The braid indices of the reverse parallel links of alternating knots (Q6614593)

The braid index \(b(L)\) of a link \(L\) (the minimal number of strings of a braid whose closure is \(L\)) has received considerable study since the 1980s. This partly originates from a lower bound due to the second author [\textit{H. R. Morton}, Math. Proc. Camb. Philos. Soc. 99, 107--109 (1986; Zbl 0588.57008)] and Franks-Williams [\textit{J. Franks} and \textit{R. F. Williams}, Trans. Am. Math. Soc. 303, 97--108 (1987; Zbl 0647.57002)]. This Morton-Franks-Williams (MFW) inequality remains the main algebraic method in studying the braid index up to date, despite alternatives and modifications (including its ``cablings'', and the geometric methods of Birman and Menasco).\N\NWhen \(P(v,z)\) is the HOMFLY-PT polynomial, then \N\[\N\operatorname{MFW}(L):=\frac{1}{2}\operatorname{span}_v P(L)+1\le b(L)\,, \N\tag{1}\N\] \Nwhere the left hand-side is the \emph{MFW bound} for \(b(L)\).\N\NIt had been early observed that the inequality is often exact (i.e., an equality), and it was temporarily conjectured that this is always the case for alternating links. However, Murasugi-Przytycki found some counterexamples, which for knots start with 18 crossings. Thus the problem to determine the braid index is not completely resolved even for alternating links.\N\NNevertheless, there have been extensive partial results. Beside those the authors mention in their introduction, there are the Murasugi results for 2-bridge and fibered alternating links [\textit{K. Murasugi}, Trans. Am. Math. Soc. 326, No. 1, 237--260 (1991; Zbl 0751.57008)], the second of which was extended by the first author, Hetyei and Liu [\textit{Y. Diao} et al., Math. Proc. Camb. Philos. Soc. 168, No. 3, 415--434 (2020; Zbl 1446.57002)]. Also, it is known by the reviewer that $(1)$ is exact for alternating links of braid index up to 5 [\textit{A. Stoimenow}, Properties of closed 3-braids and braid representations of links. Cham: Springer (2017; Zbl 1433.57001)], and for alternating \emph{knots} up to genus 4 [\textit{A. Stoimenow}, Diagram genus, generators, and applications. Boca Raton, FL: CRC Press (2016; Zbl 1336.57016)]. (The Murasugi-Przytycki examples have braid index 6 and the knots among them genus at least 6.)\N\NWhat the authors prove in the present paper is that the braid index of the \emph{reverse 2-parallel} \(K_f\) of framing \(f\) of every alternating knot \(K\) can be determined by an explicit formula, which expresses \(b(K_f)\) in terms of \(f\) and the checkerboard coloring of a reduced alternating diagram of \(K\). They establish that $(1)$ is exact in all these cases.\N\N(A) Lower bound. To determine \(\operatorname{MFW}(K_f)\), two main tools are invoked. The first is Rudolph's congruence [\textit{L. Rudolph}, Math. Proc. Camb. Philos. Soc. 107, No. 2, 319--327 (1990; Zbl 0703.57005)], stated in Theorem 2.2, which relates \(P(K_f)\) modulo 2 to the Kauffman polynomial \(F(K)\). For the purpose of applying $(1)$, write \(\operatorname{MFW}_{\!\!\bmod 2}(L)\) for the bound obtained from \(\operatorname{MFW}(L)\) when the polynomial \(P(L)\) has its coefficients reduced modulo 2, so that\N\[\N\operatorname{MFW}_{\!\!\bmod 2}(L)\le \operatorname{MFW}(L)\le b(L)\,.\N\]\N\NThe second tool (Theorem 2.3) is a refinement of Thistlethwaite's work [\textit{M. B. Thistlethwaite}, Invent. Math. 93, No. 2, 285--296 (1988; Zbl 0645.57007); see in particular Corollary 1.1(iv)] in the case of alternating links, due to \textit{P. R. Cromwell} [Banach Cent. Publ. 42, 57--64 (1998; Zbl 0904.57004)]. It exhibits an earlier result known to \textit{Y. Yokota} [Topology Appl. 65, No. 3, 229--236 (1995; Zbl 0836.57006)],\N\[\N\operatorname{span}_a F(K)=c(K)\N\]\Nfor an alternating link \(K\) (with \(c(K)\) being the crossing number of \(K\)), through explicit \emph{odd} coefficients \N\[\N[F(K)]_{z^{k_1}a^{l_1}}= [F(K)]_{z^{k_2}a^{l_2}}=1 \mbox{\ \ with\ \ \(l_2-l_1=c(K)\).} \N\tag{2}\N\]\N\NWhen \(D\) is a reduced alternating diagram of \(K\), and \(D\) is not a \((2,n)\)-torus knot diagram (we may assume \(n\) odd), then \(k_1,k_2>1\). This avoids potential cancellations and easily implies that the numbers stated in the authors' abstract are lower bounds for \(b(K_f)\).\N\NHowever, if \(D\) is a \((2,n)\)-torus knot diagram, then one of \(k_1,k_2\) in $(2)$ is \(1\), and there is exactly one framing \(f\) for which \(\operatorname{MFW}_{\!\!\bmod 2} (K_f)< c(K)+2\). The authors then go into a calculation with the skein algebra (Theorem 2.7) to show that the relevant coefficient of \(P(K_f)\) is \(-1\), and not \(+1\) (pay attention to their normalization), so that still \(\operatorname{MFW}(K_f)\ge c(K)+2\).\N\N(B) Upper bound. This is done, as usual, by exhibiting explicit braid representatives of the links \(K_f\). For this purpose, the authors apply a construction considered by \textit{I. J. Nutt} [Math. Proc. Camb. Philos. Soc. 126, No. 1, 77--98 (1999; Zbl 0918.57003)] on suitably chosen grid diagrams/arc presentations of \(K\).\N\NThe paper also contains some comments on rope length and Birman-Menasco-Ng's study of the braid index of satellite links.\N\NRemark: The following refers to forthcoming work of the reviewer with Gyo Taek Jin and Hwa Jeong Lee.\N\NThe sum in Theorem 2.7 extends only over \(i\le 4\), not \(i<2n\). This property, which yields a very peculiar ``panhandle'' of the \(P\) polynomials in question, is not evident at all \emph{a priori}. It can be established by a computer calculation. This behavior appears to extend to other torus knots, and understanding its reasons opens a topic of further research.\N\NAnother way to avoid the complexity of the proof of Theorem 2.7 is to use Bennequin's inequality. This will not confirm that $(1)$ is always exact, but still determine the braid index of \(K_f\) as stated by the authors.\N\NWe also show that the authors' main result can be extended to a certain larger class of knots, including all prime knots up to 10 crossings except \(10_{132}\). The values of the braid index and the conditions on the three cases are expressed in terms of the maximal Thurston-Bennequin invariant and arc index.