Related papers: Census L-space knots are braid positive, except fo…
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…
We consider the classical pretzel knots $P(a_1, a_2, a_3)$, where $a_1, a_2, a_3$ are positive odd integers. By using continuous paths of elliptic $\mathrm{SL}_2(\mathbb R)$-representations, we show that (i) the 3-manifold obtained by…
Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving…
We study the twisted Alexander polynomial $\Delta_{K,\rho}$ of a knot $K$ associated to a non-abelian representation $\rho$ of the knot group into $SL_2(\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every…
The $L$-space conjecture asserts the equivalence, for prime 3-manifolds, of three properties: not being an $L$-space, having a left-orderable fundamental group, and admitting a co-oriented taut foliation. We investigate these properties for…
A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the…
We show, using Mellit's recent results, that K\'alm\'an's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids.
We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $\delta$ are fibered. We classify links which admit…
A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a…
Let {T_n} be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran, Harvey, and Horn. It is known that for each n not equal to 1 the quotient group T_n/T_{n+1} has infinite rank…
Let $K$ be a tame knot embedded in $\mathbf{S}^3$. We address the problem of finding the minimal degree non-cyclic cover $p:X \rightarrow \mathbf{S}^3 \smallsetminus K$. When $K$ has non-trivial Alexander polynomial we construct finite…
If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q)…
The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…
In this short note, we show that the twisted Alexander polynomial associated to a parabolic SL(2,C)-representation detects genus and fibering of the twist knots. As a corollary, a conjecture of Dunfield, Friedl and Jackson is proved for the…
We prove that the Alexander polynomials of certain families of alternating 4-braid knots satisfy Fox's Trapezoidal Conjecture. Moreover, we give explicit formulas for the signature and for the first 4 coefficients of the Alexander…
By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this…
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We…
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that…
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the…