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7 1

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6 3.gif

6_3

7 2.gif

7_2

Contents

7 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,6,-3,7,-4,1,-5,2,-6,3,-7,4/goTop.html 7_1's page] at Knotilus!

Visit 7 1's page at the original Knot Atlas!

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 (lexing error): \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 3
Topological 4 genus 3
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(7,1))
Rasmussen s-Invariant -6

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

Polynomial invariants

Alexander polynomial t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1
Conway polynomial z^6+5 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, -6 }
Jones polynomial - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3}
HOMFLY-PT polynomial (db, data sources) a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6
Kauffman polynomial (db, data sources) a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6
The A2 invariant -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}
The G2 invariant q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50}