yui_link/inv_link/

inv_link.rs

//! [`InvLink`]: a [`Link`](crate::Link) with an involution `τ`, given by an edge
//! bijection and the induced map on nodes. Strong invertibility (`τ` reverses the
//! orientation) and 2-periodicity (`τ` preserves it) are decided from the orientation.

use std::collections::HashMap;

use delegate::delegate;
use itertools::Itertools;
use yui_core::algo::KeyedUnionFind;
use crate::{Node, Edge, Link, Path, Slot, State, PDCodeX};

// Involutive link
#[derive(Debug, Clone)]
pub struct InvLink {
    inner: Link,
    e_map: HashMap<Edge, Edge>,
    x_map: HashMap<Node, Node>
}

impl InvLink {
    pub fn new<I>(inner: Link, e_map: I) -> InvLink
    where I: IntoIterator<Item = (Edge, Edge)> {
        assert!(inner.loops().is_empty(), "free loops are not supported");
        let e_map: HashMap<Edge, Edge> = e_map.into_iter().collect();

        // `e_map` must be an involution of the whole edge set.
        let link_edges = inner.edges();
        let missing = link_edges.iter().filter(|e| !e_map.contains_key(e)).collect_vec();
        assert!(missing.is_empty(), "e_map does not cover edges {missing:?}");

        let extra = e_map.keys().filter(|e| !link_edges.contains(e)).sorted().collect_vec();
        assert!(extra.is_empty(), "e_map maps edges {extra:?}, which are not in the link");

        for &e in &link_edges {
            let f = e_map[&e];
            assert!(link_edges.contains(&f), "e_map sends edge {e} to {f}, not an edge of the link");
            assert_eq!(e_map[&f], e, "e_map is not involutive: {e} ↦ {f} ↦ {}", e_map[&f]);
        }

        // ... and each node must have a unique corresponding node.
        let x_map: HashMap<Node, Node> = inner.nodes().map(|x| {
            let y = Self::find_tau_x(x, &inner, &e_map);
            (x.clone(), y.clone())
        }).collect();

        assert_eq!(x_map.len(), inner.n_nodes(), "the diagram has duplicate nodes");

        for (x, y) in &x_map {
            let z = &x_map[y];
            assert_eq!(z, x, "e_map induces a non-involutive node map: {x} ↦ {y} ↦ {z}");
        }

        Self { inner, e_map, x_map }
    }

    // τ is a rotation about an axis in the plane, so it reverses the normal — the cyclic order of a
    // node's four slots must run backwards for τ to preserve orientation (as `preserves_dir_at` reads it).
    fn find_tau_x<'a>(x: &'a Node, inner: &'a Link, e_map: &HashMap<Edge, Edge>) -> &'a Node {
        let matches = |y: &Node| Slot::ALL.into_iter().any(|k|
            Slot::ALL.into_iter().all(|s| y.edge(k.shift(4 - s.index())) == e_map[&x.edge(s)])
        );
        let cands = inner.nodes().filter(|y| matches(y)).collect_vec();

        match cands.as_slice() {
            [y] => {
                assert_eq!(x.node_type(), y.node_type(), "e_map changes the type of node {x}");
                y
            },
            [] => panic!("e_map sends node {x} to no node of the link"),
            _  => panic!("e_map does not determine the image of node {x}: {} nodes match", cands.len())
        }
    }

    pub fn from_symmetric_pd_code<I1>(pd_code: I1) -> Self
    where I1: IntoIterator<Item = PDCodeX> {
        // the base point defaults to the least edge, which the symmetric convention puts on the axis.
        Self::si_knot_from(Link::from_pd_code(pd_code))
    }

    // A strongly invertible knot, given a diagram based on its axis. τ reverses the traversal, so
    // walking both ways from the base point pairs each edge with its image.
    pub fn si_knot_from(inner: Link) -> InvLink {
        assert!(inner.is_knot(), "expected a knot, found {} components", inner.n_comps());
        let base = inner.base_pt().expect("the diagram needs a base point on the axis");
        let comps = inner.comps();
        let seq = comps[0].edges();
        let n = seq.len();
        let k = seq.iter().position(|&e| e == base).expect("the base point is not on a strand");

