7 4

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7 3.gif

7_3

7 5.gif

7_5

7 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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Visit 7 4 at Knotilus!

Simplest version of Endless knot symbol.

Celtic or pseudo-Celtic knot
Mongolian ornament
Susan Williams' medallion [1], the "Endless knot" of Buddhism [2]
Ornamental "Endless knot"
a knot seen at the Castle of Kornik [3]
A 7-4 knot reduced from TakaraMusubi with 9 crossings [4]
TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5]
Buddhist Endless Knot
Ornamental Endless Knot
Albrecht Dürer knot, 16th-century
A laser cut by Tom Longtin [6]
Unicursal hexagram of occultism
Logo of the raelian sect
Lissajous curve : x=cos 3t , y=sin 2t, z=sin 7t
French europa stamp 2023


Knot presentations

Planar diagram presentation X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
Gauss code 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
Dowker-Thistlethwaite code 6 10 12 14 4 2 8
Conway Notation [313]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 9, width is 4,

Braid index is 4

7 4 ML.gif 7 4 AP.gif
[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]

[edit Notes on presentations of 7 4]

Knot 7_4.
A graph, knot 7_4.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 2
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number 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 \text{$\$$Failed}}
Hyperbolic Volume 5.13794
A-Polynomial See Data:7 4/A-polynomial

[edit Notes for 7 4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -2

[edit Notes for 7 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 15, 2 }
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: {9_2,}

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

Vassiliev invariants

V2 and V3: (4, 8)
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 2 is the signature of 7 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567χ
17       1-1
15        0
13     21 -1
11    1   1
9   12   1
7  21    1
5  1     1
312      -1
11       1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials