9 22

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9_21

9 23.gif

9_23

Contents

9 22.gif
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Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code 1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code 4 8 10 14 2 16 18 6 12
Conway Notation [211,3,2]


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

Length is 9, width is 4,

Braid index is 4

9 22 ML.gif 9 22 AP.gif
[{6, 12}, {1, 9}, {11, 4}, {12, 10}, {8, 3}, {9, 7}, {5, 8}, {7, 11}, {4, 2}, {3, 6}, {2, 5}, {10, 1}]

[edit Notes on presentations of 9 22]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index \{4,7\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-8]
Hyperbolic Volume 10.6207
A-Polynomial See Data:9 22/A-polynomial

[edit Notes for 9 22's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for 9 22's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 43, 2 }
Jones polynomial -q^6+3 q^5-5 q^4+7 q^3-7 q^2+7 q-6+4 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+6 z^2 a^{-2} -2 z^2 a^{-4} -6 z^2+2 a^2+4 a^{-2} - a^{-4} -4
Kauffman polynomial (db, data sources) z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+7 z^6 a^{-2} +6 z^6 a^{-4} +2 z^6-7 a z^5-16 z^5 a^{-1} -4 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-23 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -15 z^4+7 a z^3+10 z^3 a^{-1} -2 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +5 a^2 z^2+17 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +16 z^2-2 a z-2 z a^{-1} +z a^{-3} +z a^{-5} -2 a^2-4 a^{-2} - a^{-4} -4
The A2 invariant q^{10}+q^8+q^4-2 q^2-1- q^{-4} +3 q^{-6} +2 q^{-10} - q^{-14} + q^{-16} - q^{-18}
The G2 invariant q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-3 q^{36}-3 q^{34}+14 q^{32}-20 q^{30}+25 q^{28}-17 q^{26}+2 q^{24}+19 q^{22}-35 q^{20}+43 q^{18}-35 q^{16}+14 q^{14}+11 q^{12}-35 q^{10}+41 q^8-32 q^6+10 q^4+12 q^2-28+24 q^{-2} -14 q^{-4} -11 q^{-6} +29 q^{-8} -38 q^{-10} +31 q^{-12} -9 q^{-14} -20 q^{-16} +46 q^{-18} -56 q^{-20} +50 q^{-22} -25 q^{-24} -5 q^{-26} +37 q^{-28} -52 q^{-30} +54 q^{-32} -30 q^{-34} +5 q^{-36} +23 q^{-38} -34 q^{-40} +28 q^{-42} -9 q^{-44} -11 q^{-46} +25 q^{-48} -25 q^{-50} +12 q^{-52} +8 q^{-54} -28 q^{-56} +36 q^{-58} -33 q^{-60} +17 q^{-62} +2 q^{-64} -21 q^{-66} +28 q^{-68} -29 q^{-70} +24 q^{-72} -11 q^{-74} +8 q^{-78} -14 q^{-80} +14 q^{-82} -11 q^{-84} +8 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100}