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7 4

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7 3.gif

7_3

7 5.gif

7_5

Contents

7 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-5,6,-7,2,-1,3,-4,7,-6,5,-2,4,-3/goTop.html 7_4's page] at Knotilus!

Visit 7 4's page at the original Knot Atlas!

Simplest version of Endless knot symbol.

Celtic or pseudo-Celtic knot
a knot seen at the Castle of Kornik [1]
Susan Williams' medallion [2], the "Endless knot" of Buddhism [3]
Ornamental "Endless knot"
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

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 \{3,4\}
Nakanishi index 1
Maximal Thurston-Bennequin number Failed to parse (lexing error): \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 1
Topological 4 genus 1
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(7,4))
Rasmussen s-Invariant -2

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

Polynomial invariants

Alexander polynomial 4 t+4 t^{-1} -7
Conway polynomial 4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 15, 2 }
Jones polynomial -q^8+q^7-2 q^6+3 q^5-2 q^4+3 q^3-2 q^2+q
HOMFLY-PT polynomial (db, data sources) - a^{-8} +z^2 a^{-6} +2 z^2 a^{-4} +2 a^{-4} +z^2 a^{-2}
Kauffman polynomial (db, data sources) z^6 a^{-6} +z^6 a^{-8} +2 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-4} -3 z^4 a^{-8} +2 z^3 a^{-3} -2 z^3 a^{-5} -8 z^3 a^{-7} -4 z^3 a^{-9} +z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8}
The A2 invariant  q^{-2} - q^{-4} + q^{-8} + q^{-10} +2 q^{-12} + q^{-14} + q^{-16} - q^{-20} - q^{-24} - q^{-26}
The G2 invariant  q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-22} +4 q^{-24} -2 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} -2 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-44} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} - q^{-56} +3 q^{-58} + q^{-60} -2 q^{-62} +6 q^{-64} -3 q^{-66} +4 q^{-68} +2 q^{-70} -3 q^{-72} +4 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} - q^{-82} + q^{-84} -2 q^{-86} +2 q^{-88} -3 q^{-92} - q^{-94} -2 q^{-96} -4 q^{-102} +2 q^{-104} -3 q^{-106} + q^{-108} + q^{-110} -4 q^{-112} +3 q^{-114} -2 q^{-116} + q^{-118} -2 q^{-122} +2 q^{-124} + q^{-128}