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🪢Knot Theory

9.2 Classification of knots up to certain crossing numbers

3 min readLast Updated on July 22, 2024

Crossing number is a crucial concept in knot classification. It's the minimum number of crossings in any diagram of a knot, helping determine complexity. Knots with the same crossing number are grouped together in knot tables.

Up to 10 crossings, there are 165 distinct knots. These include familiar examples like the trefoil knot (3₁) and figure-eight knot (4₁). Prime knots can't be decomposed, while composite knots are formed by combining prime knots.

Knot Classification by Crossing Number

Crossing number in knot classification

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  • Minimum number of crossings in any diagram of a knot KK denoted as c(K)c(K)
  • Determines knot complexity and serves as primary means of classification
  • Knots with same crossing number grouped together in knot tables
  • Invariant property remains constant under ambient isotopy (continuous deformations)

Knot classification up to 10 crossings

  • Knots denoted by nmn_m, where nn is crossing number and mm is index within that crossing number
    • Trefoil knot (313_1) first knot with 3 crossings
  • 165 distinct knots up to 10 crossings (not including mirror images)
    • 0 crossings: unknot (010_1)
    • 3 crossings: trefoil knot (313_1)
    • 4 crossings: figure-eight knot (414_1)
    • 5 crossings: 515_1 and 525_2
    • 6 crossings: 616_1, 626_2, and 636_3
    • 7 crossings: 717_1 to 777_7
    • 8 crossings: 818_1 to 8218_{21}
    • 9 crossings: 919_1 to 9499_{49}
    • 10 crossings: 10110_1 to 1016510_{165}

Prime vs composite knots

  • Prime knots cannot be decomposed into sum (connected sum) of two non-trivial knots
    • All knots up to 7 crossings are prime (trefoil knot 313_1, figure-eight knot 414_1)
  • Composite knots formed by connected sum of two or more prime knots
    • Denoted by prime knots used in composition (square knot 31#313_1 \# 3_1 composed of two trefoil knots)
    • First appear in knot tables at 8 crossings (8208_{20} and 8218_{21})
  • Knot tables typically list prime knots first followed by composite knots

Knot Invariants and Symmetries

Knot invariants for differentiation

  • Properties that remain constant under ambient isotopy used to distinguish knots with same crossing number
  • Tricolorability: knot diagram can be colored with three colors such that at each crossing either all three colors are present or only one color is present
    • Trefoil knot (313_1) is tricolorable, figure-eight knot (414_1) is not
  • Knot polynomials: algebraic expressions associated with knot that are invariant under ambient isotopy (Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial)
  • Knot signature: integer invariant derived from knot's Seifert matrix
    • Trefoil knot (313_1) has signature -2, figure-eight knot (414_1) has signature 0

Symmetries and chirality of knots

  • Symmetries are transformations that preserve knot type
    • Rotational symmetry: knot unchanged by rotations about an axis
      • Trefoil knot (313_1) has no rotational symmetry, figure-eight knot (414_1) has 180-degree rotational symmetry
    • Mirror symmetry: knot unchanged by reflection across a plane
      • Amphichiral knots equivalent to their mirror image (figure-eight knot 414_1)
      • Chiral knots not equivalent to their mirror image (trefoil knot 313_1 with distinct left-handed and right-handed forms)
  • Knot tables often include information about symmetries and chirality of each knot


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.