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This paper is a brief overview of recent results by the authors relating
colored Jones polynomials to geometric topology. The proofs of these results
appear in the papers [arXiv:1002.0256] and [arXiv:1108.3370], while this survey
focuses on the main ideas and examples.
Contexts in source publication
Context 1
... and state graphs. Associated to a diagram D and a crossing x of D are two link diagrams, each with one fewer crossing than D. These are obtained by removing the crossing x, and reconnecting the diagram in one of two ways, called the A-resolution and B-resolution of the crossing, shown in Figure 1. For each crossing of D, we may make a choice of A-resolution or B-resolution, and end up with a crossing-free diagram. ...
Context 2
... components of this diagram are called state circles. For each crossing x of D, attach an edge from the state circle on one side of the crossing to the other, as in the dashed lines of Figure 1. Denote the resulting graph by H σ . ...
Context 3
... this example, it turns out that the manifold M A = S 3 S A contains a single essential product disk. This disk D is shown in Figure 10. Observe that this lone EPD corresponds to the single 2-edge loop in G A . ...
Context 4
... the actual hyperbolic volume of the link in Figure 9 is vol(S 3 K) ≈ 11.3407. Turning to the all-B resolution, Figure 11 shows the graphs H B , G B , and G ′ B , as well as the state surface S B . This time, G ′ B is a tree, thus Theorem 3.2 (applied to a reflected diagram) implies S B is a fiber. ...
Context 5
... important point to note is that even though S 3 K is fibered, it does contain surfaces (such as S A ) with quite a lot of guts. Conversely, having |β ′ K | > 0, as we do here, Figure 11. The graphs H B , G B , and G ′ B for the link of Figure 9, and the corresponding surface S B . ...
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... Figure 12, the state surface for this example is shown lying below the projection plane (although soup cans have been smoothed off at their sharp edges in this figure). ...
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... form the disk that we cut along, we isotope the disk by pushing it under the projection plane slightly, keeping its boundary on the state surface S A , so that it meets the link a minimal number of times. Indeed, because S A itself lies on or below the projection plane Figure 12. The surface S A is hanging below the projection plane. ...
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... disk lies slightly below the projection plane everywhere. Figure 13 gives an example. ...
Context 9
... a result, each lower polyhedron is combinatorially identical to the checkerboard polyhedron of an alternating sub-diagram of D(K), where the sub-diagram corresponds to region R. This is illustrated in Figure 14, for the knot diagram of Figure 4. ...
Context 10
... ideal vertices of the upper polyhedron correspond to strands of the link between undercrossings. In Figure 16, the surface S A is shown in green and gold. There are four ideal edges meeting a crossing, labeled e 1 through e 4 in this figure. ...
Context 11
... surface S A runs through the crossing in a twisted rectangle. Looking again at Figure 16, note that the gold portion of S A at the top of that figure is not cut off at the crossing by an ideal edge terminating at v 2 . Instead, it follows e 4 through the crossing and along the underside of the figure, between e 4 and the link. ...
Context 12
... visualize these thin portions of shaded face between an edge and the link as tentacles, and sketch them onto H A running from the top-right of a segment to the base of the segment, and then along an edge of H A (Figure 17). Similarly for the bottom-left. ...
Context 13
... Figure 17, we have drawn the tentacle by removing a bit of edge of H A . When we do this for each segment, top-right and bottom-left, the remaining connected components of H A correspond exactly to ideal vertices. ...
Context 14
... we observed above, edges terminate at undercrossings. Figure 18 shows the portions of ideal edges sketched schematically onto H A . ...
Context 15
... that a tentacle will continue past segments of H A on the opposite side of the state circle without terminating, spawning new tentacles, but will terminate at a segment on the same side of the state surface. This is shown in Figure 19. Since H A is a finite graph, the process terminates. ...
Context 16
... every non-prime arc indicates that a lower polyhedron violates Definition 5.2. See Figure 21. ...
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Citations
... This is the construction of the famous Jones polynomial [93]. Some surveys on the Jones polynomial are [1,71,113]. ...
... If we restrict to hyperbolic Montesinos links, lower bounds for volumes that depend on the combinatorics of the link diagram are provided in [10] and [14]. Other categories of links with lower bounds include prime link diagrams with twist-reduced diagrams of twist number at least two and all twist regions containing at least seven crossings [12], homogeneously adequate links [13], and positive braids [14,15]. ...
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... 2. An adequate knot (and thus in particular an alternating knot) has Jones period p K = 1, two Jones slopes and two corresponding Jones surfaces each with a single boundary component [6,4]. Note, that the characteristic Jones surfaces are spanning surfaces that are often non-orientable. ...
We show that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure 8 knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial
... Here, we abuse symbols RI − and S − because Observation 2 holds. In the following figures (Figs. [8][9][10][11], dotted curves indicate connections of curves. Fig. 8. ...
... Now we can choose D that is a reduced alternating diagram of K. Recall Facts 11-13. Readers can refer to [8]. In the following, let t(D), v 3 , and v 8 be as in Notation 3. ...
... Futer, Kalfagianni and Purcell gave results analogous to Fact 11 using theory of guts. For example, they stated Fact 12 [8]. ...
In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).
... 2. An adequate knot (and thus in particular an alternating knot) has Jones period p K = 1, two Jones slopes and two corresponding Jones surfaces each with a single boundary component [6,4]. Note, that the characteristic Jones surfaces are spanning surfaces that are often non-orientable. ...
We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial.
... The following theorem, proved in [37], provides an estimate that has proved useful for lower bounds on the volume of knot complements. Earlier results in the same vein include an asymptotic estimate by Neumann and Zagier [70], as well as a cone-deformation estimate by Hodgson and Kerckhoff [51]. ...
... Theorem 4.12 leads to diagrammatic volume bounds for several classes of hyperbolic links. For example, the following theorem from [37] gives a double-sided volume bound similar to Theorem 4.10. ...
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.
... Thistlethwaite [Thi88] later studied the Kauffman polynomial of adequate links. Semi-adequate links play an important role in hyperbolic geometry (see [FKP15], [FKP14] and references within) and in the study of the colored Jones polynomial (see [Arm13,FKP13,KL14,Kal18]). ...
The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.
... The following theorem, proved in [37], provides an estimate that has proved useful for lower bounds on the volume of knot complements. Earlier results in the same vein include an asymptotic estimate by Neumann and Zagier [70], as well as a cone-deformation estimate by Hodgson and Kerckhoff [51]. ...
... Theorem 4.12 leads to diagrammatic volume bounds for several classes of hyperbolic links. For example, the following theorem from [37] gives a double-sided volume bound similar to Theorem 4.10. ...
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.
... Next, we apply the notion of partial duality, as defined by Chmutov in [4], to the ribbon graph associated to the black checkerboard state of a checkerboard-colored link diagram in order to give necessary and sufficient conditions for a link diagram to be σ-adequate and σ-homogeneous (also called homogeneously adequate) with respect to a given state. The family of links with homogeneously adequate diagrams has been studied by a number of authors ( [2], [6], [7], [19]). Theorem 1.5. ...
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... Adequate knots form a large class of knots that behaves well with respect to Jones-type knot invariants and has nice topological and geometric properties [1,2,6,8,7,9,10,11,12,24,27]. Several well known classes of knots are adequate; these include all alternating knots and Conway sums of strongly alternating tangles. ...
