Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

5 2

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5 1.gif

5_1

6 1.gif

6_1

Contents

5 2.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,1,-3,4,-5,2,-4,3/goTop.html at Knotilus!

Visit 5 2's page at the original Knot Atlas!

5_2 is also known as the 3-twist knot.


3D depiction
Simple square depiction

Knot presentations

Planar diagram presentation X1425 X3849 X5,10,6,1 X9,6,10,7 X7283
Gauss code -1, 5, -2, 1, -3, 4, -5, 2, -4, 3
Dowker-Thistlethwaite code 4 8 10 2 6
Conway Notation [32]


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

Length is 6, width is 3,

Braid index is 3

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

[edit Notes on presentations of 5 2]

Knot 5_2.
A graph, knot 5_2.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index \{3,4\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][1]
Hyperbolic Volume 2.82812
A-Polynomial See Data:5 2/A-polynomial

[edit Notes for 5 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(5,2))
Rasmussen s-Invariant -2

[edit Notes for 5 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t+2 t^{-1} -3
Conway polynomial 2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, -2 }
Jones polynomial - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1}
HOMFLY-PT polynomial (db, data sources) -a^6+a^4 z^2+a^4+a^2 z^2+a^2
Kauffman polynomial (db, data sources) a^7 z^3-2 a^7 z+a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^3-2 a^5 z+a^4 z^4-a^4 z^2+a^4+a^3 z^3+a^2 z^2-a^2
The A2 invariant -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2
The G2 invariant q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2 q^{50}-q^{48}+2 q^{46}+q^{44}+q^{40}+q^{34}+2 q^{24}+q^{20}+q^{14}+q^{10}