9.2 Classification of knots up to certain crossing numbers
3 min read•Last 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 K denoted as 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 nm, where n is crossing number and m is index within that crossing number
Trefoil knot (31) first knot with 3 crossings
165 distinct knots up to 10 crossings (not including mirror images)
0 crossings: unknot (01)
3 crossings: trefoil knot (31)
4 crossings: figure-eight knot (41)
5 crossings: 51 and 52
6 crossings: 61, 62, and 63
7 crossings: 71 to 77
8 crossings: 81 to 821
9 crossings: 91 to 949
10 crossings: 101 to 10165
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 31, figure-eight knot 41)
Composite knots formed by connected sum of two or more prime knots
Denoted by prime knots used in composition (square knot 31#31 composed of two trefoil knots)
First appear in knot tables at 8 crossings (820 and 821)
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 (31) is tricolorable, figure-eight knot (41) 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 (31) has signature -2, figure-eight knot (41) 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 (31) has no rotational symmetry, figure-eight knot (41) has 180-degree rotational symmetry
Mirror symmetry: knot unchanged by reflection across a plane
Amphichiral knots equivalent to their mirror image (figure-eight knot 41)
Chiral knots not equivalent to their mirror image (trefoil knot 31 with distinct left-handed and right-handed forms)
Knot tables often include information about symmetries and chirality of each knot