Symmetric quandle colorings and ribbon concordance

A surface-link is a closed surface smoothly embedded in $\mathbb{R}^{4}$ up to ambient isotopy. A surface-knot is a connected surface-link. A 2-knot is a surface-knot diffeomorphic to $S^{2}$. A surface-knot is unknotted if it bounds a handlebody or is the connected sum of some number of unknotted projective planes. Broken sheet diagrams of the two unknotted projective planes, distinguished by their self-intersection numbers or normal Euler numbers, are shown in Figure 1.

Let $F_{0}$ and $F_{1}$ be surface-links with components of the same genus and orientability. If there is a concordance $C$ $(\cong F_{0}\times[0,1]\cong F_{1}\times[0,1])$ in $S^{4}\times[0,1]$ between $F_{1}\subset S^{4}\times\{1\}$ and $F_{0}\subset S^{4}\times\{0\}$ such that the restriction to $C$ of the projection $S^{4}\times[0,1]\to[0,1]$ is a Morse function with critical points of index 0 and 1 only, then $F_{1}$ is ribbon concordant to $F_{0}$ denoted

A singular link diagram is an immersed link diagram in the plane with crossings and traverse double points called vertices. At each vertex assign a marker, a local choice of two non-adjacent regions in the complement of the vertex. Such a marked singular link diagram is called a ch-diagram [CKS, kamada2017surface, kamadakim, ch]. One of the two smoothings of a vertex connects the two regions of the marker, the positive resolution $L^{+}$, and one separates the marker’s regions, the negative resolution $L^{-}$, see Figure 3. If $L^{-}$ and $L^{+}$ are unlinks, then the ch-diagram is said to be admissible. Admissible ch-diagrams represent surface-links and induce broken sheet diagrams and every surface-link defines an admissible ch-diagram.

Including the 3 Reidemeister moves of classical link diagrams, there are 8 moves on ch-diagrams called Yoshikawa moves [ch, kamada2017surface]. Two admissible ch-diagrams represent equivalent surface-links if and only if they are Yoshikawa move equivalent. The ch-index of a ch-diagram is the number of marked vertices plus the number of crossings. The ch-index of a surface-link $F$, denoted ch($F$), is the minimum of ch-indices ch$(D)$ among all admissible ch-diagrams $D$ representing $F$. A sufrace-link $F$ is said to be weakly prime if $F$ is not the connected sum of any two surfaces $F_{1}$ and $F_{2}$ such that ch$(F_{i})$ $<$ ch$(F)$. Yoshikawa classified weakly prime surface-links whose ch-index is 10 or less in [ch]. He generated a table of their representative ch-diagrams up to orientation and mirror. His notation is of the form $I_{k}^{g_{1},\dots,g_{c}}$ where $I$ is the surface-link’s ch-index and $|g_{1}|,\dots,|g_{c}|$ are the genera of its components with $g_{i}<0$ implying the component is non-orientable.

There is a translation of Reidemeister moves in a motion picture to sheets in a broken sheet diagram. Associate the time parameter of a Reidemeister move with the height of a local broken sheet diagram. A Reidemeister III move gives a triple point diagram, a Reidemeister I move corresponds to a branch point, and a Reidemeister II move corresponds to a maximum or minimum of a double point curve, see [kamadakim]. Triple points of the induced broken sheet diagram are in correspondence with the Reidemeister III moves in the motion picture. Include sheets containing saddles for each saddle point in the motion picture to generate a broken sheet diagram of the entire surface-link.

Consider an admissible ch-diagram. There are finite sequences of Reidemeister moves that take $L^{-}$ and $L^{+}$ to crossing-less diagrams $O^{-}$ and $O^{+}$. Translate these Reidemeister moves to a broken sheet diagram. In between the still of $L^{-}$ and $L^{+}$ and for each marked vertex include sheets containing the saddles traced by transitioning from $L^{-}$ to $L^{+}$ in the local picture of Figure 3. These saddles are pictured in Figure 5. Finally, cap-off $O^{-}$ and $O^{+}$ with embedded disks to produce a broken sheet diagram.

