Three twist knot - Alchetron, The Free Social Encyclopedia

Arf invariant 0 Braid no. 3 Crosscap no. 2		Braid length 6 Bridge no. 2 Crossing no. 5

In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 5₂ knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

Δ ( t ) = 2 t − 3 + 2 t − 1 ,

its Conway polynomial is

∇ ( z ) = 2 z 2 + 1 ,

and its Jones polynomial is

V ( q ) = q − 1 − q − 2 + 2 q − 3 − q − 4 + q − 5 − q − 6 .

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.

References

Three-twist knot Wikipedia

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