9 1

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8 21.gif

8_21

9 2.gif

9_2

Contents

9 1.gif
(KnotPlot image)

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9_1 should perhaps be called "The Nonafoil Knot", following the trefoil knot, the cinquefoil knot and (maybe) the septafoil knot. The next in the series is K11a367. See also T(9,2).


Interlaced form of 9/2 star polygon or "nonagram"
Decorative interlaced form of 9/2 star polygon or "nonagram"
Alternate interlaced form of 9/2 star polygon or "nonagram"

Knot presentations

Planar diagram presentation X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9
Gauss code -1, 6, -2, 7, -3, 8, -4, 9, -5, 1, -6, 2, -7, 3, -8, 4, -9, 5
Dowker-Thistlethwaite code 10 12 14 16 18 2 4 6 8
Conway Notation [9]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

Length is 9, width is 2,

Braid index is 2

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

[edit Notes on presentations of 9 1]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [-18][7]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:9 1/A-polynomial

[edit Notes for 9 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 4
Topological 4 genus 4
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for 9 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4}
Conway polynomial z^8+7 z^6+15 z^4+10 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, -8 }
Jones polynomial  q^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{10}-6 z^4 a^{10}-10 z^2 a^{10}-4 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+20 z^2 a^8+5 a^8
Kauffman polynomial (db, data sources) z a^{17}+z^2 a^{16}+z^3 a^{15}-z a^{15}+z^4 a^{14}-2 z^2 a^{14}+z^5 a^{13}-3 z^3 a^{13}+z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-z a^{11}+z^8 a^{10}-7 z^6 a^{10}+16 z^4 a^{10}-14 z^2 a^{10}+4 a^{10}+z^7 a^9-6 z^5 a^9+10 z^3 a^9-4 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-20 z^2 a^8+5 a^8
The A2 invariant -q^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14}
The G2 invariant q^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+q^{86}+q^{84}+2 q^{82}+2 q^{80}+q^{78}+q^{74}+q^{72}+q^{70}