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[[Category:Knot Page]]
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Revision as of 21:00, 29 August 2005

9 45.gif

9_45

9 47.gif

9_47

9 46.gif Visit 9 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 46's page at Knotilus!

Visit 9 46's page at the original Knot Atlas!

9_46 is also known as the pretzel knot P(3,3,-3).


9 46 Further Notes and Views

Knot presentations

Planar diagram presentation X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code 1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code 4 10 -14 -12 -16 2 -6 -18 -8
Conway Notation [3,3,21-]

Minimum Braid Representative:

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

Length is 9, width is 4.

Braid index is 4.

A Morse Link Presentation:

9 46 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 1
Bridge index 3
Super bridge index 4
Nakanishi index 2
Maximal Thurston-Bennequin number [-7][-1]
Hyperbolic Volume 4.7517
A-Polynomial See Data:9 46/A-polynomial

[edit Notes for 9 46's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 46's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 9, 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: {6_1, K11n67, K11n97, K11n139, ...}

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

Vassiliev invariants

V2 and V3: (-2, 3)
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 9 46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-10χ
1      22
-1      11
-3    11 0
-5   1   -1
-7   1   -1
-9 11    0
-11       0
-131      1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials