10 89

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10 88.gif

10_88

10 90.gif

10_90

10 89.gif
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Knot presentations

Planar diagram presentation X4251 X12,8,13,7 X8394 X2,9,3,10 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,16,19,15 X16,11,17,12 X10,17,11,18
Gauss code 1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5
Dowker-Thistlethwaite code 4 8 14 12 2 16 20 18 10 6
Conway Notation [.21.210]


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

Length is 12, width is 5,

Braid index is 5

10 89 ML.gif 10 89 AP.gif
[{3, 10}, {6, 2}, {1, 3}, {5, 8}, {7, 9}, {8, 11}, {10, 6}, {12, 7}, {11, 4}, {2, 5}, {4, 12}, {9, 1}]

[edit Notes on presentations of 10 89]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-10][-2]
Hyperbolic Volume 15.5661
A-Polynomial See Data:10 89/A-polynomial

[edit Notes for 10 89's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 89's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 99, -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: {}

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

Vassiliev invariants

V2 and V3: (1, -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 -2 is the signature of 10 89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-10123χ
5          1-1
3         4 4
1        51 -4
-1       84  4
-3      96   -3
-5     87    1
-7    79     2
-9   58      -3
-11  27       5
-13 15        -4
-15 2         2
-171          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials