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4 1

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3 1.gif

3_1

5 1.gif

5_1

Contents

4 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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

Visit 4 1's page at the original Knot Atlas!

4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. [1] .

For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

Square depiction
Alternate square depiction
3D depiction
In "figure 8" form
A Neli-Kolam with 3x2 dot array[1]
In curved symmetrical form
Quasi-Celtic depiction
Symmetrical from parametric equation
Thurston's Trick [2]

Non-prime (compound) versions

Knot presentations

Planar diagram presentation X4251 X8615 X6374 X2738
Gauss code 1, -4, 3, -1, 2, -3, 4, -2
Dowker-Thistlethwaite code 4 6 8 2
Conway Notation [22]


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

Length is 4, width is 3,

Braid index is 3

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

[edit Notes on presentations of 4 1]

Knot 4_1.
A graph, knot 4_1.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index 3
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-3]
Hyperbolic Volume 2.02988
A-Polynomial See Data:4 1/A-polynomial

[edit Notes for 4 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(4,1))
Rasmussen s-Invariant 0

[edit Notes for 4 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t- t^{-1} +3
Conway polynomial 1-z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, 0 }
Jones polynomial q^2+ q^{-2} -q- q^{-1} +1
HOMFLY-PT polynomial (db, data sources) a^2+ a^{-2} -z^2-1
Kauffman polynomial (db, data sources) a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1
The A2 invariant q^8+q^6-1+ q^{-6} + q^{-8}
The G2 invariant q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38}