# 4 1

(Redirected from Figure Eight) (KnotPlot image) See the full Rolfsen Knot Table. Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 4 1 at Knotilus! 4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g.  . For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

### Knot presentations

 Planar diagram presentation X4251 X8615 X6374 X2738 Gauss code 1, -4, 3, -1, 2, -3, 4, -2 Dowker-Thistlethwaite code 4 6 8 2 Conway Notation 

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-3] Hyperbolic Volume 2.02988 A-Polynomial See Data:4 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $\textrm{ConcordanceGenus}(\textrm{Knot}(4,1))$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial $-t- t^{-1} +3$ Conway polynomial $1-z^2$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 5, 0 } Jones polynomial $q^2+ q^{-2} -q- q^{-1} +1$ HOMFLY-PT polynomial (db, data sources) $a^2+ a^{-2} -z^2-1$ Kauffman polynomial (db, data sources) $a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1$ The A2 invariant $q^8+q^6-1+ q^{-6} + q^{-8}$ The G2 invariant $q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38}$