10 124

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10 123.gif

10_123

10 125.gif

10_125

Contents

10 124.gif
(KnotPlot image)

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10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).

If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle Q_{30}. See [1].

10_124 is not k-colourable for any k. See The Determinant and the Signature.

Torus knot T(5,3) form

Knot presentations

Planar diagram presentation X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X2837
Gauss code 1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code 4 8 -14 2 -16 -18 -20 -6 -10 -12
Conway Notation [5,3,2-]


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

Length is 10, width is 3,

Braid index is 3

10 124 ML.gif 10 124 AP.gif
[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 1}]

[edit Notes on presentations of 10 124]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [7][-15]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:10 124/A-polynomial

[edit Notes for 10 124'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 10 124's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4}
Conway polynomial z^8+7 z^6+14 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 1, 8 }
Jones polynomial -q^{10}+q^6+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 a^{-12}
Kauffman polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12}
The A2 invariant  q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-28} -2 q^{-30} -2 q^{-32} - q^{-34} + q^{-40}
The G2 invariant  q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} +2 q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} +2 q^{-88} +2 q^{-90} +2 q^{-92} +2 q^{-94} + q^{-96} +2 q^{-98} + q^{-100} + q^{-104} + q^{-106} - q^{-112} - q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} -2 q^{-128} -2 q^{-130} -2 q^{-132} - q^{-134} - q^{-136} -2 q^{-138} -2 q^{-140} - q^{-142} - q^{-146} - q^{-148} + q^{-160} + q^{-162} + q^{-168} + q^{-170} + q^{-180}