10 5: Difference between revisions

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Revision as of 20:06, 29 August 2005

10 4.gif

10_4

10 6.gif

10_6

10 5.gif Visit 10 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 5's page at Knotilus!

Visit 10 5's page at the original Knot Atlas!

10 5 Quick Notes


10 5 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X7,17,8,16 X9,19,10,18 X15,7,16,6 X17,9,18,8 X19,11,20,10 X11,2,12,3
Gauss code -1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3
Dowker-Thistlethwaite code 4 12 14 16 18 2 20 6 8 10
Conway Notation [6112]

Minimum Braid Representative:

BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

10 5 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [0][-12]
Hyperbolic Volume 7.37394
A-Polynomial See Data:10 5/A-polynomial

[edit Notes for 10 5's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 33, 4 }
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: (4, 7)
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 4 is the signature of 10 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-101234567χ
19          1-1
17         1 1
15        21 -1
13       21  1
11      32   -1
9     22    0
7    23     1
5   22      0
3  13       2
1 11        0
-1 1         1
-31          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials