10 49

From Knot Atlas
Jump to navigationJump to search

10 48.gif

10_48

10 50.gif

10_50

10 49.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 49 at Knotilus!


Knot presentations

Planar diagram presentation X1425 X3849 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X7283
Gauss code -1, 10, -2, 1, -3, 9, -10, 2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4
Dowker-Thistlethwaite code 4 8 14 2 16 18 6 20 10 12
Conway Notation [41,21,2]


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

Length is 11, width is 4,

Braid index is 4

10 49 ML.gif 10 49 AP.gif
[{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 9}, {4, 1}, {5, 10}, {6, 2}, {7, 5}, {8, 6}, {12, 7}, {1, 8}]

[edit Notes on presentations of 10 49]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-17][5]
Hyperbolic Volume 11.4532
A-Polynomial See Data:10 49/A-polynomial

[edit Notes for 10 49's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -6

[edit Notes for 10 49's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 59, -6 }
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: (7, -16)
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 -6 is the signature of 10 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-10-9-8-7-6-5-4-3-2-10χ
-5          11
-7         21-1
-9        3  3
-11       32  -1
-13      63   3
-15     43    -1
-17    56     -1
-19   34      1
-21  25       -3
-23 13        2
-25 2         -2
-271          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials