In knot theory (the mathematical study of curved, closed loops), knots are characterized by their crossing numbers—that is, the number...

Let's begin by understanding the meaning of this sentence. We'll use our understanding of pause points and segment the sentence as shown - understanding and assimilating the meaning of each segment bit by bit!

Sentence Structure

• In knot theory
  • (the mathematical study of curved, closed loops),
• knots are characterized by their crossing numbers—
  • that is,
  • the number of times the knotted thread crosses over itself.
• The trefoil knot and the figure-eight knot,
  • each with a crossing number below five,
    • (?) among the simplest possible knots
      • with the fewest number of crossings.

Understanding the Meaning

The first sentence gives us background:

• 'In knot theory (the mathematical study of curved, closed loops)'
  • Sets up the topic - we're talking about the mathematical field that studies knots
• 'knots are characterized by their crossing numbers'
  • Knots are identified/categorized by something called crossing numbers
• 'that is, the number of times the knotted thread crosses over itself'
  • This explains what a crossing number means - it counts how many times the thread crosses over itself

Now the second sentence:

• 'The trefoil knot and the figure-eight knot'
  • This introduces TWO specific types of knots
• 'each with a crossing number below five'
  • Both of these knots have crossing numbers less than five

Now here's where we need to fill in the blank:

• 'The trefoil knot and the figure-eight knot ______ among the simplest possible knots'

Let's look at our choices:

• The choices are: was, are, has been, is
• These vary in whether they're singular or plural, and in tense

What do we notice about the subject here?

• The subject is "The trefoil knot and the figure-eight knot"
• This is TWO things connected by "and" - a compound subject
• When we have two subjects joined by "and," we're talking about multiple things
• Multiple things need a PLURAL verb

Looking at our choices:

• "was" = singular
• "are" = plural ✓
• "has been" = singular
• "is" = singular

So we need: are

The complete meaning is:

• These two specific knots (the trefoil and the figure-eight), which both have low crossing numbers (below five),
• ARE among the simplest possible knots - they're in the category of knots with the fewest crossings.

GRAMMAR CONCEPT APPLIED

Compound Subjects with "And" Take Plural Verbs

When you have two or more subjects connected by "and," they form what's called a compound subject in grammar terms. Compound subjects with "and" always require plural verbs because you're talking about multiple things together.

The Pattern:

• Subject 1 + and + Subject 2 → plural verb
• Example: "The cat and the dog are sleeping."
• Two subjects (cat, dog) joined by "and" = plural verb "are"

Important note: Even when descriptive phrases come between the subject and verb, the rule stays the same:

• "The cat, along with all her kittens, and the dog are sleeping."
• Still two main subjects = still plural verb

In this question:

• "The trefoil knot and the figure-eight knot" = two subjects
• Connected by "and" = compound subject
• Requires plural verb = "are"
• The phrase "each with a crossing number below five" is just a descriptor - it doesn't change the fact that we have two subjects