L11n15

L11n14

L11n16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n15 at Knotilus!

Link Presentations

[edit Notes on L11n15's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X8493 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, -7, 8, 11, -2, 3, 10, -8, 7, -9, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}}$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^9-a^9 z^{-1} +z^3 a^7+3 z a^7+2 a^7 z^{-1} -z a^5-a^5 z^{-1} +a^3 z^{-1} -z a-a z^{-1}$ (db)
Kauffman polynomial	$-z^8 a^{10}+7 z^6 a^{10}-15 z^4 a^{10}+11 z^2 a^{10}-3 a^{10}-z^9 a^9+7 z^7 a^9-15 z^5 a^9+11 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+14 z^6 a^8-31 z^4 a^8+26 z^2 a^8-7 a^8-z^9 a^7+7 z^7 a^7-16 z^5 a^7+16 z^3 a^7-8 z a^7+2 a^7 z^{-1} -z^8 a^6+7 z^6 a^6-16 z^4 a^6+15 z^2 a^6-4 a^6-z^5 a^5+5 z^3 a^5-5 z a^5+a^5 z^{-1} -z a^3+a^3 z^{-1} -a^2-z a+a z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                   1 1 -2                 1 2 1 -4               1 1 1 1 -6             1 2     1 -8           1 1 1     1 -10         1 2 1       0 -12       1 1           0 -14       1 1           0 -16   1 1               0 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$

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.

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