Journal of Knot Theory and Its Ramifications

Let $S_g$ be the closed-oriented surface of genus $g$ and let ${\text{Mod}}^{\pm }(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that ${\text{Mod}}^{\pm }(S_g)$ can be generated by two torsion elements. One of these generators is of order $2$, and the other one is of order $4g+2$.

We introduce a Kauffman–Jones type polynomial ${{ℒ}}_{\gamma }(A)$ for a curve $\gamma$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial ${{ℒ}}_{\gamma }(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $\gamma$. As an application, we obtain an estimate in terms of the span of ${{ℒ}}_{\gamma }(A)$ for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of ${{ℒ}}_{\gamma }(A)$ and show some computational results on the span of ${{ℒ}}_{\gamma }(A)$.

Let $M_i$ $(i=1,2)$ be a perfect $3$-manifold, $F_i$ be a component of $\partial M_i$, $f:F_1\to F_2$ be a homeomorphic map, $M=M_1{\cup }_f{M}_2$ and $F=M_1\cap M_2(\cong F_i)$. In this paper, we show that if $d(M_i)\ge 3$ $(i=1,2)$ and $d(f)\ge 2$, then $g(M)=g(M_1)+g(M_2)-g(F)$. As a corollary, if $F_i^j$ $(i=1,2,1\le j\le n)$ is a component of $\partial M_i$, $f_j:F_1^j\to F_2^j$ is a homeomorphic map, $M=M_1{\cup }_{f_1,\dots ,f_n}{M}_2$, ${\bigcup }_{j=1}^n{F}^j=M_1\cap M_2(\cong F_i^j)$, $d(M_i)\ge 3$ $(i=1,2)$ and $d(f_j)\ge 2$ $(1\le j\le n)$, then $g(M)=g(M_1)+g(M_2)-g(F^1)-\cdots -g(F^n)+n-1$.

Let $S_1$ and $S_2$ be connected orientable surfaces of genus $g_1,g_2\ge 3$, $n_1,n_2\ge 0$ punctures, and empty boundary. Let also $\phi :{\mathscr{H}}{𝒯}(S_1)\to {\mathscr{H}}{𝒯}(S_2)$ be an edge-preserving alternating map between their Hatcher–Thurston graphs. We prove that $g_1\le g_2$ and that there is also a multicurve of cardinality $g_2-g_1$ contained in every element of the image. We also prove that if $n_1=0$ and $g_1=g_2$, then the map $\tilde{\phi }$ obtained by filling the punctures of $S_2$, is induced by a homeomorphism of $S_1$.

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q^s(s{l}_2)$ denote the quantum group with large center corresponding to the Lie algebra $s{l}_2ℂ$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q^s(s{l}_2)$ is an $N$-dimensional irreducible representation $\rho :U_q^s(s{l}_2)\to M_N(ℂ)$, so that $\rho (E^N)=aId$ with $a\ne 0$, $\rho (F^N)=0$ and $\rho (K^N)=Id$. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for $(1,1)$-tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev’s invariant.

If $X$ is an orientable, strongly minimal $P{D}_4$-complex and ${\pi }_1(X)$ has one end, then it has no nontrivial locally finite normal subgroup. Hence, if $\pi$ is a 2-knot group, then (a) if $\pi$ is virtually solvable, then either $\pi$ has two ends or $\pi \cong \mathrm{\Phi }$, with presentation $〈a,t|ta=a^{2}t〉$, or $\pi$ is torsion-free and polycyclic of Hirsch length 4 (b) either $\pi$ has two ends, or $\pi$ has one end and the center $\zeta \pi$ is torsion-free, or $\pi$ has infinitely many ends and $\zeta \pi$ is finite, and (c) the Hirsch–Plotkin radical $\sqrt{\pi }$ is nilpotent.

In this paper, we show that if 0-surgery along a ribbon knot $𝔰$ and 0-surgery along another knot $𝔟$ give diffeomorphic 3-manifolds then $𝔟$ has to be a slice knot. Moreover, they have diffeomorphic slice disk exteriors $D^4-N(D_{𝔰}^2)\cong D^4-N(D_{𝔟}^2)$ for some ribbon disk $D_{𝔰}^2$ and slice disk $D_{𝔟}^2$, where $N(D_{𝔰}^2)$ and $N(D_{𝔟}^2)$ are tubular neighborhoods of $D_{𝔰}^2$ and $D_{𝔟}^2$, respectively.

We give explicit formulas for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors’ previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge’s knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers.

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard $(p,q)$-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, that is, $(p-1)(q-1)/2$.

Let $A$ be a nonzero complex number which is not a root of unity. Let $F$ be a compact oriented surface, the $\mathrm{\text{SU}}(3)$-skein space of $F$, $S_A(F)$, is the vector space over $ℂ$ generated by framed oriented links (including framed oriented trivalent graphs in $F\times I$) quotient by the $\mathrm{\text{SU}}(3)$-skein relations due to Kuperberg [Spiders for rank $2$ Lie algebra, Comm. Math. Phys. 180(1) (1996) 109–151]. For closed $F$, with genus greater than $1$, we construct a local diffeomorphism invariant trace on $S_A(F)$ when $A$ is a positive real number not equal to $1$.