10 109

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10 108.gif

10_108

10 110.gif

10_110

10 109.gif
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Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code 6 10 14 16 2 18 4 20 8 12
Conway Notation [2.2.2.2]


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

Length is 10, width is 3,

Braid index is 3

10 109 ML.gif 10 109 AP.gif
[{3, 13}, {2, 6}, {4, 7}, {6, 12}, {5, 3}, {1, 4}, {13, 11}, {12, 8}, {7, 9}, {8, 10}, {9, 5}, {11, 2}, {10, 1}]

[edit Notes on presentations of 10 109]


Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.9002
A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 109'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: {}

Same Jones Polynomial (up to mirroring, ): {10_81,}

Vassiliev invariants

V2 and V3: (3, 0)
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 10 109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        51 -4
5       62  4
3      75   -2
1     86    2
-1    68     2
-3   57      -2
-5  26       4
-7 15        -4
-9 2         2
-111          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials