7₁ knot - Alchetron, The Free Social Encyclopedia

Arf invariant 0 Braid no. 2 Crosscap no. 1		Braid length 7 Bridge no. 2 Crossing no. 7

In knot theory, the 7₁ knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil.

The 7₁ knot is invertible but not amphichiral. Its Alexander polynomial is

Δ ( t ) = t 3 − t 2 + t − 1 + t − 1 − t − 2 + t − 3 ,

its Conway polynomial is

∇ ( z ) = z 6 + 5 z 4 + 6 z 2 + 1 ,

and its Jones polynomial is

V ( q ) = q − 3 + q − 5 − q − 6 + q − 7 − q − 8 + q − 9 − q − 10 .

References

7₁ knot Wikipedia

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