8 18

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8 17.gif

8_17

8 19.gif

8_19

Contents

8 18.gif
(KnotPlot image)

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According to Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 57, a flat ribbon or strip can be tightly folded into a heptagonal 8_18 knot (just as it can be tightly folded into a pentagonal trefoil knot).


Logo of the International Guild of Knot Tyers [1]
A charity logo in Porto [2]
A laser cut by Tom Longtin [3]
Knot in (pseudo-)Celtic decorative form
Less symmetrical
Within outer circle
Impossible figure
Mongolian ornament
Jump rope knot
Belt design
Bondage knot
Spheric depiction
A "Hungarian Knot", decorating French Military uniforms.

Knot presentations

Planar diagram presentation X6271 X8394 X16,11,1,12 X2,14,3,13 X4,15,5,16 X10,6,11,5 X12,7,13,8 X14,10,15,9
Gauss code 1, -4, 2, -5, 6, -1, 7, -2, 8, -6, 3, -7, 4, -8, 5, -3
Dowker-Thistlethwaite code 6 8 10 12 14 16 2 4
Conway Notation [8*]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 8, width is 3,

Braid index is 3

8 18 ML.gif 8 18 AP.gif
[{3, 10}, {2, 8}, {4, 9}, {5, 3}, {1, 4}, {7, 2}, {8, 6}, {10, 7}, {9, 5}, {6, 1}]

[edit Notes on presentations of 8 18]

Knot 8_18.
A graph, knot 8_18.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index 4
Nakanishi index 2
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 12.3509
A-Polynomial See Data:8 18/A-polynomial

[edit Notes for 8 18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus [1,3]
Rasmussen s-Invariant 0

[edit Notes for 8 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 45, 0 }
Jones polynomial q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+z^4 a^{-2} -3 z^4+a^2 z^2+z^2 a^{-2} -z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 3 a z^7+3 z^7 a^{-1} +6 a^2 z^6+6 z^6 a^{-2} +12 z^6+4 a^3 z^5+3 a z^5+3 z^5 a^{-1} +4 z^5 a^{-3} +a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-4 a^3 z^3-9 a z^3-9 z^3 a^{-1} -4 z^3 a^{-3} +3 a^2 z^2+3 z^2 a^{-2} +6 z^2+a z+z a^{-1} +a^2+ a^{-2} +3
The A2 invariant q^{12}-2 q^{10}-q^6-q^4+4 q^2+1+4 q^{-2} - q^{-4} - q^{-6} -2 q^{-10} + q^{-12}
The G2 invariant q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+8 q^{58}-4 q^{56}-5 q^{54}+23 q^{52}-36 q^{50}+48 q^{48}-38 q^{46}+7 q^{44}+28 q^{42}-67 q^{40}+84 q^{38}-71 q^{36}+29 q^{34}+17 q^{32}-58 q^{30}+77 q^{28}-56 q^{26}+8 q^{24}+34 q^{22}-59 q^{20}+45 q^{18}-6 q^{16}-45 q^{14}+81 q^{12}-81 q^{10}+64 q^8-11 q^6-48 q^4+97 q^2-111+97 q^{-2} -48 q^{-4} -11 q^{-6} +64 q^{-8} -81 q^{-10} +81 q^{-12} -45 q^{-14} -6 q^{-16} +45 q^{-18} -59 q^{-20} +34 q^{-22} +8 q^{-24} -56 q^{-26} +77 q^{-28} -58 q^{-30} +17 q^{-32} +29 q^{-34} -71 q^{-36} +84 q^{-38} -67 q^{-40} +28 q^{-42} +7 q^{-44} -38 q^{-46} +48 q^{-48} -36 q^{-50} +23 q^{-52} -5 q^{-54} -4 q^{-56} +8 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66}