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)

L6n1

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L6a5.gif

L6a5

L7a1.gif

L7a1

Contents

L6n1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L6n1's page at Knotilus.

Visit L6n1's page at the original Knot Atlas.

L6n1 is 6^3_3 in Rolfsen's table of links. It makes three fibers in the Hopf fibration.

One modern form of the Germanic Valknut
Basic depiction
Coat of arms of Suchy, Vaud, Switzerland
Rich Schwartz' "98"
Episcopal coat of arms of Dom Jacinto Bergmann (Brazil)
Basic symmetrical depiction
Heraldic badge of Admiral Lord Boyce as Lord Warden of the Cinque Ports
Canadian trade-union federation emblem.

Link Presentations

[edit Notes on L6n1's Link Presentations]

Planar diagram presentation X6172 X12,8,9,7 X4,12,1,11 X5,11,6,10 X3845 X9,3,10,2
Gauss code {1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L6n1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{w-u v}{\sqrt{u} \sqrt{v} \sqrt{w}} (db)
Jones polynomial q^4+q^2+2 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^{-2} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial z^4 a^{-2} +z^4 a^{-4} +z^3 a^{-1} +z^3 a^{-3} -4 z^2 a^{-2} -4 z^2 a^{-4} -3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=0 is the signature of L6n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234χ
9    11
7    11
5  1  1
31    1
131   2
-12    2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=0 {\mathbb Z}^{2} {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.

L6a5.gif

L6a5

L7a1.gif

L7a1