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8 17

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8 16.gif

8_16

8 18.gif

8_18

Contents

8 17.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 17's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8_17's page at Knotilus!

Visit 8 17's page at the original Knot Atlas!

A knot in Brian Sanderson's Garden [1]

Knot presentations

Planar diagram presentation X6271 X14,8,15,7 X8394 X2,13,3,14 X12,5,13,6 X4,9,5,10 X16,12,1,11 X10,16,11,15
Gauss code 1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 7, -5, 4, -2, 8, -7
Dowker-Thistlethwaite code 6 8 12 14 4 16 2 10
Conway Notation [.2.2]


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

Length is 8, width is 3,

Braid index is 3

8 17 ML.gif 8 17 AP.gif
[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}]

[edit Notes on presentations of 8 17]

Knot 8_17.
A graph, knot 8_17.

Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 10.9859
A-Polynomial See Data:8 17/A-polynomial

[edit Notes for 8 17's three dimensional invariants] 8_17 is the first negatively amphicheiral knot in the Rolfsen Table. Namely, it is equal to the inverse of its mirror, yet it is different from both its inverse and its mirror.

Four dimensional invariants

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

[edit Notes for 8 17's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+4 t^2-8 t+11-8 t^{-1} +4 t^{-2} - t^{-3}
Conway polynomial -z^6-2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 37, 0 }
Jones polynomial q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 q^{-2} -3 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^2+ a^{-2} -1
Kauffman polynomial (db, data sources) 2 a z^7+2 z^7 a^{-1} +4 a^2 z^6+4 z^6 a^{-2} +8 z^6+3 a^3 z^5+2 a z^5+2 z^5 a^{-1} +3 z^5 a^{-3} +a^4 z^4-6 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} -14 z^4-4 a^3 z^3-6 a z^3-6 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +8 z^2+a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^2- a^{-2} -1
The A2 invariant q^{12}-q^{10}+q^8-q^4+2 q^2-1+2 q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12}
The G2 invariant q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+4 q^{58}-2 q^{56}-4 q^{54}+14 q^{52}-21 q^{50}+26 q^{48}-20 q^{46}+3 q^{44}+17 q^{42}-36 q^{40}+47 q^{38}-38 q^{36}+17 q^{34}+10 q^{32}-32 q^{30}+41 q^{28}-29 q^{26}+6 q^{24}+18 q^{22}-31 q^{20}+23 q^{18}-2 q^{16}-24 q^{14}+44 q^{12}-47 q^{10}+33 q^8-5 q^6-27 q^4+53 q^2-63+53 q^{-2} -27 q^{-4} -5 q^{-6} +33 q^{-8} -47 q^{-10} +44 q^{-12} -24 q^{-14} -2 q^{-16} +23 q^{-18} -31 q^{-20} +18 q^{-22} +6 q^{-24} -29 q^{-26} +41 q^{-28} -32 q^{-30} +10 q^{-32} +17 q^{-34} -38 q^{-36} +47 q^{-38} -36 q^{-40} +17 q^{-42} +3 q^{-44} -20 q^{-46} +26 q^{-48} -21 q^{-50} +14 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66}