8 5

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8 4.gif

8_4

8 6.gif

8_6

Contents

8 5.gif
(KnotPlot image)

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8 5 is also known as the pretzel knot P(3,3,2).


Symmetric alternative representation
Pretzel P(3,3,2) form

Knot presentations

Planar diagram presentation X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15
Gauss code 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code 6 8 12 2 14 16 4 10
Conway Notation [3,3,2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 5]

Knot 8_5.
A graph, knot 8_5.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index \{4,6\}
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-11]
Hyperbolic Volume 6.99719
A-Polynomial See Data:8 5/A-polynomial

[edit Notes for 8 5's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 4 }
Jones polynomial q^8-2 q^7+3 q^6-4 q^5+3 q^4-3 q^3+3 q^2-q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} +4 a^{-2} -5 a^{-4} +2 a^{-6}
Kauffman polynomial (db, data sources) z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +4 z^6 a^{-4} +3 z^6 a^{-6} -3 z^5 a^{-3} +z^5 a^{-5} +4 z^5 a^{-7} -5 z^4 a^{-2} -15 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} -10 z^3 a^{-5} -8 z^3 a^{-7} +2 z^3 a^{-9} +8 z^2 a^{-2} +15 z^2 a^{-4} +4 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +7 z a^{-5} +4 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6}
The A2 invariant 1+ q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-24}
The G2 invariant  q^{-2} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +7 q^{-16} -5 q^{-18} +5 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -5 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} +2 q^{-42} -4 q^{-44} +2 q^{-46} -4 q^{-48} -2 q^{-50} +2 q^{-52} -9 q^{-54} +4 q^{-56} -7 q^{-58} + q^{-62} -7 q^{-64} +7 q^{-66} -7 q^{-68} +4 q^{-70} +2 q^{-72} -5 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} + q^{-88} - q^{-90} +4 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-100} +3 q^{-102} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} -3 q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134}