10 65

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10 64.gif

10_64

10 66.gif

10_66

10 65.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code -1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8
Dowker-Thistlethwaite code 4 10 14 16 2 18 20 8 6 12
Conway Notation [31,3,21]


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

Length is 11, width is 4,

Braid index is 4

10 65 ML.gif 10 65 AP.gif
[{9, 2}, {1, 7}, {4, 8}, {7, 9}, {10, 13}, {8, 12}, {13, 11}, {3, 5}, {6, 4}, {5, 10}, {2, 6}, {12, 3}, {11, 1}]

[edit Notes on presentations of 10 65]


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 [-2][-10]
Hyperbolic Volume 12.0765
A-Polynomial See Data:10 65/A-polynomial

[edit Notes for 10 65's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 65's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 63, 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: {10_77, K11n71, K11n75,}

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

Vassiliev invariants

V2 and V3: (4, 7)
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 65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-101234567χ
17          1-1
15         1 1
13        41 -3
11       41  3
9      54   -1
7     64    2
5    45     1
3   46      -2
1  25       3
-1 13        -2
-3 2         2
-51          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials