10 109

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10 108.gif

10_108

10 110.gif

10_110

Contents

10 109.gif
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Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code 6 10 14 16 2 18 4 20 8 12
Conway Notation [2.2.2.2]


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

Length is 10, width is 3,

Braid index is 3

10 109 ML.gif 10 109 AP.gif
[{3, 13}, {2, 6}, {4, 7}, {6, 12}, {5, 3}, {1, 4}, {13, 11}, {12, 8}, {7, 9}, {8, 10}, {9, 5}, {11, 2}, {10, 1}]

[edit Notes on presentations of 10 109]


Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.9002
A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109'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 10 109's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+10 t^2-17 t+21-17 t^{-1} +10 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+6 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, 0 }
Jones polynomial -q^5+3 q^4-7 q^3+11 q^2-13 q+15-13 q^{-1} +11 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +14 z^4-6 a^2 z^2-6 z^2 a^{-2} +15 z^2-3 a^2-3 a^{-2} +7
Kauffman polynomial (db, data sources) 2 a z^9+2 z^9 a^{-1} +5 a^2 z^8+5 z^8 a^{-2} +10 z^8+5 a^3 z^7+6 a z^7+6 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6+a^5 z^5-8 a^3 z^5-16 a z^5-16 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} -5 z^4 a^{-4} +22 z^4-2 a^5 z^3+4 a^3 z^3+13 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -18 z^2+a^5 z-a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7
The A2 invariant -q^{14}+q^{12}-3 q^{10}+q^8-q^4+5 q^2-1+5 q^{-2} - q^{-4} + q^{-8} -3 q^{-10} + q^{-12} - q^{-14}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+14 q^{66}-32 q^{64}+51 q^{62}-62 q^{60}+51 q^{58}-20 q^{56}-39 q^{54}+113 q^{52}-171 q^{50}+187 q^{48}-138 q^{46}+19 q^{44}+124 q^{42}-249 q^{40}+297 q^{38}-239 q^{36}+89 q^{34}+90 q^{32}-231 q^{30}+264 q^{28}-177 q^{26}+13 q^{24}+150 q^{22}-232 q^{20}+187 q^{18}-37 q^{16}-151 q^{14}+301 q^{12}-336 q^{10}+248 q^8-53 q^6-170 q^4+353 q^2-417+353 q^{-2} -170 q^{-4} -53 q^{-6} +248 q^{-8} -336 q^{-10} +301 q^{-12} -151 q^{-14} -37 q^{-16} +187 q^{-18} -232 q^{-20} +150 q^{-22} +13 q^{-24} -177 q^{-26} +264 q^{-28} -231 q^{-30} +90 q^{-32} +89 q^{-34} -239 q^{-36} +297 q^{-38} -249 q^{-40} +124 q^{-42} +19 q^{-44} -138 q^{-46} +187 q^{-48} -171 q^{-50} +113 q^{-52} -39 q^{-54} -20 q^{-56} +51 q^{-58} -62 q^{-60} +51 q^{-62} -32 q^{-64} +14 q^{-66} + q^{-68} -8 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80}