        let e_map = (0..n).map(|i| (seq[(k + i) % n], seq[(k + n - i) % n])).collect_vec();
        let l = Self::new(inner.clone(), e_map);
        assert!(l.is_oriented(), "the diagram is not oriented");
        if let Some(x) = l.inner.nodes().find(|x| l.preserves_dir_at(x) != Some(false)) {
            panic!("τ does not reverse the orientation at node {x} — the diagram is not symmetric there");
        }
        l
    }

    pub fn inner(&self) -> &Link {
        &self.inner
    }

    /// The symmetric PD code — the inner diagram's PD, with edges numbered so the strong inversion is
    /// the standard `e ↦ (n+1-e)%n+1`. Feeding this to `from_symmetric_pd_code` (or the `ykh` CLI)
    /// reconstructs the same `InvLink`. The builders (`from_symmetric_pd_code`, `conn_sum`,
    /// `whitehead_double`) already reindex to this involution, so an arbitrary `e_map` is rejected.
    pub fn pd_code(&self) -> Vec<PDCodeX> {
        let n = self.inner.n_edges() as Edge;
        assert!(
            self.inner.edges().into_iter().all(|e| self.inv_edge(e) == (n + 1 - e) % n + 1),
            "pd_code requires the standard involution `e ↦ (n+1-e)%n+1`; reindex the InvLink first"
        );
        self.inner.pd_code()
    }

    // delegate methods from Link

    delegate! {
        to self.inner {
            pub fn is_empty(&self) -> bool;
            pub fn is_knot(&self) -> bool;
            pub fn is_oriented(&self) -> bool;
            pub fn writhe(&self) -> i32;
            pub fn n_nodes(&self) -> usize;
            pub fn nodes(&self) -> impl Iterator<Item = &Node>;
            pub fn node(&self, i: usize) -> &Node;
            pub fn crossings(&self) -> impl Iterator<Item = &Node>;
            pub fn n_crossings(&self) -> usize;
            pub fn n_signed_crossings(&self) -> (usize, usize);
            pub fn n_edges(&self) -> usize;
            pub fn edges(&self) -> Vec<Edge>;
            pub fn n_comps(&self) -> usize;
            pub fn comps(&self) -> Vec<Path>;
            pub fn seifert_state(&self) -> State;
            pub fn seifert_circles(&self) -> Vec<Path>;
            pub fn base_pt(&self) -> Option<Edge>;
        }
    }

    pub fn with_base_pt(mut self, e: Edge) -> Self {
        assert!(self.is_on_axis(e), "base_pt {e} must be on-axis (fixed by involution)");
        self.inner = self.inner.with_base_pt(e);
        self
    }

    pub fn inv_edge(&self, e: Edge) -> Edge {
        self.e_map.get(&e).cloned().unwrap()
    }

    pub fn inv_node(&self, x: &Node) -> &Node {
        self.x_map.get(x).unwrap()
    }

    // `e` meets the axis, i.e. is fixed by τ.
    pub fn is_on_axis(&self, e: Edge) -> bool {
        self.inv_edge(e) == e
    }

    pub fn on_axis_edges(&self) -> Vec<Edge> {
        self.edges().into_iter().filter(|&e| self.is_on_axis(e)).collect()
    }

    // The axis lies in the projection plane (as opposed to an intravergent diagram, where it is
    // perpendicular): a line separates the plane, so no cluster of off-axis nodes is τ-invariant.
    pub fn is_transvergent(&self) -> bool {
        let off_axis = self.inner.nodes().filter(|&x| self.inv_node(x) != x).collect_vec();

        // two off-axis nodes are in one cluster iff they share an edge the axis does not meet
        let shares_edge = |x: &Node, y: &Node|
            x.edges().iter()
                .filter(|&&e| !self.is_on_axis(e))
                .any(|e| y.edges().contains(e));

        let mut uf = KeyedUnionFind::from_iter(off_axis.iter().copied());
        for (i, &x) in off_axis.iter().enumerate() {
            for &y in &off_axis[..i] {
                if shares_edge(x, y) {
                    uf.union(&x, &y);
                }
            }
        }

        uf.into_disjoint().into_iter().all(|group|
            group.first().is_none_or(|&rep| !group.contains(&self.inv_node(rep)))
        )
    }

    // A strong inversion reverses the orientation.
    pub fn is_strongly_invertible(&self) -> bool {
        self.is_oriented()
            && self.inner.nodes().all(|x| self.preserves_dir_at(x) == Some(false))
    }

