10 70

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10 69.gif

10_69

10 71.gif

10_71

10 70.gif
(KnotPlot image)

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Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X5,16,6,17 X11,19,12,18 X13,1,14,20 X19,13,20,12 X17,15,18,14 X15,6,16,7
Gauss code -1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 8, -7, 9, -10, 5, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 6 14 12
Conway Notation [22,3,2+]


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

Length is 10, width is 5,

Braid index is 5

10 70 ML.gif 10 70 AP.gif
[{13, 10}, {11, 9}, {10, 12}, {3, 11}, {8, 4}, {9, 7}, {5, 8}, {7, 13}, {4, 6}, {2, 5}, {1, 3}, {12, 2}, {6, 1}]

[edit Notes on presentations of 10 70]

Knot 10_70.
A graph, knot 10_70.
A part of a knot and a part of a graph.

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 [-3][-9]
Hyperbolic Volume 12.5109
A-Polynomial See Data:10 70/A-polynomial

[edit Notes for 10 70's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 70's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 67, 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: (-3, -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 70. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
15          11
13         2 -2
11        41 3
9       52  -3
7      64   2
5     55    0
3    56     -1
1   46      2
-1  14       -3
-3 14        3
-5 1         -1
-71          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials