Important Knot Diagrams

Why This Matters

Knot theory sits at the intersection of topology, algebra, and geometry—and understanding key knot diagrams is essential for grasping how mathematicians classify and distinguish these objects. You're being tested on your ability to recognize knots by their invariants, understand crossing number, linking number, and tricolorability, and explain why certain knots are fundamentally different despite looking similar. These diagrams aren't just pretty pictures; they're the primary tool for computing invariants and proving equivalence (or inequivalence) between knots.

Don't just memorize what each knot looks like—know what concept each diagram illustrates. Can you explain why the trefoil is nontrivial? Why the Borromean rings demonstrate a property that pairwise linking misses? The exam will push you to connect specific diagrams to broader principles like prime decomposition, chirality, and the distinction between knots and links. Master the "why" behind each example, and you'll be ready for any comparison question they throw at you.

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The Foundation: Trivial and Prime Knots

Every knot can be decomposed into prime knots—knots that cannot be expressed as the connected sum of two nontrivial knots. Understanding prime knots is like understanding prime numbers: they're the building blocks of all knot complexity.

Unknot (Trivial Knot)

• Crossing number of zero—the only knot that can be deformed to a simple circle without any crossings
• Baseline for all invariants—every knot invariant is calibrated against the unknot (e.g., the Jones polynomial equals 1)
• Recognition problem—determining whether a complex diagram represents the unknot is computationally difficult, highlighting why invariants matter

Trefoil Knot

• Simplest nontrivial knot with crossing number 3—cannot be simplified to fewer crossings through any sequence of Reidemeister moves
• First example of chirality—the left-handed and right-handed trefoils are distinct knots (mirror images that cannot be deformed into each other)
• Tricolorable—can be colored with three colors following the tricolorability rules, proving it's not equivalent to the unknot

Figure-Eight Knot

• Crossing number of 4—the unique knot with exactly four crossings
• Amphichiral (achiral)—equivalent to its mirror image, contrasting sharply with the chiral trefoil
• Hyperbolic knot—its complement admits a complete hyperbolic structure, making it fundamental in geometric topology

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Higher Crossing Number Knots

As crossing number increases, knots become more numerous and harder to distinguish. These examples show how invariants become essential tools when visual inspection fails.

Cinquefoil Knot

• Crossing number of 5—a $(5,2)$ torus knot, meaning it wraps around a torus 5 times in one direction and 2 times in the other
• Torus knot family—belongs to the well-understood class where invariants can be computed directly from the $(p,q)$ parameters
• Chiral like the trefoil—its mirror image is a distinct knot, reinforcing the pattern that most torus knots lack amphichirality

Stevedore Knot

• Crossing number of 6—specifically knot $6_1$ in standard tables
• Ribbon knot—bounds a self-intersecting disk in 4-dimensional space, connecting to the slice-ribbon conjecture
• Unknotting number of 1—changing just one crossing converts it to the unknot, despite its apparent complexity

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Composite Knots and Binding Failures

Not every knot is prime—some are connected sums of simpler knots. Understanding composite knots helps explain why certain "knots" fail in practical applications.

Square Knot

• Composite knot—the connected sum of a trefoil and its mirror image ($3_1 \# \overline{3_1}$)
• Crossing number of 6—but unlike prime 6-crossing knots, it decomposes into simpler pieces
• Amphichiral composite—despite being made of chiral components, the square knot equals its mirror image

Granny Knot

• Also composite—the connected sum of two identical trefoils ($3_1 \# 3_1$), both left-handed or both right-handed
• Chiral unlike the square knot—this is the key distinction, despite both being "trefoil + trefoil"
• Same crossing number, different knot—demonstrates that crossing number alone cannot distinguish knots

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Links: When Multiple Components Interact

Links extend knot theory to multiple closed curves. The key question becomes: how are the components entangled with each other?

Hopf Link

• Simplest nontrivial link—two unknotted circles with linking number $\pm 1$
• Linking number demonstration—the fundamental example for teaching how to compute and interpret this invariant
• Each component is an unknot—the complexity comes entirely from how they're linked, not from the individual curves

Whitehead Link

• Linking number of zero—despite being clearly linked, the algebraic count of crossings cancels out
• Not equivalent to the unlink—proves that linking number is an incomplete invariant (necessary but not sufficient)
• Boundary link—each component bounds a surface (Seifert surface) that doesn't intersect the other component

Borromean Rings

• Three-component link with pairwise linking number zero—any two rings form the unlink, yet all three together are inseparable
• Brunnian link—removing any single component causes the remaining two to become unlinked
• Higher-order linking—demonstrates that pairwise invariants miss global entanglement, motivating Milnor invariants

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Quick Reference Table

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Self-Check Questions

1. Which two knots both have crossing number 6 but differ in chirality, and what accounts for this difference?

2. The Whitehead link and Hopf link are both two-component links—what invariant distinguishes them, and why does this matter for understanding invariant limitations?

3. Compare and contrast the trefoil and figure-eight knots in terms of crossing number, chirality, and tricolorability. Which would you use to demonstrate that a knot can equal its mirror image?

4. If an FRQ asks you to explain why pairwise linking numbers are insufficient to detect all linking, which diagram provides the strongest example and why?

5. Both the square knot and granny knot are connected sums of trefoils—explain precisely how their constructions differ and what topological property this affects.