    // A 2-periodic link carries an involution preserving the orientation.
    pub fn is_2periodic(&self) -> bool {
        self.is_oriented()
            && self.inner.nodes().all(|x| self.preserves_dir_at(x) == Some(true))
    }

    // Whether τ keeps the strands running the same way at `x`. The rotation reverses the cyclic
    // order of a node's slots, sending both incoming slots to incoming ones, or both to outgoing.
    fn preserves_dir_at(&self, x: &Node) -> Option<bool> {
        let y = self.inv_node(x);
        let (p, q) = x.incoming()?;
        let k = Slot::ALL.into_iter().find(|k|
            Slot::ALL.into_iter().all(|s| y.edge(k.shift(4 - s.index())) == self.inv_edge(x.edge(s)))
        )?;

        let is_in = |s: Slot| {
            let img = k.shift(4 - s.index());
            y.incoming().is_some_and(|(a, b)| img == a || img == b)
        };
        match (is_in(p), is_in(q)) {
            (true, true) => Some(true),
            (false, false) => Some(false),
            _ => None,
        }
    }

    pub fn mirror(&self) -> Self {
        Self {
            inner: self.inner.mirror(),
            e_map: self.e_map.clone(),
            x_map: self.x_map.iter().map(|(x, y)|
                (x.mirror(), y.mirror())
            ).collect(),
        }
    }

    // Reverse the orientation. `τ` is untouched: it is a map of edges, and reversing renames
    // nothing — so a strong inversion stays one.
    pub fn reversed(&self) -> Self {
        Self {
            inner: self.inner.reversed(),
            e_map: self.e_map.clone(),
            x_map: self.x_map.iter().map(|(x, y)|
                (x.reversed(), y.reversed())
            ).collect(),
        }
    }

    // Equivariant connected sum: splice self's other on-axis edge to other's base point, so self's
    // base point survives as the sum's.
    pub fn conn_sum(&self, other: &InvLink) -> InvLink {
        let base = self.base_pt().expect("self needs a base point");
        let self_e = self.on_axis_edges().into_iter()
            .find(|&e| e != base)
            .expect("self needs a second on-axis edge");
        let other_e = other.base_pt().expect("other needs a base point");
        self.conn_sum_at(other, self_e, other_e)
    }

    // Equivariant connected sum: splice along on-axis edges (`is_on_axis`) of each summand,
    // then recover the combined strong inversion by reindexing to the standard involution.
    pub fn conn_sum_at(&self, other: &InvLink, self_e: Edge, other_e: Edge) -> InvLink {
        assert!(self.is_knot() && other.is_knot(), "connected sum requires knots");
        assert!(self.is_strongly_invertible() && other.is_strongly_invertible(),
            "connected sum requires strongly invertible knots");
        assert_ne!(Some(self_e), self.base_pt(), "the splice must not consume self's base point");
        assert_eq!(self.inv_edge(self_e), self_e, "self_e {self_e} must be on-axis");
        assert_eq!(other.inv_edge(other_e), other_e, "other_e {other_e} must be on-axis");

        // `Link::conn_sum_at` carries self's base point through the splice, and it is on-axis.
        let inner = self.inner.conn_sum_at(other.inner(), self_e, other_e);
        Self::si_knot_from(inner)
    }
}

impl InvLink {
    pub fn load(name: &str) -> Result<InvLink, Box<dyn std::error::Error>> {
        let json = yui_core::util::data_dir::load_json("inv_link", name)?;
        let data: Vec<PDCodeX> = serde_json::from_str(&json)?;
        Ok(InvLink::from_symmetric_pd_code(data))
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::Slot;
    use crate::misc::det;

    #[test]
    fn pd_code_roundtrip() {
        // 5_1 as a symmetric PD (standard involution by construction); emit + reparse must recover
        // the same diagram and the same strong inversion.
        let l = InvLink::from_symmetric_pd_code([[1,7,2,6],[3,9,4,8],[5,1,6,10],[7,3,8,2],[9,5,10,4]]);
        let l2 = InvLink::from_symmetric_pd_code(l.pd_code());
        assert_eq!(l.pd_code(), l2.pd_code());
        for e in l.inner().edges() {
            assert_eq!(l.inv_edge(e), l2.inv_edge(e), "involution differs at edge {e}");
        }
    }

    #[test]
    fn test_data_diagrams() {
        // the bundled symmetric diagrams parse, and are the size Lamm's tables give.
        for (name, n) in [("3_1", 3), ("4_1", 4), ("4_1a", 4), ("4_1b", 4), ("6_3", 7), ("6_3a", 8), ("7_7b", 7)] {
            assert_eq!(InvLink::test_data(name).n_crossings(), n, "{name}");
        }
    }

    #[test]
    fn inv_edge() {
        let l = InvLink::test_data("3_1");

        assert_eq!(l.inv_edge(1), 1);
        assert_eq!(l.inv_edge(2), 6);
        assert_eq!(l.inv_edge(3), 5);
        assert_eq!(l.inv_edge(4), 4);
        assert_eq!(l.inv_edge(5), 3);
        assert_eq!(l.inv_edge(6), 2);
    }

    #[test]
    fn inv_node() {
        let l = InvLink::test_data("3_1");
        let nodes = l.inner.nodes().collect_vec();

        assert_eq!(l.inv_node(nodes[0]), nodes[1]);
        assert_eq!(l.inv_node(nodes[1]), nodes[0]);
        assert_eq!(l.inv_node(nodes[2]), nodes[2]);
    }

    #[test]
    fn mirror_keeps_the_involution() {
        let k = InvLink::test_data("3_1");
        let m = k.mirror();
        assert_eq!(m.n_crossings(), k.n_crossings());
        for e in k.edges() {
            assert_eq!(m.inv_edge(e), k.inv_edge(e), "involution differs at edge {e}");
        }
        for (x, y) in k.inner().nodes().zip(m.inner().nodes()) {
            assert_eq!(m.inv_node(y), &k.inv_node(x).mirror(), "τ differs at node {x:?}");
        }
    }

    #[test]
    #[should_panic(expected = "e_map does not cover edges [3, 4, 5, 6]")]
    fn new_names_the_uncovered_edges() {
        let _ = InvLink::new(Link::test_data("3_1"), [(1, 1), (2, 2)]);
    }

    #[test]
    #[should_panic(expected = "e_map sends node")]
    fn new_rejects_an_edge_map_no_rotation_realizes() {
        // the identity is an edge-involution, but no π-rotation of the trefoil fixes every edge:
        // at a crossing with four distinct edges, no reversal of the slots is the identity.
        let l = Link::test_data("3_1");
        let e_map: Vec<_> = l.edges().into_iter().map(|e| (e, e)).collect();
        let _ = InvLink::new(l, e_map);
    }

    #[test]
    fn nodes_sharing_an_edge_set() {
        // both crossings of L2a1 carry the same four edges, so an edge-set match cannot tell them
        // apart; the slot arrangement picks the rotation that swaps them.
        let l = Link::from_pd_code([[4, 1, 3, 2], [2, 3, 1, 4]]);
        let il = InvLink::new(l, [(1, 1), (2, 2), (3, 3), (4, 4)]);

        let (x0, x1) = (il.node(0), il.node(1));
        assert_eq!(il.inv_node(x0), x1);
        assert_eq!(il.inv_node(x1), x0);
    }

    #[test]
    fn transvergent() {
        // 5_1's symmetric PD: the axis lies in the plane, so the off-axis crossings fall into
        // clusters that τ pairs up.
        let l = InvLink::from_symmetric_pd_code([[1,7,2,6],[3,9,4,8],[5,1,6,10],[7,3,8,2],[9,5,10,4]]);
        assert!(l.is_transvergent());

        // L2a1 rotated about an axis perpendicular to the plane: no edge is fixed, and the two
        // crossings form a single cluster that τ maps onto itself.
        let m = InvLink::new(Link::from_pd_code([[4,1,3,2],[2,3,1,4]]), [(1,2),(2,1),(3,4),(4,3)]);
        assert!(m.on_axis_edges().is_empty());
        assert!(!m.is_transvergent());
    }

    #[test]
    fn twist_unknot() {
        // the 1-crossing unknot: the identity is the π-rotation through the crossing, a strong
        // inversion; swapping the two edges is a rotation missing the knot, so 2-periodic.
        let l = Link::from_pd_code([[0, 1, 1, 0]]);

        let si = InvLink::new(l.clone(), [(0, 0), (1, 1)]);
        assert_eq!(si.on_axis_edges(), vec![0, 1]);
        assert!(si.is_strongly_invertible());
        assert!(!si.is_2periodic());

        let per = InvLink::new(l, [(0, 1), (1, 0)]);
        assert!(per.on_axis_edges().is_empty());
        assert!(!per.is_strongly_invertible());
        assert!(per.is_2periodic());
    }

    #[test]
    fn involution_is_a_strong_inversion() {
        // `new` only checks that the edge map is an involution; these are the conditions making it
        // a π-rotation about an axis in the plane.
        fn check(name: &str, l: &InvLink) {
            let inner = l.inner();
            assert_eq!(l.on_axis_edges().len(), 2, "{name}: the axis must meet the knot twice");
            assert!(l.is_strongly_invertible(), "{name}: τ must reverse the orientation");

            for x in inner.nodes() {
                let y = l.inv_node(x);
                assert_eq!(y.node_type(), x.node_type(), "{name}: τ changed a crossing type");
                assert_eq!(l.inv_node(y), x, "{name}: τ is not an involution on nodes");

                // the rotation reverses the cyclic order of a crossing's slots: s ↦ (k - s) mod 4, k odd.
                let k = (0..4).find(|&k|
                    Slot::ALL.iter().all(|&s|
                        y.edge(Slot::from((k + 4 - s.index()) % 4)) == l.inv_edge(x.edge(s))
                    )
                );
                assert!(matches!(k, Some(1) | Some(3)), "{name}: τ does not reverse the slot order at {x}");
            }
        }

        for name in ["3_1", "4_1", "6_3"] {
            check(name, &InvLink::test_data(name));
        }
        check("sym_pretzel(-3,3,-3)", &InvLink::sym_pretzel(-3, 3, -3));
        check("sym_wh+(3_1)", &InvLink::whitehead_double(&InvLink::test_data("3_1"), true, 0));
    }

    #[test]
    #[should_panic(expected = "must be on-axis")]
    fn with_base_pt_rejects_an_off_axis_edge() {
        // 3_1's axis meets it at edges 1 and 4; edge 2 is swapped with 6 by tau.
        let _ = InvLink::test_data("3_1").with_base_pt(2);
    }

    #[test]
    fn strong_inversions_are_not_2periodic() {
        // the two cases are exclusive: tau reverses the orientation, so it cannot preserve it.
        for name in ["3_1", "4_1", "6_3", "5_2a", "7_7b"] {
            let k = InvLink::test_data(name);
            assert!(k.is_strongly_invertible(), "{name}");
            assert!(!k.is_2periodic(), "{name}");
        }
    }

    #[test]
    fn reindexed_keeps_strong_inversion() {
        // renumbering the symmetric trefoil from either on-axis edge must leave the standard
        // e ↦ (n+1-e)%n+1 a valid τ. Starting from edge 1 is a no-op; edge 4 is the real case.
        let il = InvLink::test_data("3_1");
        let n = il.n_edges() as Edge;

        for start in il.on_axis_edges() {
            let r = il.inner().reindexed(start, 1);
            assert_eq!(r.edges(), (1..=n).collect::<Vec<Edge>>());

            let e_map: Vec<_> = r.edges().into_iter().map(|e| (e, (n + 1 - e) % n + 1)).collect();
            let re = InvLink::new(r, e_map);
            assert!(re.is_strongly_invertible(), "renumbered from edge {start}");
        }
    }

    #[test]
    fn conn_sum_at_depends_on_which_side_of_the_axis() {
        // The companion's two on-axis edges are the two sides of its axis, and splicing to one or
        // the other gives different diagrams in general — the equivariant connected sum is not
        // determined by the two knots alone. `conn_sum` fixes the convention: other's base point.
        // (For 4_1, 5_2a, 6_1a, 7_2a the two sides happen to agree; 6_3 is a companion where they
        // do not, so this also pins which side `conn_sum` takes.)
        let k1 = InvLink::test_data("3_1");
        let k2 = InvLink::test_data("6_3");

        let axis = k2.on_axis_edges();
        assert_eq!(axis, vec![1, 8]);
        assert_eq!(k2.base_pt(), Some(axis[0]));

