable double cover in is essential in the sense as defined.
Let be an essential and orientable surface with non-empty boundary in . A fraction is a boundary slope of if represents the homology class of in , where and are the canonical meridian and longitude basis of . The boundary slope of an essential non-orientable surface is that of its orientable double cover.
Garoufalidis showed in [Gar11] that since the colored Jones polynomial is -holonomic [GL05], the functions and , where is the maximum degree of , are quadratic quasi-polynomials viewed as functions from . This means that there exist integers , and rational numbers for each , such that for all ,
We consider the sets and . An element is called a Jones slope. We also consider the sets and . We may now state the Strong Slope Conjecture.
An essential surface in satisfying the conditions described in the conjecture is called a Jones surface.
The difference in our convention from [Gar11, KT15] is that in this paper the asterisk indicates the corresponding quantity from the maximum degree, rather than the minimum degree, of the th colored Jones polynomial , while indicates the corresponding quantities from the minimum degree. Also, instead of substituting we substitute for the colored Jones, see Definition 2.2 for our choice of the normalization convention.
The Strong Slope Conjecture is currently known for alternating knots [Gar11], adequate knots [FKP13], which is a generalization of alternating knots by Definition 1.7, iterated -cables of torus knots and iterated cables of adequate knots [KT15], and families of 3-tangle pretzel knots [LvdV]. It is also known for all knots with up to 9 crossings [Gar11, KT15, How] and an infinite family of arborescent non-Montesinos knots [HD]. The Slope Conjecture is also known for 2-fusion knots [GvdV16].
Given a Kauffman state on a link diagram , we may form the -state surface, denoted by , by filling in the disjoint circles in with disks, and replacing each segment recording the previous location of the crossing by half-twisted bands. See Figure 3.
For a near-alternating knot with a diagram of , the surface is essential by [OR12, Theorem 2.15] and is given by the state surface where chooses the -resolution on the crossings corresponding to the single edge with negative weight in , and the -resolution everywhere else. We compute the boundary slope and Euler characteristic of this surface and show that it matches with and .
Let be a link admitting a near-alternating diagram with a single negative twist region of weight , then the surface is essential with 1 boundary component such that each component has slope and
To see the Jones surface with boundary slope matching and matching , we use the fact that a near-alternating link is -adequate, see Lemma 4.3, as defined below.
A link diagram is -adequate (resp. -adequate) if its all- (resp. all-) state graph (resp. ) has no one-edged loops. A link is semi-adequate (- or -adequate) if it admits a diagram that is - or -adequate. If a link admits a diagram that is both - and -adequate, then we say that is adequate.
Note that alternating knots form a subset of adequate knots.
It is well known that for any link diagram , we have , and the first equality is achieved when is -adequate, while the second equality is achieved when is -adequate. This follows from [LT88], [Lic97, Lemma 5.4], and [FKP13]. Therefore, if is -adequate (resp. -adequate) then there is a single Jones slope in (resp. in ).
If admits an -(resp. -)adequate diagram, then [Oza11] implies that the all- (resp. all-) state surface is essential. An all- or all- state surface was shown by [FKP13] to realize , or , respectively. As noted, a near-alternating diagram is -adequate, so the all- state surface of realizes the Jones slope of and . The surface and the all- state surface of a near-alternating diagram verify the Strong Slope Conjecture for these knots.
Numerical evidence, particularly those from 3-string pretzel knots [LvdV] and fusion knots [GvdV16], suggests that the graphical conditions imposed on a near-alternating knot diagram are the best possible to ensure that Jones slope are integral and realized by state surfaces. In other words, if a knot diagram where is a 2-connected, finite, weighted planar graph without one-edged loops with a single negative edge of weight , so that the quantities and still make sense, we expect that implies that the Jones slope is rational, or, it is not realized by a state surface. We will address this in a future project.
1.3. Generalization to Murasugi sums
We extend Theorem 1 by restricting to certain Murasugi sums, or planar star product, of a near-alternating diagram with an -adequate diagram. We consider a general version of the planar star product (Murasugi sum) of two link diagrams and .
Definition 1.9 (Compare with [Mp89]).
Let and be the all- state graphs of two links diagrams and , respectively. If we glue and along a vertex, we obtain a new graph called the star product of and . The new graph uniquely determines a link diagram which we denote by . We say that is a Murasugi sum, or planar star product of and , and we write .
Theorem 1 generalizes with some restrictions on the Murasugi sum.
