10 82

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10 81.gif

10_81

10 83.gif

10_83

Contents

10 82.gif
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Knot presentations

Planar diagram presentation X1627 X7,16,8,17 X3948 X15,3,16,2 X5,15,6,14 X9,5,10,4 X11,18,12,19 X13,20,14,1 X17,10,18,11 X19,12,20,13
Gauss code -1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code 6 8 14 16 4 18 20 2 10 12
Conway Notation [.4.2]


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

Length is 10, width is 3,

Braid index is 3

10 82 ML.gif 10 82 AP.gif
[{12, 3}, {1, 5}, {11, 4}, {5, 2}, {6, 12}, {3, 7}, {2, 6}, {4, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 1}]

[edit Notes on presentations of 10 82]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-9][-3]
Hyperbolic Volume 12.4315
A-Polynomial See Data:10 82/A-polynomial

[edit Notes for 10 82's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus [2,4]
Rasmussen s-Invariant -2

[edit Notes for 10 82's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+4 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +4 t^{-3} - t^{-4}
Conway polynomial -z^8-4 z^6-4 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, -2 }
Jones polynomial q^3-3 q^2+5 q-7+10 q^{-1} -10 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+4 a^4 z^4-12 a^2 z^4+4 z^4+4 a^4 z^2-8 a^2 z^2+4 z^2+1
Kauffman polynomial (db, data sources) 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+4 a^5 z^7-a^3 z^7-2 a z^7+3 z^7 a^{-1} +4 a^6 z^6-8 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -14 z^6+3 a^7 z^5-3 a^5 z^5-4 a^3 z^5-8 a z^5-10 z^5 a^{-1} +a^8 z^4-4 a^6 z^4+10 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +14 z^4-4 a^7 z^3-2 a^5 z^3+5 a^3 z^3+10 a z^3+7 z^3 a^{-1} -a^8 z^2-5 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -6 z^2+a^7 z+2 a^5 z-2 a z-z a^{-1} +1
The A2 invariant q^{20}-q^{18}+q^{16}-2 q^{14}-q^{12}+q^{10}-q^8+4 q^6-q^4+2 q^2- q^{-2} + q^{-4} - q^{-6} + q^{-8}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-22 q^{88}+35 q^{86}-38 q^{84}+35 q^{82}-25 q^{80}+6 q^{78}+16 q^{76}-37 q^{74}+57 q^{72}-60 q^{70}+47 q^{68}-18 q^{66}-20 q^{64}+52 q^{62}-67 q^{60}+53 q^{58}-17 q^{56}-28 q^{54}+53 q^{52}-53 q^{50}+16 q^{48}+36 q^{46}-76 q^{44}+82 q^{42}-56 q^{40}-q^{38}+64 q^{36}-107 q^{34}+116 q^{32}-84 q^{30}+30 q^{28}+38 q^{26}-85 q^{24}+106 q^{22}-88 q^{20}+49 q^{18}+4 q^{16}-52 q^{14}+75 q^{12}-60 q^{10}+23 q^8+28 q^6-66 q^4+68 q^2-36-21 q^{-2} +70 q^{-4} -95 q^{-6} +84 q^{-8} -38 q^{-10} -21 q^{-12} +69 q^{-14} -87 q^{-16} +78 q^{-18} -43 q^{-20} +2 q^{-22} +27 q^{-24} -41 q^{-26} +39 q^{-28} -26 q^{-30} +13 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46}