10 88

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10 87.gif

10_87

10 89.gif

10_89

Contents

10 88.gif
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Square quasi-symmetrical depiction.

Knot presentations

Planar diagram presentation X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20
Gauss code 1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2
Dowker-Thistlethwaite code 4 8 12 14 2 16 20 18 10 6
Conway Notation [.21.21]


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

Length is 10, width is 5,

Braid index is 5

10 88 ML.gif 10 88 AP.gif
[{3, 10}, {11, 5}, {9, 4}, {10, 6}, {5, 8}, {6, 2}, {1, 3}, {7, 9}, {8, 12}, {2, 11}, {12, 7}, {4, 1}]

[edit Notes on presentations of 10 88]


Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 15.6466
A-Polynomial See Data:10 88/A-polynomial

[edit Notes for 10 88's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for 10 88's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+8 t^2-24 t+35-24 t^{-1} +8 t^{-2} - t^{-3}
Conway polynomial -z^6+2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1
Kauffman polynomial (db, data sources) 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1
The A2 invariant -q^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16}
The G2 invariant q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80}