Suppose is a link with a diagram that is a Murasugi sum of a near-alternating diagram with a single negative twist region of weight , and an -adequate diagram , such that the circle in along which the Murasugi sum is formed has no one-edged loops, then
An essential spanning surface for with boundary slope and realizing may be formed by taking the Murasugi sum of two spanning surfaces , for the links and as follows [Oza11].
Let be a spanning surface for a link . Suppose that there exists a 2-sphere decomposing into two 3-balls such that is a disk. Put for . Then we say that has a Murasugi decomposition into and and denote it by . Conversely, we say that is obtained from and by a Murasugi sum along a disk .
Suppose a knot is a Murasugi sum of a near-alternating diagram and an -adequate diagram , such that the circle in along which the Murasugi sum is formed has no one-edged loops. Let be the 2-connected, weighted planar graph from which we obtain . The Jones slope is realized by a Murasugi sum of the surface and the all- state surface for , and
As for the question of whether a near-alternating knot can admit an -adequate diagram, we show, using the Kauffman polynomial, that a near-alternating knot cannot admit a diagram that is both - and -adequate.
A near-alternating knot does not admit an adequate diagram.
It is an interesting question whether the colored Jones polynomial can be used to obstruct the existence of an -adequate diagram for a near-alternating knot. The criterion from [Lee16] may be applied if there is information restricting the number of positive crossings in a diagram. We will pursue this question in a future project.
We would like to remark that by [ABB92, Theorem 3.1], every near-alternating link admits an almost-alternating diagram, and it is not known whether every almost-alternating link is semi-adequate.
1.4. Stable coefficients and Coarse volume
Let be the coefficient of of the reduced colored Jones polynomial , where is the th colored Jones of the unknot, and let be the coefficient of , so that are the first, second, penultimate, and last coefficient of , respectively.
Let , the first th coefficient (resp. last th coefficient) of the reduced colored Jones polynomial is stable if (resp. ) for all .
It is known that for an adequate knot, the first and last th coefficient are stable for all [Arm13]. The cases and have first been shown by [DL06]. They also give explicit formulas for the coefficients from the all- and all- state graphs of an adequate diagram of a knot. These have been used to give a two-sided volume bound for alternating knots [DL07]. Futer, Kalfagianni, and Purcell have these coefficients to give two-sided bounds on the volume of a hyperbolic, adequate knot [FKP13]. These results establish that for an adequate knot, the stable coefficients of the colored Jones polynomial are coarsely related to the volume of the knot as defined below.
Let be functions from some (infinite) set to the non-negative reals. We say that and are coarsely related if there exist universal constants and such that
The Coarse Volume Conjecture [FKP13, Question 10.13] asks whether there exists a function of the coefficients of the colored Jones polynomials of every knot , such that for hyperbolic knots, is coarsely related to hyperbolic volume . Here the infinite set is taken to be the set of hyperbolic knots.
We show that a near-alternating knot has stable first, second, last, and penultimate coefficients which are determined by state graphs of a near-alternating diagram. We give a two-sided bound on the volume of a highly twisted, near-alternating knot based on these coefficients. To simplify notation we will just write for , for , for , and for .
Let be a graph without one-edged loops, an edge is called multiple if there is another edge in . The reduced graph of , denoted by , is obtained from keeping the same vertices but replacing each set of multiple edges between a pair of vertices by a single edge. The first Betti number of a graph, denoted by , is the number , where is the number of vertices of , is the number of edges of , and is the number of connected components of .
Let be a link admitting a near-alternating diagram , where is a finite 2-connected, weighted planar graph with a single negatively-weighted edge of weight . The first and second coefficient, , respectively, of the reduced colored Jones polynomial of a near-alternating link are stable. Write and . We have and , where is the Kauffman state corresponding to the state surface and is the first Betti number of the reduced graph of . The last and penultimate coefficient, , respectively, are also stable, and we write and . We have and .
If is such that the near-alternating diagram is prime and twist-reduced with more than 7 crossings in each twist region, then is hyperbolic, and
for a constant . Here is the volume of a regular ideal tetrahedron. In other words, stable coefficients of are coarsely related to the hyperbolic volume of .
The second stable coefficient is computed in terms of the Euler characteristic of the state surface in a formula similar to those given in [DL06, DL07] for adequate knots. Numerical experiments suggest that more coefficients of the reduced colored Jones polynomial should be stable. However, we do not pursue this question in this paper. For the two-sided bound on volume, we use volume estimates based on the twist numbers of a knot developed in [FKP08] using the works of Adams, Agol, Lackenby, and Thurston. For other examples of volume estimates for links admitting different types of diagrams, see [BMPW15] and [Gia15, Gia16].
