5 1

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4 1.gif


5 2.gif


5 1.gif
(KnotPlot image)

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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.
Knotted epitrochoid
Sum of two 5_1s, Vienna, orthodox church

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

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
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -4

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

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 5, -4 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {[[10_132]], }

Same Jones Polynomial (up to mirroring, ): {[[10_132]], }

Vassiliev invariants

V2 and V3: (3, -5)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where -4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-3     11
-5     11
-7   1  1
-9      0
-11 11   0
-13      0
-151     -1
Integral Khovanov Homology

(db, data source)


The Coloured Jones Polynomials