10 8

From Knot Atlas
Jump to navigationJump to search

10 7.gif

10_7

10 9.gif

10_9

10 8.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 8 at Knotilus!


Knot presentations

Planar diagram presentation X1627 X7,16,8,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X9,18,10,19 X11,20,12,1 X17,8,18,9 X19,10,20,11
Gauss code -1, 6, -4, 5, -3, 1, -2, 9, -7, 10, -8, 3, -5, 4, -6, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code 6 14 12 16 18 20 4 2 8 10
Conway Notation [514]


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

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 8]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][-1]
Hyperbolic Volume 6.08323
A-Polynomial See Data:10 8/A-polynomial

[edit Notes for 10 8's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 29, -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: (-3, 4)
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 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-101234χ
5          11
3           0
1        21 1
-1       1   -1
-3      32   1
-5     22    0
-7    22     0
-9   22      0
-11  12       -1
-13 12        1
-15 1         -1
-171          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials