8 18

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8 17.gif

8_17

8 19.gif

8_19

8 18.gif
(KnotPlot image)

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According to Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 57, a flat ribbon or strip can be tightly folded into a heptagonal 8_18 knot (just as it can be tightly folded into a pentagonal trefoil knot).


Logo of the International Guild of Knot Tyers [1]
A charity logo in Porto [2]
A laser cut by Tom Longtin [3]
Knot in (pseudo-)Celtic decorative form
Less symmetrical
Within outer circle
Impossible figure
Mongolian ornament
Jump rope knot
Belt design
Bondage knot
Spheric depiction
A "Hungarian Knot", decorating French Military uniforms.
Carpet swatter.
Geodesic of the prolate ellipsoid.
Obtained with an epitrochoid.


Knot presentations

Planar diagram presentation X6271 X8394 X16,11,1,12 X2,14,3,13 X4,15,5,16 X10,6,11,5 X12,7,13,8 X14,10,15,9
Gauss code 1, -4, 2, -5, 6, -1, 7, -2, 8, -6, 3, -7, 4, -8, 5, -3
Dowker-Thistlethwaite code 6 8 10 12 14 16 2 4
Conway Notation [8*]


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

Length is 8, width is 3,

Braid index is 3

8 18 ML.gif 8 18 AP.gif
[{3, 10}, {2, 8}, {4, 9}, {5, 3}, {1, 4}, {7, 2}, {8, 6}, {10, 7}, {9, 5}, {6, 1}]

[edit Notes on presentations of 8 18]

Knot 8_18.
A graph, knot 8_18.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index 4
Nakanishi index 2
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 12.3509
A-Polynomial See Data:8 18/A-polynomial

[edit Notes for 8 18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 0

[edit Notes for 8 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 45, 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: {9_24, K11n85, K11n164,}

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

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 8 18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234χ
9        11
7       3 -3
5      31 2
3     43  -1
1    53   2
-1   35    2
-3  34     -1
-5 13      2
-7 3       -3
-91        1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials