Journal of Knot Theory and Its Ramifications

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth–Szabó correction terms, as smooth double sliceness obstructions, vanish for both knots. Recently, Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such a knot that is distinct from Meier’s knots modulo doubly slice knots.

In this paper, we use theory of embedded graphs on oriented and compact $PL$-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter’s circles. For the case of signed Gauss words, we use a generating set of $H_1(S_w,\mathbb{Z})$, given in [G. Cairns and D. Elton, The Planarity problem for signed Gauss world, J. Knots Theor. Ramif. 2(4) (1993) 359–367], and the intersection pairing of immersed $PL$-normal curves to present a short solution of the signed Gauss word problem. We relate this solution with the one given by Cairns and Elton. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus $PL$-surface.

We connect the characterization of signed Gauss paragraph with the recognition virtual links problem. Also we present a combinatorial algorithm to compute, in an easier way, skew-symmetric graded matrices [V. Turaev, Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] for virtual knots through the concept of triplets [M. Toro and J. Rodríguez, Triplets associated to virtual knot diagrams, Rev. Integración (2011)]. Therefore, we can prove that the Kishino’s knot is not classical, moreover, we prove that the virtual knots of the family $B$ given in [H. A. Dye, Virtual knots undetected by $1$ and $2$-strand bracket polynomials, Topol. Appl. 153 (2005) 141–160] are not classical knots.

Let $D(K)$ be the positively clasped untwisted Whitehead double of a knot $K$, and $T_{p,q}$ be the $(p,q)$ torus knot. We show that $D(T_{2,2m+1})$ and $D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group ${𝒞}$ for each $m\ge 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$ generate a $\mathbb{Z}\oplus \mathbb{Z}$ summand in the subgroup of ${𝒞}$ generated by topologically slice knots. We use the concordance invariant $\delta$ of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the $\tau$-invariant of knot Floer theory and the $s$-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of $D(T_{2,2m+1})$ for any $m\ge 1$ generalizing a result for $T_{2,3}$ of Hedden, Kim and Livingston.

A $2n$-component non-splittable spatial graph $G=G_1\cup \cdots \cup G_{2n}$ in $S^3$ possibly admits a smooth involution on its exterior whose fixed point set is a closed surface $F$ of genus $g\ge 1$. It is known that $n\ge g$ holds if $G$ is a graph link in $S^3$. In this paper, we consider $G$ to be a hyperbolic spatial graph in a closed orientable 3-manifold $M$. We first study the case where $M$ is the 3-sphere, and provide a method for constructing $G$ and $F$ with $a_1+\cdots +a_n>g>1$, where $1-a_i$ is the Euler characteristic of $G_i$ for $1\le i\le n$. Next, we study the case where $M$ is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.

Let $M_i=V_i{\cup }_{S_i}{W}_i$ $(i=1,2)$ be a Heegaard splitting of compact orientable $3$-manifold, and $M=V{\cup }_{S}W=(V_1{\cup }_{S_1}{W}_1){\cup }_F(V_2{\cup }_{S_2}{W}_2)$ be an amalgamation along an essential surface $F$. In this paper, we show that if there is an essential disk $D_2$ in $V_2$, such that $D_2$ cuts $V_2$ into two product $I$-bundles, one of which is $F\times [0,1]$, and $d(S_i)\ge 3$ $(i=1,2)$, then $M=V{\cup }_{S}W$ is unstabilized and $S$ is uncritical. We also give a sufficient condition for $S$ to be critical.

In this paper, the properties of the Kauffman bracket skein module (KBSM) of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the $3$-sphere.

We prove that the degree of the Hilbert polynomial of the HOMFLYPT homology of a closed braid $B$ is $l-1$, where $l$ is the number of components of $B$. This controls the growth of the HOMFLYPT homology with respect to its polynomial grading. The Hilbert polynomial also reveals a link polynomial hidden in the HOMFLYPT polynomial.

We establish a simple formula to compute the incidence signs in the spectral sequence used by Vassiliev to define finite-type cohomology classes in the space of long knots. We consider only the subcomplex of tree diagrams, which carries all the cohomology in low degrees and is the most difficult to compute. On the way, it is shown that by getting rid of the most complicated ingredients in the boundary maps, one obtains an isomorphic complex.

We give an intrinsic (not diagrammatic) characterization of almost alternating knots by generalizing Howie and Greene’s characterization of alternating knots. We also introduce a notion of a spanning surface defect by using the Gordon–Litherland pairing of (possibly non-orientable) spanning surfaces of the knot, and show that it is an alternating distance, a quantity which measures to what extent the knot is far from alternating.

In this paper, we prove that embeddings of right-angled Artin group $A_1$ on the complement of a linear forest into another right-angled Artin group $A_2$ can be reduced to full embeddings of the defining graph of $A_1$ into the extension graph of the defining graph of $A_2$.

The Tait conjecture states that alternating reduced diagrams of links in $S^3$ have the minimal number of crossings. It has been proved in 1987 by Thistlethwaite, Kauffman and Murasugi studying the Jones polynomial. In [A. Carrega, The Tait conjecture in $S^1\times S^2$, J. Knot Theory Ramifications 25(11) (2016) 1650063], the author proved an analogous result for alternating links in $S^1\times S^2$ giving a complete answer to this problem. In this paper, we extend the result to alternating links in the connected sum ${\#}_g(S^1\times S^2)$ of $g$ copies of $S^1\times S^2$. In $S^1\times S^2$ and ${\#}_2(S^1\times S^2)$, the appropriate version of the statement is true for ${\mathbb{Z}}_2$-homologically trivial links, and the proof also uses the Jones polynomial. Unfortunately, in the general case, the method provides just a partial result and we are not able to say if the appropriate statement is true.