10 86

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10 85.gif

10_85

10 87.gif

10_87

10 86.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 86's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 86 at Knotilus!

Warning. There is a mixup in the original (1976) Rolfsen table between the pictures and the invariants of the knots 10_83 and 10_86. That mixup lead to a similar mixup here. In the new (2003) edition of Rolfsen's book the mixup was corrected and on August 17, 2004, it was corrected here (actually in Dror's original Knot Atlas) consistently with Rolfsen's correction. In the years between 1976 and 2003 other authors fixed the problem in different ways and our enumeration here may be different than theirs. Dror would like to thank Z-X. Tao for telling him about the (now corrected) mixup here and A. Stoimenow for telling him about the mixup in Rolfsen's original table.

Knot presentations

Planar diagram presentation X6271 X16,8,17,7 X8394 X2,15,3,16 X14,5,15,6 X4,9,5,10 X20,14,1,13 X10,20,11,19 X18,12,19,11 X12,18,13,17
Gauss code 1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 9, -10, 7, -5, 4, -2, 10, -9, 8, -7
Dowker-Thistlethwaite code 6 8 14 16 4 18 20 2 12 10
Conway Notation [.31.2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.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 86 ML.gif 10 86 AP.gif
[{2, 12}, {1, 7}, {11, 4}, {12, 8}, {7, 9}, {8, 10}, {5, 3}, {4, 6}, {9, 5}, {6, 2}, {3, 11}, {10, 1}]

[edit Notes on presentations of 10 86]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-7]
Hyperbolic Volume 14.3413
A-Polynomial See Data:10 86/A-polynomial

[edit Notes for 10 86's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 86's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 85, 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: {K11a190,}

Same Jones Polynomial (up to mirroring, ): {10_60,}

Vassiliev invariants

V2 and V3: (-1, -1)
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 10 86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
13          11
11         2 -2
9        41 3
7       62  -4
5      74   3
3     76    -1
1    77     0
-1   58      3
-3  36       -3
-5 15        4
-7 3         -3
-91          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials