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 1

From Knot Atlas
Jump to: navigation, search


4 1.gif

4_1

5 2.gif

5_2

Contents

5 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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

Visit 5 1's page at the original Knot Atlas!

An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).


A kolam of a 2x3 dot array
The VISA Interlink Logo [1]
Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]
The Utah State Parks logo
As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).
Non-pentagonal shape.
Pentagram of circles.
Alternate pentagram of intersecting circles.
3D-looking rendition.
Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation X1627 X3849 X5,10,6,1 X7283 X9,4,10,5
Gauss code -1, 4, -2, 5, -3, 1, -4, 2, -5, 3
Dowker-Thistlethwaite code 6 8 10 2 4
Conway Notation [5]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

Length is 5, width is 2,

Braid index is 2

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

[edit Notes on presentations of 5 1]

Knot 5_1.
A graph, knot 5_1.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index 3
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][3]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:5 1/A-polynomial

[edit Notes for 5 1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 5 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2+ t^{-2} -t- t^{-1} +1
Conway polynomial z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, -4 }
Jones polynomial - q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2}
HOMFLY-PT polynomial (db, data sources) a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4
Kauffman polynomial (db, data sources) a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4
The A2 invariant -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6
The G2 invariant q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}