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  1. Knot Theory

    Linked via "fundamental group"

    Alexander Polynomial
    The Alexander polynomial, $\DeltaK(t)$, introduced by James Alexander in 1923, is a Laurent polynomial in one variable, $t$. It is derived from the Alexander matrix, which is formed using the fundamental group of the knot complement, $\pi1(\mathbb{R}^3 \setminus K)$.
    $$ \Delta_K(t) = \det(A - t A^T) $$

  2. Knot Theory

    Linked via "fundamental group"

    $$ \Delta_K(t) = \det(A - t A^T) $$
    where $A$ is the Alexander matrix, derived from the Wirtinger presentation of the fundamental group. While powerful, the Alexander polynomial is not a complete invariant; for instance, the unknot and the mirror image of the knot $61$ share the same Alexander polynomial (though this is often confused with the fact that $\DeltaK(t) = \Delta_{K^}(-t^{-1})$ for the mirror image $K^$ [4]).
    Jones Polynomial

  3. Topology

    Linked via "fundamental group"

    Homotopy theory studies deformations of maps. Two continuous maps $f, g: X \to Y$ are homotopic if one can be continuously deformed into the other. This relationship partitions the space of maps into equivalence classes called homotopy classes.
    The fundamental group, $\pi1(X, x0)$, based at a point $x0$, is the set of homotopy classes of closed loops' starting and ending at $x0$. This […

  4. Topology

    Linked via "fundamental group"

    The fundamental group, $\pi1(X, x0)$, based at a point $x0$, is the set of homotopy classes of closed loops' starting and ending at $x0$. This group' captures the presence of 1-dimensional "holes" in the space $X$. A space is simply connected if its fundamental group is trivial (the trivial group).
    It is a cornerstone result of [algebraic topology](/entries/algebraic-…

  5. Torus

    Linked via "fundamental group"

    The torus (plural: tori or toruses) is a topological space that resembles the surface of a donut or an inner tube. Mathematically, it is the Cartesian product of two circles, $S^1$/) $\times$ $S^1$/). It is an essential object in algebraic topology and differential geometry, frequently serving as the simplest non-trivial example of a compact manifold with a nontrivial fundamental group [1]. The torus possesses a defin…