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Revision as of 20:00, 29 August 2005

8 3.gif

8_3

8 5.gif

8_5

8 4.gif Visit 8 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 4's page at Knotilus!

Visit 8 4's page at the original Knot Atlas!

8 4 Quick Notes



Somewhat symmetric representation

Knot presentations

Planar diagram presentation X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15
Gauss code 1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code 6 10 12 16 14 4 2 8
Conway Notation [413]

Minimum Braid Representative:

BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.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.

A Morse Link Presentation:

8 4 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-3]
Hyperbolic Volume 5.50049
A-Polynomial See Data:8 4/A-polynomial

[edit Notes for 8 4's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 19, -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: {...}

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

Vassiliev invariants

V2 and V3: (-3, 1)
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 8 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234χ
7        11
5         0
3      21 1
1     1   -1
-1    22   0
-3   22    0
-5  11     0
-7 12      1
-9 1       -1
-111        1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials