7 1

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6 3.gif

6_3

7 2.gif

7_2

7 1.gif
(KnotPlot image)

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7_1 should perhaps be called "The Septafoil Knot", following the trefoil knot and the cinquefoil knot. See also T(7,2).


Interlaced form of 7/2 star polygon or "septagram"
Decorative interlaced form of 7/2 star polygon or "septagram"
3D depiction
Heptagram of intersecting circles.

Knot presentations

Planar diagram presentation X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7
Gauss code -1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4
Dowker-Thistlethwaite code 8 10 12 14 2 4 6
Conway Notation [7]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

Length is 7, width is 2,

Braid index is 2

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

[edit Notes on presentations of 7 1]

Knot 7_1.
A graph, knot 7_1.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number Failed to parse (syntax error): {\displaystyle \text{$\$$Failed}}
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:7 1/A-polynomial

[edit Notes for 7 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant -6

[edit Notes for 7 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 7, -6 }
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: (6, -14)
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 -6 is the signature of 7 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-10χ
-5       11
-7       11
-9     1  1
-11        0
-13   11   0
-15        0
-17 11     0
-19        0
-211       -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials