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6 1

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5 2.gif

5_2

6 2.gif

6_2

Contents

6 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6_1's page at Knotilus!

Visit 6 1's page at the original Knot Atlas!

6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1).


A Kolam of a 3x3 dot array
3D depiction

Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X5,12,6,1 X11,6,12,7
Gauss code -1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5
Dowker-Thistlethwaite code 4 8 12 10 2 6
Conway Notation [42]


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

Length is 7, width is 4,

Braid index is 4

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

[edit Notes on presentations of 6 1]

knot 6_1.
A graph, knot 6_1

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index \{3,4\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-3]
Hyperbolic Volume 3.16396
A-Polynomial See Data:6 1/A-polynomial

[edit Notes for 6 1's three dimensional invariants]
6_1 is a ribbon knot (drawings by Yoko Mizuma):

a ribbon diagram
isotopy to a ribbon
6_1 is doubly slice, by Scott Carter
Scott Carter notes that 6_1 is doubly slice - it bounds two distinct slice disks. He says: "this was spoken of in Fox's Example 10, 11, and 12 in a Quick Trip through Knot Theory ... BTW, the cover of Carter and Saito's Knotted Surfaces and Their Diagrams contains an illustration of such a slice disk". A picture is on the right.

Four dimensional invariants

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

[edit Notes for 6 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t+5-2 t^{-1}
Conway polynomial 1-2 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial q^2-q+2-2 q^{-1} + q^{-2} - q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-z^2 a^2-a^2-z^2+ a^{-2}
Kauffman polynomial (db, data sources) a^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+a^4+a^2- a^{-2}
The A2 invariant q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8}
The G2 invariant q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38}