4 1

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3 1.gif

3_1

5 1.gif

5_1

4 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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Visit 4 1 at Knotilus!

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]
Cylindrical depiction

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
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 5, 0 }
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: {}

Same Jones Polynomial (up to mirroring, ): {K11n19,}

Vassiliev invariants

V2 and V3: (-1, 0)
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 0 is the signature of 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-1012χ
5    11
3     0
1  11 0
-1 11  0
-3     0
-51    1
Integral Khovanov Homology

(db, data source)

  
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials