L9a32

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L9a31.gif

L9a31

L9a33.gif

L9a33

Contents

L9a32.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a32 at Knotilus!

L9a32 is 9^2_{40} in the Rolfsen table of links.

A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")
Symmetric version, with three lines touching at center
Symmetric version, with three lines touching at cicrumference

Link Presentations

[edit Notes on L9a32's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X18,13,7,14 X14,9,15,10 X10,17,11,18 X16,5,17,6 X2738 X4,11,5,12 X6,15,1,16
Gauss code {1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}
A Braid Representative
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A Morse Link Presentation L9a32 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^4-3 t(1) t(2)^3+3 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1) t(2)^2} (db)
Jones polynomial \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{9}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -3 a^7 z^3-4 a^7 z-2 a^7 z^{-1} -3 a^5 z^3-3 a^5 z-a^3 z^3 (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-10 a^{11} z^5+9 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-a^{10} z^6-12 a^{10} z^4+14 a^{10} z^2-3 a^{10}+9 a^9 z^7-26 a^9 z^5+22 a^9 z^3-11 a^9 z+3 a^9 z^{-1} +2 a^8 z^8+5 a^8 z^6-21 a^8 z^4+14 a^8 z^2-3 a^8+6 a^7 z^7-10 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 z^3 (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 \
-9-8-7-6-5-4-3-2-10χ
-2         11
-4        31-2
-6       3  3
-8      43  -1
-10     53   2
-12    34    1
-14   45     -1
-16  24      2
-18 13       -2
-20 2        2
-221         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L9a31.gif

L9a31

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L9a33