3-braid Knots Research Articles

3-braid knots with maximal 4-genus

We classify 3-braid knots whose topological 4-genus coincides with their Seifert genus, using McCoy’s twisting method and the Xu normal form. In addition, we give upper bounds for the topological 4-genus of positive and strongly quasipositive 3-braid knots.

A note on closed 3-braid knot shadows

A knot shadow is a diagram without over/ under information at all crossings. Many knot types are obtained from a knot shadow by assigning over/ under information to each crossing. The purpose of the paper is to observe numbers of knot types obtained from closed 3-braid knot shadows. As a result, we discover that given an arbitary odd integer [Formula: see text], there exists a closed 3-braid knot shadow [Formula: see text] such that the number of [Formula: see text] torus knots is more than that of trivial knots in the knots obtained from [Formula: see text] by assigning over/ under information to each crossing.

Colored HOMFLY-PT polynomials of quasi-alternating 3-braid knots

Obtaining a closed form expression for the colored HOMFLY-PT polynomials of knots from 3-strand braids carrying arbitrary SU(N) representation is a challenging problem. In this paper, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter by Qˆ3(m1,−m2,n,ℓ). This family of knots not only generalizes the well-known class of weaving knots but also contains an infinite family of quasi-alternating knots. Interestingly, we obtain a closed form expression for the HOMFLY-PT polynomial of Qˆ3(m1,−m2,n,ℓ) using a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knots Qˆ3(1,−1,n,±1). For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behavior of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the [r]-colored HOMFLY-PT polynomials of quasi alternating knots for small values of r. Remarkably, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to mth Lucas numbers, denoted hereafter as Lm,2n. At the end, we verify that the reformulated invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions F12[a,b,c;z].

3-braid knots do not admit purely cosmetic surgeries

A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. Using recent work of Hanselman, we show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries.

Crossing number of an alternating knot and canonical genus of its whitehead double

A conjecture proposed by J. Tripp in 2002 and modified by T. Nakamura in 2006 states that the crossing number of any alternating knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including (2,m) torus knots, 2-bridge knots, algebraic alternating knots, alternating pretzel knots and so on. In this paper, we prove that the conjecture is not true for any alternating 3-braid knot which is the connected sum of two torus knots of type (2,m) and (2,n) with odd integers m,n≥3. This results in a new modified conjecture that the crossing number of any prime alternating knot coincides with the canonical genus of its Whitehead double. We prove that this modified conjecture holds for all prime alternating 3-braid knots in addition to the known classes above.

Taut foliations, positive 3‐braids, and the L‐space conjecture

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot $P(-2,3,7)$, and indeed along every pretzel knot $P(-2,3,q)$, for $q$ a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 1-bridge braid in $S^3$, where $r <g(K)$.

Donaldson's theorem, Heegaard Floer homology, and knots with unknotting number one

We establish an obstruction to unknotting an alternating knot by a single crossing change. The obstruction is lattice-theoretic in nature, and combines Donaldson's diagonalization theorem with an obstruction developed by Ozsváth and Szabó using Heegaard Floer homology. As an application, we enumerate the alternating 3-braid knots with unknotting number one, and show that each has an unknotting crossing in its standard alternating diagram.

UNKNOTTING NUMBER AND GENUS OF 3-BRAID KNOTS

Let u(K) and g(K) denote the unknotting number and the genus of a knot K, respectively. For a 3-braid knot K, we show that u(K) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T ( 3 , l ) , l ≡ 1 , 2 ( mod 6 ) , and T ( 4 , 5 ) . Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.

Heegaard Floer homology and genus one, one-boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, ϕ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T2-bundle over S1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.

Bennequin’s inequality and the positivity of the signature

. We use an algorithm for special diagrams to prove a Bennequin type inequality tor the signature of an arbitrary link diagram, related to its Murasugi sum decomposition. We apply this inequality to show that the signature of a non-trivial positive 3-braid knot is greater than its genus, and that the signature of a positive braid link is minorated by an increasing function of its negated Euler characteristic. The latter property is conjectured to extend to positive links.

Property P for knots admitting certain Gabai disks

We show that if a knot has a minimal spanning surface that admits certain Gabai disks, then a manifold obtained by ±1-Dehn surgery along the knot has infinite fundamental group. As one of the applications we extend and simplify a recent result of Menasco and Zhang that closed 3-braid knots have Property P. Other applications are given.

Link invariant of the Izergin - Korepin model

The link invariant associated with the Izergin - Korepin 19-vertex model is deduced using the method of statistical mechanics. It is shown that the Izergin - Korepin model leads to an invariant which is precisely the 3-state Akutsu - Wadati polynomial, previously known only for 2- and 3-braid knots. We give a table of the invariant for all knots and links up to seven crossings.

Beyond the triangle and uniqueness relations: non-zeta counterterms at large $N$ from positive knots

Counterterms that are not reducible to ζn are generated by 3F2 hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots (4, 3) = 819 and (5, 3) = 10124, are found in anomalous dimensions at O(1/N3) in the large-N limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pade resummations of e-expansions, which are compared with analytical results in 3 dimensions. The O(1/N3) results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in 1/N one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots 10139 and 10152 are not generated by such self-energy insertions.