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8 18

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8 17.gif

8_17

8 19.gif

8_19

Contents

8 18.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_18's page at Knotilus!

Visit 8 18's page at the original Knot Atlas!

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
Jump rope knot
Belt design
Bondage knot

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}