# 7 4

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 4 at Knotilus! Simplest version of Endless knot symbol.
 Celtic or pseudo-Celtic knot Mongolian ornament Susan Williams' medallion [1], the "Endless knot" of Buddhism [2] Ornamental "Endless knot" a knot seen at the Castle of Kornik [3] A 7-4 knot reduced from TakaraMusubi with 9 crossings [4] TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5] Buddhist Endless Knot Ornamental Endless Knot Albrecht Dürer knot, 16th-century A laser cut by Tom Longtin [6] Unicursal hexagram of occultism Logo of the raelian sect Lissajous curve : x=cos 3t , y=sin 2t, z=sin 7t French europa stamp 2023

### Knot presentations

 Planar diagram presentation X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9 Gauss code 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3 Dowker-Thistlethwaite code 6 10 12 14 4 2 8 Conway Notation [313]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]
 Knot 7_4. A graph, knot 7_4.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 2 Super bridge index $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (lexing error): \text{$\$\$Failed} Hyperbolic Volume 5.13794 A-Polynomial See Data:7 4/A-polynomial

### Four dimensional invariants

 Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $\textrm{ConcordanceGenus}(\textrm{Knot}(7,4))$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial $4 t+4 t^{-1} -7$ Conway polynomial $4 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 15, 2 } Jones polynomial $-q^8+q^7-2 q^6+3 q^5-2 q^4+3 q^3-2 q^2+q$ HOMFLY-PT polynomial (db, data sources) $- a^{-8} +z^2 a^{-6} +2 z^2 a^{-4} +2 a^{-4} +z^2 a^{-2}$ Kauffman polynomial (db, data sources) $z^6 a^{-6} +z^6 a^{-8} +2 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-4} -3 z^4 a^{-8} +2 z^3 a^{-3} -2 z^3 a^{-5} -8 z^3 a^{-7} -4 z^3 a^{-9} +z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8}$ The A2 invariant $q^{-2} - q^{-4} + q^{-8} + q^{-10} +2 q^{-12} + q^{-14} + q^{-16} - q^{-20} - q^{-24} - q^{-26}$ The G2 invariant $q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +4 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} -2 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-44} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} - q^{-56} +3 q^{-58} + q^{-60} -2 q^{-62} +6 q^{-64} -3 q^{-66} +4 q^{-68} +2 q^{-70} -3 q^{-72} +4 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} - q^{-82} + q^{-84} -2 q^{-86} +2 q^{-88} -3 q^{-92} - q^{-94} -2 q^{-96} -4 q^{-102} +2 q^{-104} -3 q^{-106} + q^{-108} + q^{-110} -4 q^{-112} +3 q^{-114} -2 q^{-116} + q^{-118} -2 q^{-122} +2 q^{-124} + q^{-128}$