10 135

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10 134.gif

10_134

10 136.gif

10_136

10 135.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3849 X9,15,10,14 X12,5,13,6 X6,13,7,14 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X7283
Gauss code -1, 10, -2, 1, 4, -5, -10, 2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 -12 2 14 18 -6 20 10 16
Conway Notation [221,21,2-]


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

Length is 11, width is 4,

Braid index is 4

10 135 ML.gif 10 135 AP.gif
[{6, 8}, {7, 9}, {8, 12}, {11, 6}, {1, 10}, {9, 11}, {5, 2}, {4, 1}, {3, 5}, {12, 4}, {2, 7}, {10, 3}]

[edit Notes on presentations of 10 135]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-4]
Hyperbolic Volume 10.6872
A-Polynomial See Data:10 135/A-polynomial

[edit Notes for 10 135's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 135's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 37, 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: {10_34,}

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

Vassiliev invariants

V2 and V3: (3, -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 10 135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123χ
7        2-2
5       2 2
3      32 -1
1     42  2
-1    34   1
-3   33    0
-5  13     2
-7 13      -2
-9 1       1
-111        -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials