Secant-quandle: an invariant of braids and knots

Abstract

We construct a novel invariant of braids and knots, secant-quandle (SQ),with generic secants serving as generators and generic horizontal trisecants serving as relations, i.e., $SQ = Γ\left< \mathcal{S}_M\mid \mathcal{S}_T,\mathcal{E}_M \right>$, where $M$ is a braid or link.

Figures (9)

• Figure 1: Trisecants divide $R_{ij}$ into several film-frame .
• Figure 2: Birman knot and generators of $K_{2n}$.
• Figure 3: Type 2
• Figure 4: Trisecant isotopy moves.
• Figure 5: $\Lambda_3$: Case 1.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Proposition 1
• Proposition 2
• Lemma 1
• proof
• Lemma 2