8 1

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7 7.gif

7_7

8 2.gif

8_2

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8 1 Quick Notes


8 1 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7
Gauss code -1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code 4 10 16 14 12 2 8 6
Conway Notation [62]

Minimum Braid Representative:

BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 10, width is 5.

Braid index is 5.

A Morse Link Presentation:

8 1 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-3]
Hyperbolic Volume 3.42721
A-Polynomial See Data:8 1/A-polynomial

[edit Notes for 8 1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 13, 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, ): {K11n70, ...}

Vassiliev invariants

V2 and V3: (-3, 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 0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012χ
5        11
3         0
1      21 1
-1     11  0
-3    11   0
-5   11    0
-7   1     -1
-9 11      0
-11         0
-131        1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials