L9n15

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L9n14.gif

L9n14

L9n16.gif

L9n16

Contents

L9n15.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n15 at Knotilus!

This is a Seifert-fibered link: it is the trefoil plus the core of one solid torus.

L9n15 is 9^2_{49} in the Rolfsen table of links.

The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Photo by sameold2008

Link Presentations

[edit Notes on L9n15's Link Presentations]

Planar diagram presentation X8192 X16,11,17,12 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X9,14,10,15 X13,18,14,7 X17,4,18,5
Gauss code {1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 8}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L9n15 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4+t(1) t(2)^2+1}{t(1) t(2)^2} (db)
Jones polynomial -\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{11} (-z)-a^{11} z^{-1} +a^9 z^5+6 a^9 z^3+9 a^9 z+3 a^9 z^{-1} -a^7 z^7-7 a^7 z^5-15 a^7 z^3-11 a^7 z-2 a^7 z^{-1} (db)
Kauffman polynomial a^{12}+z a^{11}-a^{11} z^{-1} -z^6 a^{10}+6 z^4 a^{10}-9 z^2 a^{10}+3 a^{10}-z^7 a^9+7 z^5 a^9-15 z^3 a^9+12 z a^9-3 a^9 z^{-1} -z^6 a^8+6 z^4 a^8-9 z^2 a^8+3 a^8-z^7 a^7+7 z^5 a^7-15 z^3 a^7+11 z a^7-2 a^7 z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-6-5-4-3-2-10χ
-6      11
-8      11
-10    1  1
-12  1    1
-14  11   0
-1611     0
-1811     0
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6 i=-4
r=-6 {\mathbb Z} {\mathbb Z}
r=-5 {\mathbb Z} {\mathbb Z}
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}
r=-1
r=0 {\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).

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L9n14.gif

L9n14

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L9n16