Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

6 3

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6 2.gif

6_2

7 1.gif

7_1

Contents

6 3.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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

Visit 6_3's page at Knotilus!

Visit 6 3's page at the original Knot Atlas!

3D depiction

Knot presentations

Planar diagram presentation X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837
Gauss code 1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3
Dowker-Thistlethwaite code 4 8 10 2 12 6
Conway Notation [2112]


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

Length is 6, width is 3,

Braid index is 3

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

[edit Notes on presentations of 6 3]

Knot 6_3.
A graph, knot 6_3.

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index \{3,4\}
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-4]
Hyperbolic Volume 5.69302
A-Polynomial See Data:6 3/A-polynomial

[edit Notes for 6 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for 6 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-3 t+5-3 t^{-1} + t^{-2}
Conway polynomial z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 13, 0 }
Jones polynomial -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3}
HOMFLY-PT polynomial (db, data sources) z^4-a^2 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3
The A2 invariant -q^{10}+2 q^2+1+2 q^{-2} - q^{-10}
The G2 invariant q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52}