An $(X,\rho)$-coloring of a ch-diagram is a $(X,\rho)$-coloring of the link diagram such that the arcs meeting at a vertex are given the same color and their two orientations related by a basic inversion are shown in Figure 6. There is a one-to-one correspondence between colorings of ch-diagram and the colorings of the surface-link it represents. The Reidemeister moves in the trivialization of $L^{-}$ and $L^{+}$ uniquely extend the ch-diagram’s coloring to the induced broken sheet diagram. Kamada, Kim, and Leem give a detailed account of coloring ch-diagrams and calculating their symmetric quandle cocycle invariant in [kamadakim].

Let $D_{0}$ and $D_{1}$ be broken sheet diagrams of the surface-knots $F_{0}$ and $F_{1}$. Let $\cup_{i=1}^{n}S_{i}$ be a disjoint union of broken sheet diagrams each representing an unknotted surface-knot. Suppose that $D_{1}$ is obtained from $D_{0}\cup(\cup_{i=1}^{n}S_{i})$ by surgery along sufficiently thin 1-handles $\cup_{j=1}^{m}h_{j}$ that intersect $D_{0}\cup(\cup_{i=1}^{n}S_{i})$ along meridian 2-disks of $h_{j}$. Then,

The symmetric quandle 3-cocycle invariant can also be used to obstruct ribbon concordance. For every symmetric quandle $(X,\rho)$ and $(X,\rho)$-set $Y$ there is an associated chain and cochain complex. Symmetric quandle 3-cocycles are cocycles of this cochain complex and represent cohomology classes of $H^{3}_{Q,\rho}(X;A)_{Y}$ for any abelian group $A$, see [carter2009symmetric, kamada2017surface, sym].

Let $C$ be an $(X,\rho)_{Y}$-coloring of a broken sheet diagram $D$. For a triple point $t$ of $D$, choose 1 of the 8 3-dimensional complementary regions around $t$ and call the region specified. There are 12 semi-sheets around a triple point. Let $S_{B}$, $S_{M}$, and $S_{T}$ be the 3 of them that face the specified region, where $S_{B}$, $S_{M}$, and $S_{T}$ are semi-sheets of the bottom, middle, and top sheet respectively. Let $n_{B}$, $n_{M}$, and $n_{T}$ be the normal orientations of $S_{B}$, $S_{M}$, and $S_{T}$. Through basic inversions, it is assumed that each normal orientation points away from the specified region. Let $x_{1},x_{2},$ and $x_{3}$ be the elements of $X$ assigned to the semi-sheets $S_{B}$, $S_{M}$, and $S_{T}$ whose normal orientations $n_{B}$, $n_{M}$, and $n_{T}$ point away from the specified region. Additionally, let $y$ be the color of the specified region. The color of the triple point $t$ is $C_{t}=(y,x_{1},x_{2},x_{3}).$

Let $\phi:\mathbb{Z}(Y\times X^{3})\to A$ be a symmetric quandle 3-cocycle of $(X,\rho)_{Y}$ with coefficients in $A$. For a $(X,\rho)_{Y}$-coloring $C$, the ${\it\phi-weight}$ of the triple point $t$ is defined by $\varepsilon\phi(y,x_{1},x_{2},x_{3})$ such that $\varepsilon$ is +1 (or -1) if the triple of the normal orientations $(n_{T},n_{M},n_{B})$ is (or is not) coherent with the orientation of $S^{3}\subset\mathbb{R}^{4}$ at the triple point and is denoted $W_{\phi}(t;C)$. The triple point of Figure 8 is positive. The ${\it\phi-weight}$ of a diagram $D$ with respect to a symmetric quandle coloring $C$ is

where $t$ runs over all triple points of $D$. The value $W_{\phi}(D;C)$ is an invariant of an $(X,\rho)_{Y}$-colored surface-link $(F,C)$ [sym, oshiro]. Denote $W_{\phi}(D;C)$ by $W_{\phi}(F;C)$.

The symmetric quandle cocycle invariant of a broken sheet diagram $D$ representing a surface-link $F$ is the multi-set

The multi-set $\Phi_{\phi}(D)$ is an invariant of the $(X,\rho)_{Y}$-colored surface-link $F$ [sym]. Since the invariant is independent of broken sheet representative, denote $\Phi_{\phi}(D)$ by $\Phi_{\phi}(F)$.

The Kinoshita conjecture posits that all knotted projective planes have a trivial projective summand.

I would like to thank Maggie Miller for helpful discussion and guidance.