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The Knot Semiring

The Knot Semiring is a specialized subclass of the Monoid Algebra Semiring for working with formal linear combinations of knots (often called a Skein Module) under the connected sum operation (\(\#\)). It allows us to use the algebraic machinery of algebrax to reason about knots and their compositions over various coefficient rings (like integers, real numbers, or even polynomials).

Mathematical Definition

The Knot Semiring is defined over the set of formal sums of knots. It is parameterized by a coefficient semiring, which governs the arithmetic of the coefficients.

  • Set (\(S\)): Formal sums of knots, represented as dictionaries from knot identifiers (strings) to coefficients of a generic type T.
    • Example: {'3_1': 2, '4_1': -1} represents the formal sum \(2 \cdot 3_1 - 1 \cdot 4_1\) over the integers.
  • Coefficient Semiring (\(R_C\)): An underlying semiring that defines addition and multiplication for the coefficients. Defaults to StandardSemiring(int).
  • Addition (\(+\)): Formal addition of two sums. This corresponds to combining the dictionaries and using the coefficient semiring's addition for common knots.
  • Multiplication (\(\cdot\)): The connected sum (\(\#\)) of knots, distributed over the formal addition. The coefficients are multiplied using the coefficient semiring's multiplication.
  • Additive Identity (\(0\)): The empty set, an empty dictionary {}.
  • Multiplicative Identity (\(1\)): The unknot, represented as {'U': R_C.one}, where R_C.one is the multiplicative identity of the coefficient semiring.

Knot Representation

To handle composite knots, we use a specific string notation:

  • Prime Knots: Identified by their standard notation (e.g., '3_1' for the trefoil, '4_1' for the figure-eight knot).
  • The Unknot: Represented by the string 'U'.
  • Composite Knots: Formed by joining the identifiers of their prime knot components with a # symbol. To ensure the operation is commutative, the components are sorted alphabetically.
    • Example: The connected sum of the trefoil (3_1) and the figure-eight (4_1) is represented by the string '3_1#4_1'.

Implementation in algebrax

The KnotSemiring is a generic class that defaults to using integer coefficients.

Example 1: Default (Integer Coefficients)

This is the most basic case, forming a Skein module over \(\mathbb{Z}\).

import algebrax as ax

# Initialize the semiring (defaults to integer coefficients)
knot_semiring = ax.semiring.KnotSemiring()

# Define two formal sums of knots
# a = 2 * (3_1) + 1 * (4_1)
a = {'3_1': 2, '4_1': 1}

# b = 1 * (3_1) - 1 * (5_2)
b = {'3_1': 1, '5_2': -1}

# --- Operations ---

# Addition: (2*3_1 + 4_1) + (3_1 - 5_2) = 3*3_1 + 4_1 - 5_2
added = knot_semiring.add(a, b)
# Result: {'3_1': 3, '4_1': 1, '5_2': -1}
print(f"Addition over Integers: {added}")

# Multiplication (Connected Sum): (2*3_1 + 4_1) # 3_1
# = 2 * (3_1 # 3_1) + 1 * (4_1 # 3_1)
# = 2 * (3_1#3_1) + 1 * (3_1#4_1)
multiplied = knot_semiring.mul(a, {'3_1': 1})
# Result: {'3_1#3_1': 2, '3_1#4_1': 1}
print(f"Multiplication over Integers: {multiplied}")

Example 2: Custom Coefficient Semiring (Real Numbers)

By passing a different semiring to the constructor, we can work with other coefficient types.

import algebrax as ax

# 1. Define the coefficient semiring (floats)
float_semiring = ax.semiring.StandardSemiring(float)

# 2. Initialize the ax.semiring.KnotSemiring with the float semiring
knot_semiring_float = ax.semiring.KnotSemiring(float_semiring)

# a = 0.5 * (3_1)
a = {'3_1': 0.5}

# b = 0.5 * (3_1)
b = {'3_1': 0.5}

# Addition: 0.5*3_1 + 0.5*3_1 = 1.0*3_1
added = knot_semiring_float.add(a, b)
# Result: {'3_1': 1.0}
print(f"Addition over Floats: {added}")

# Multiplication: (0.5*3_1) # (0.5*3_1) = 0.25 * (3_1#3_1)
multiplied = knot_semiring_float.mul(a, b)
# Result: {'3_1#3_1': 0.25}
print(f"Multiplication over Floats: {multiplied}")

Use Cases

The generic nature of the KnotSemiring allows it to model various algebraic structures in topology:

  • Skein Modules: Using integer or polynomial coefficients to study knot invariants.
  • Quantum Topology: Using complex coefficients to compute values of knot polynomials at roots of unity.
  • Probabilistic Models: Using a probability semiring for coefficients to model stochastic topological processes.