Related papers: Upsilon invariants of L-space cable knots
Ozsvath-Stipsicz-Szabo recently defined a one-parameter family, upsilon of K at t, of concordance invariants associated to the knot Floer complex. We compare their invariant to the {-1, 0, 1}-valued concordance invariant epsilon, which is…
The Upsilon invariant is a concordance invariant in knot Floer homology. F\"{o}ldv\'{a}ri reconstructed the Upsilon invariant using grid homology. We prove that the Upsilon invariant in knot Floer homology and one in grid homology are…
Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise…
We compute the involutive concordance invariants for the 10- and 11-crossing (1,1)-knots.
The $\Upsilon$ invariant is a concordance invariant defined by using knot Floer homology. F\"{o}ldv\'{a}ri gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\Upsilon$ invariant for balanced spatial…
In this paper we construct an infinite family of knots with vanishing Upsilon invariant $\Upsilon$, although their secondary Upsilon invariants $\Upsilon^2$ show that they are linearly independent in the smooth knot concordance group. We…
We present a comprehensive classification of invariants of knots and links associated with irreducible representations of \uqslN{}, when the parameter of quantization $q$ is a root of unity. We demonstrate that, besides the standard…
We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…
We compute the involutive knot invariants for pretzel knots of the form P(-2,m,n) for m and n odd and greater than or equal to 3.
We give a formula for the $\tau$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $\epsilon(K)=1$. When…
According to the idea of Ozsv\'ath, Stipsicz and Szab\'o, we define the knot invariant $\Upsilon$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that…
We prove a cabling formula for the concordance invariant $\nu^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.
In this article we present formulae for q-integration on quantum spaces which could be of particular importance in physics, i.e. q-deformed Minkowski space and q-deformed Euclidean space in 3 or 4 dimensions. Furthermore, our formulae can…
We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $\Upsilon$ recently introduced by…
Considering a multi-dimensional $q$-oscillator invariant under the (non quantum) group $U(n)$, we construct a $q$-deformed Levi-Civita epsilon tensor from the inner product states. The invariance of this $q$-epsilon tensor is shown to yield…
I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.
We introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsv\' ath, Stipsicz and Szab\' o in \cite{OSS4}. By means of holomorphic triangles counting, we prove equivalence with the analytic theory for a…
The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations…
An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link $K_{rm,rn}$ is an L-space link if and only if K is an L-space knot and…