10 84

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10 83.gif

10_83

10 85.gif

10_85

10 84.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X8,12,9,11 X20,15,1,16 X16,5,17,6 X12,18,13,17 X14,8,15,7 X18,14,19,13 X6,19,7,20 X2,10,3,9
Gauss code 1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4
Dowker-Thistlethwaite code 4 10 16 14 2 8 18 20 12 6
Conway Notation [.22.2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 84 ML.gif 10 84 AP.gif
[{3, 11}, {2, 4}, {1, 3}, {7, 2}, {9, 12}, {10, 8}, {6, 9}, {11, 7}, {5, 10}, {4, 6}, {12, 5}, {8, 1}]

[edit Notes on presentations of 10 84]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 14.7099
A-Polynomial See Data:10 84/A-polynomial

[edit Notes for 10 84's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 84's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 87, 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: {K11a46, K11n184,}

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

Vassiliev invariants

V2 and V3: (2, 2)
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 10 84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-101234567χ
17          1-1
15         3 3
13        51 -4
11       63  3
9      85   -3
7     76    1
5    68     2
3   57      -2
1  27       5
-1 14        -3
-3 2         2
-51          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials