Neha Patil (Editor)

Crosscap number

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In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of

1 − χ ( S ) ,

taken over all compact, connected, non-orientable surfaces S bounding K; here χ is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one.

Examples

  • The crosscap number of the trefoil knot is 1, as it bounds a Möbius strip and is not trivial.
  • The crosscap number of a torus knot was determined by M. Teragaito.

    • The formula for the knot sum is

    C ( k 1 ) + C ( k 2 ) − 1 ≤ C ( k 1 # k 2 ) ≤ C ( k 1 ) + C ( k 2 ) .

    References

    Crosscap number Wikipedia
    (Text) CC BY-SA


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