        // self_e is forced: the other on-axis edge, since the splice may not eat self's base point.
        assert_eq!(k1.on_axis_edges(), vec![1, 4]);
        assert_eq!(k1.base_pt(), Some(1));
        let self_e = 4;

        let sums = axis.iter().map(|&e| k1.conn_sum_at(&k2, self_e, e)).collect_vec();

        for (s, e) in Iterator::zip(sums.iter(), axis.iter()) {
            assert!(s.is_knot(), "other_e = {e}");
            assert!(s.is_strongly_invertible(), "other_e = {e}");
            let base = s.base_pt().expect("the sum keeps a base point");
            assert_eq!(s.inv_edge(base), base, "other_e = {e}: base point is off-axis");
            assert_eq!(s.n_crossings(), k1.n_crossings() + k2.n_crossings(), "other_e = {e}");
            assert_eq!(det(s.inner()), det(k1.inner()) * det(k2.inner()), "other_e = {e}");
        }

        let canon = |k: &InvLink| k.inner().reindexed_canon();
        assert_ne!(canon(&sums[0]), canon(&sums[1]), "the two sides must give different diagrams");
        assert_eq!(canon(&k1.conn_sum(&k2)), canon(&sums[0]), "conn_sum splices at other's base point");
    }

    #[test]
    fn inv_link_reversed() {
        let k = InvLink::test_data("3_1");
        let r = k.reversed();

        assert!(r.inner().is_oriented());
        assert!(r.is_strongly_invertible(), "reversing does not disturb the axis");
        assert_eq!(r.writhe(), k.writhe());
        assert_eq!(r.on_axis_edges(), k.on_axis_edges(), "τ is a map of edges, unchanged");
        for e in k.inner().edges() {
            assert_eq!(r.inv_edge(e), k.inv_edge(e));
        }
        assert_eq!(r.reversed().inner(), k.inner(), "reversing twice is the identity");
    }

    #[test]
    fn dms_construction() {
        // The equivariant `2K # (-2K)` of Dai-Mallick-Stoffregen (`references/DMS.pdf` §1.2) — the
        // same diagrams as `Link::conn_sum`'s `dms_construction`, with the splice edges no longer
        // given by hand: `conn_sum` takes self's other on-axis edge each time.
        let k = InvLink::from_symmetric_pd_code([[1,4,2,5],[5,2,6,3],[3,6,4,1]]);
        assert_eq!(k.on_axis_edges(), vec![1, 4]);

        // `2K`: the connected sum of the underlying link is already the flip diagram, based on the
        // band, so τ comes back by traversal. The band's two sides are the new axis.
        let k2 = InvLink::si_knot_from(k.inner().conn_sum(k.inner()));
        assert_eq!(k2.inner().pd_code(), Link::from_pd_code([
            [1,4,2,5],[5,2,6,3],[3,6,4,7],[7,10,8,11],[11,8,12,9],[9,12,10,1],
        ]).pd_code());
        assert_eq!(k2.on_axis_edges(), vec![1, 7]);

        // `2K # m(K)`, spliced at edge 7.
        let mk = k.mirror();
        let with_mk = k2.conn_sum(&mk);
        assert_eq!(with_mk.inner().pd_code(), Link::from_pd_code([
            [1,4,2,5],[5,2,6,3],[3,6,4,13],[7,10,8,11],[11,8,12,9],[9,12,10,1],
            [16,14,17,13],[14,18,15,17],[18,16,7,15],
        ]).pd_code());
        assert_eq!(with_mk.on_axis_edges(), vec![1, 16]);

        // `2K # m(K) # r(m(K))`, spliced at edge 16.
        let dms = with_mk.conn_sum(&mk.reversed());
        assert_eq!(dms.inner().pd_code(), Link::from_pd_code([
            [1,4,2,5],[5,2,6,3],[3,6,4,13],[7,10,8,11],[11,8,12,9],[9,12,10,1],
            [16,14,17,13],[14,18,15,17],[18,19,7,15],
            [19,21,24,22],[21,23,20,24],[23,16,22,20],
        ]).pd_code());
        assert!(dms.is_strongly_invertible());
        assert_eq!(dms.base_pt(), Some(1));
    }

    #[test]
    fn conn_sum_is_equivariant() {
        // construction succeeding ⟺ from_standard_reindex found a valid τ on the sum.
        let k1 = InvLink::test_data("3_1");
        let k2 = InvLink::test_data("4_1");
        let cs = k1.conn_sum(&k2);
        assert!(cs.is_knot());
        assert!(cs.is_strongly_invertible());
        let base = cs.base_pt().expect("the sum keeps a base point");
        assert_eq!(cs.inv_edge(base), base, "the sum's base point is on-axis");
        assert_eq!(det(cs.inner()), 3 * 5, "det is multiplicative under conn sum");
    }
}