13.21.3 Tensor Contraction Diagram Notation

Tensor Contraction Diagram Notation is the specific set of visual conventions, such as line placement, arrow direction, and node shape, adopted to encode variance, index identity, and contraction structure within a contraction diagram, translating the symbolic rules of indexed notation into a consistent graphical vocabulary.

Definition

Diagram notation assigns a fixed visual meaning to each structural feature of a contraction diagram: a node represents a tensor, a leg represents an index, and an edge represents a contracted pair, with additional conventions distinguishing upper legs from lower legs so that the diagram alone conveys the same information as its corresponding indexed expression:

Conventions for Encoding Variance

Position-Based Convention

One common notation places upper legs emerging from the top of a node and lower legs emerging from the bottom, so that variance is conveyed purely by the geometric position of the leg relative to the node, without any additional marking.

Arrow-Based Convention

An alternative notation uses arrow direction along each leg, with an outward-pointing arrow denoting a contravariant index and an inward-pointing arrow denoting a covariant index, allowing legs to be drawn in any position while still conveying variance unambiguously.

Style-Based Convention

A further alternative distinguishes variance by line style, such as a solid line for upper indices and a dashed line for lower indices, useful when diagram layout constraints make consistent positional or directional placement impractical.

Consistency Requirements

Single Convention Per Diagram

Whichever convention is adopted, it must be applied uniformly across an entire diagram; mixing positional and arrow-based variance encoding within a single diagram would make the diagram ambiguous to read.

Correspondence to Written Notation

Under any chosen convention, the diagram notation must remain in exact correspondence with the underlying indexed tensor contraction notation, so that the diagram reading procedure used to recover an algebraic expression from a diagram produces a result consistent with the original expression the diagram was drawn to represent.

Diagram Illustrating Two Conventions

Purpose Within the Broader Notational System

Diagram notation exists to make the general contraction diagram representation practically usable, supplying the concrete graphical rules that turn an abstract idea of nodes and edges into a specific, readable picture, and providing the necessary conventions on which the diagram result reading procedure depends when translating a completed diagram back into an indexed algebraic expression.