10 17

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10 16.gif

10_16

10 18.gif

10_18

Contents

10 17.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X20,15,1,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X14,6,15,5 X2,12,3,11 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3
Dowker-Thistlethwaite code 6 12 14 16 18 2 4 20 8 10
Conway Notation [4114]


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

Length is 10, width is 3,

Braid index is 3

10 17 ML.gif 10 17 AP.gif
[{4, 12}, {3, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 11}, {9, 4}, {2, 5}, {12, 10}, {1, 3}, {11, 2}, {10, 1}]

[edit Notes on presentations of 10 17]


Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 8.53676
A-Polynomial See Data:10 17/A-polynomial

[edit Notes for 10 17's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 17's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+5 t^2-7 t+9-7 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+7 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 41, 0 }
Jones polynomial -q^5+2 q^4-3 q^3+5 q^2-6 q+7-6 q^{-1} +5 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-z^6 a^{-2} +7 z^6-5 a^2 z^4-5 z^4 a^{-2} +17 z^4-7 a^2 z^2-7 z^2 a^{-2} +16 z^2-2 a^2-2 a^{-2} +5
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-2 a z^7-2 z^7 a^{-1} +2 z^7 a^{-3} +2 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -18 z^6+a^5 z^5-5 a^3 z^5-5 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+11 a^2 z^4+11 z^4 a^{-2} -6 z^4 a^{-4} +34 z^4-3 a^5 z^3+2 a^3 z^3+6 a z^3+6 z^3 a^{-1} +2 z^3 a^{-3} -3 z^3 a^{-5} +3 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +3 z^2 a^{-4} -22 z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} +2 a^2+2 a^{-2} +5
The A2 invariant -q^{14}-q^{10}+q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} + q^{-8} - q^{-10} - q^{-14}
The G2 invariant q^{80}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-2 q^{68}+6 q^{66}-7 q^{64}+8 q^{62}-7 q^{60}+2 q^{58}+2 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+12 q^{48}-7 q^{46}-2 q^{44}+9 q^{42}-16 q^{40}+19 q^{38}-14 q^{36}+4 q^{34}+5 q^{32}-14 q^{30}+15 q^{28}-7 q^{26}-q^{24}+11 q^{22}-12 q^{20}+9 q^{18}+2 q^{16}-12 q^{14}+21 q^{12}-20 q^{10}+14 q^8-q^6-12 q^4+23 q^2-25+23 q^{-2} -12 q^{-4} - q^{-6} +14 q^{-8} -20 q^{-10} +21 q^{-12} -12 q^{-14} +2 q^{-16} +9 q^{-18} -12 q^{-20} +11 q^{-22} - q^{-24} -7 q^{-26} +15 q^{-28} -14 q^{-30} +5 q^{-32} +4 q^{-34} -14 q^{-36} +19 q^{-38} -16 q^{-40} +9 q^{-42} -2 q^{-44} -7 q^{-46} +12 q^{-48} -15 q^{-50} +11 q^{-52} -8 q^{-54} +2 q^{-56} +2 q^{-58} -7 q^{-60} +8 q^{-62} -7 q^{-64} +6 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80}