1.5. Sufficiently positively-twisted links
In the final section of the paper we generalize these results to links with a diagram where has more than one negative edge, so has multiple negative twist regions. We consider the effect of adding full positive twists to the positive twist regions of .
For a weighted planar graph let denote the sub-graph of consisting of the negative edges of , and let be a connected component of .
Let be a knot with a prime, twist-reduced diagram , where is 2-connected, the graph obtained by deleting all the negatively-weighted edges from remains 2-connected, and each connected component of is a single negative edge . In addition, the diagram , where is the graph obtained from by contracting along each edge , is adequate. Assume that has twist regions, and that each region contains at least 7 crossings. Let be the knot obtained from by adding full twists on two strands to every positive twist region. There exists some integer such that for all ,
the Strong Slope Conjecture is true for with spanning Jones surfaces, and
the coefficients , , , and are stable. They give the following two-sided volume bounds for :
for some constant , where is the number of maximal negative twist regions in .
In this theorem it is not determined whether is always non-negative or always non-positive in the two-sided bound, while is always positive. An example of a highly twisted link from a graph satisfying the graphical constraint of the theorem is shown below in Figure 4.
In Section 2, we give a definition of the colored Jones polynomial in terms of skein theory and summarize elementary results needed for Theorem 1, which is proven in Section 3. In Section 4, we prove Theorem 2 by computing the boundary slope and the Euler characteristic of , and we generalize a part of Theorem 1 to Murasugi sums of a near-alternating knot and an -adequate knot by proving Theorem 3 and its corollary. We show Theorem 4, which says that a near-alternating knot is not adequate in Section 5. Finally, we compute stable coefficients and give a coarse volume bound to prove Theorem 5 in Section 6. In Section 7, we prove Theorem 6.
This is a side project that grew out of a project with Roland van der Veen. I would like to thank him for our conversations which made this spin-off possible. I would also like to thank Efstratia Kalfagianni, Stavros Garoufalidis, and Oliver Dasbach for their comments and encouragement on this work, and for their hospitality during my visits. Lastly, I would like to thank Mustafa Hajij for interesting discussions on stability properties of the colored Jones polynomial, Adam Lowrance for pointing out that near-alternating knots are almost-alternating, and Joshua Howie for interesting conversations on the Strong Slope Conjecture.
2. Graphical skein theory
We follow the approach of [Lic97] in defining the Temperley-Lieb algebra. The following formulas are also found in [MV94]. Let be an orientable surface with boundary which has a finite (possibly empty) collection of points specified on . A link diagram on consists of finitely many arcs and closed curves on such that
There are finitely many transverse crossings with an over-strand and an under-strand.
The endpoints of the arcs form a subset of the specified points on .
Two link diagrams on are isotopic if they differ by a homeomorphism of isotopic to the identify. The isotopy is required to fix .
Let be a fixed complex number. The linear skein of is the vector space of formal linear sums over of isotopy classes of link diagrams in quotiented by the relations
We consider the linear skein of the disc with -points specified on its boundary. For , there is a natural multiplication operation defined by identifying the top boundary of with the bottom boundary of . This makes into an algebra , called the Temperley-Lieb algebra. The algebra is generated by crossingless matchings of points of the form shown in Figure 5.
We will denote parallel strands, the identity , also by .
Suppose that is not a th root of unity for . There is an element in called the Jones-Wenzl idempotent, which is uniquely defined by the following properties. For the original reference where the projector was defined and studied, see [Wen87].
belongs to the algebra generated by .
Let be the linear skein of the annuli with no points marked on its boundaries. The image of in obtained by joining the boundary points on the top with the those at the bottom is equal to
where is the quantum integer defined by
Let be a diagram of a link with components. For each component for of take an annuli via the blackboard framing. Let be the map that sends an element of to each in the plane. For , the th unreduced colored Jones polynomial may be defined by substituting into the bracket portion of
This definition of the colored Jones polynomial follows the convention of [KT15], except that their is such that , and we do not multiply by an extra . Note that this gives as the normalization.
The Kauffman bracket here is extended by linearity and gives the polynomial multiplying the empty diagram after reducing the diagram via skein relations. The skein is the blackboard cable of decorated by a Jones-Wenzl idempotent, which we will denote by from now on.
A triple of non-negative integers is called admissible if is even and .
Let be admissible, let be the bracket of the skein shown in Figure 11.
[Lic97, Lemma 14.5]. Let and . Also let and , then is given explicitly by the following formula.
Let be the maximum degree of a Laurent polynomial . We will mainly be concerned with the degree of the terms in the formulas above. For convenience, we will list the degrees of , , and here. They are obtained by examining the formulas.
We will be using the following lemma from [Arm13].
Let be a crossing-less diagram decorated by Jones-Wenzl idempotents , consider the skein obtained from by replacing each of the idempotents by the identity , so consists of disjoint circles. The skein is called adequate if no circle in passes through any of the regions previously decorated by an idempotent more than once.
Lemma 2.6 ([Arm13, Lemma 4]).
Let be a skein decorated by Jones-Wenzl idempotents , and be the skein obtained by replacing each Jones-Wenzl idempotent by the identity element , then
If is a crossing-less skein that is adequate, then
We also use an additional identity from [MV94].
Lemma 2.7 ([Mv94, Lemma 4]).
The slight difference with [MV94] in the coefficient multiplying the right-hand side is due to their slightly different convention for the quantum integer.
3. Jones slopes
If is a near-alternating link with a single negative twist region of weight , then
Our main strategy is to find a suitable state sum for which has a degree-dominating term. If is near-alternating, we may simplify the sum and disregard many of the terms whose skein evaluates to zero in the Kauffman bracket. This is done in Section 3.2. In Section 3.3, we highlight the term in the state sum which will be shown to be degree-dominating. The most laborious step of the proof comes from bounding the degree of a term coming from another state in the state sum. We do this in Section 3.4, where we first estimate the crossings on which chooses the -resolution by Lemma 3.7. The reason why this gives a bound on the degree is given by Lemma 3.4. This leads to the important corollary, Lemma 3.10, which we can apply to the case where is a near-alternating diagram to bound the degree of the term in the state sum corresponding to . Finally in Section 3.5 we put the estimates together to finish the proof of Theorem 7. Upon first reading the reader may skip the proof of Lemma 3.7 to get a sense of how it is applied.
3.2. Simplifying the state sum
Let be a near-alternating link diagram, which means that it has a single negative twist region of weight . We fix . Given the skein , slide the idempotents along the link strands and make copies until there are four idempotents framing the negative twist region. See Figure 13 below.
By the fusion (8) and untwisting (9) formulas, we may fuse the two strands of the negative twist region and get rid of the crossings. This results in a sum over the fusion parameter such that the triple is admissible. For a fixed consider a Kauffman state on the set of remaining crossings. Applying results in a skein that is the disjoint union of a connected component decorated by Jones-Wenzl idempotents with circles. Let
|To simplify notation let , and we write|
After isotopy, we may assume that has the form shown in Figure 14, since other states evaluate to 0 by the Kauffman bracket with a cup/cap composed with an idempotent.
We say that the Kauffman state has split strands, if after isotoping to the form in Figure 14, there are split strands connecting the top and bottom pairs of Jones-Wenzl idempotents.
To further reduce the number of terms to consider in the sum of (15), we prove the following lemma.
Consider a skein with the following local picture.
The skein is zero if .
If , then . When , the skein is not adequate since we have a circle passing through the same idempotent twice, see Figure 15 for an example. Note also that .
Now if is zero, we can slide the top two idempotents down to the bottom one by (6) and get a cap composed with a idempotent which gives 0 for the skein. When , we show by induction on that every term in the sum of the skein from repeatedly expanding the idempotent via (5) has a cap composed with an idempotent after sliding by (6). Thus, every term in the sum is zero and the skein is zero.
Suppose , there are two idempotents and therefore four terms in the sum from expanding via . This takes care of the base case: For any such that , we have that .
Now suppose that and we have that every term when evaluates to 0 by the induction hypothesis for any . We expand the pair of idempotents to get the panel of four figures in Figure 17.
If , then we are done. Otherwise, we again expand the top pair of idempotents to get another panel of 4 figures as shown in Figure 19.
The first three cases reduce to the case with . For the last one we repeat the step of Figure 18 using Lemma 2.7 to keep reducing . Then, expand the top part repeatedly as in the Figure 19 and apply the induction hypothesis to smaller , so that we can look at the last figure in the panel to determine whether we need to apply the step of Figure 18 again. We repeat these last two steps until goes to 0. ∎
3.3. The degree-dominating term in the state sum
Consider the state which chooses the -resolution at all the crossings (recall that the state is applied on the remaining crossings of after getting rid of the negative twist region using the fusion and the untwisting formulas). We have that has 0 split strands and thus for all values of except . A simple computation using Lemma 2.6 shows
The strategy to prove Theorem 7 is then to show that
for any other Kauffman state and .