L11n215

L11n214

L11n216

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n215 at Knotilus!

Link Presentations

[edit Notes on L11n215's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X6,20,7,19 X4,11,5,12 X2,13,3,14
Gauss code	{1, -11, 2, -10, 3, -9, -8, 6}, {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(2)^2 t(1)^3-t(2) t(1)^3+t(2)^3 t(1)^2-4 t(2)^2 t(1)^2+4 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+4 t(2)^2 t(1)-4 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^3 a^9-2 z a^9-a^9 z^{-1} +z^5 a^7+3 z^3 a^7+5 z a^7+3 a^7 z^{-1} +z^5 a^5+z^3 a^5-2 z a^5-2 a^5 z^{-1} -2 z^3 a^3-3 z a^3$ (db)
Kauffman polynomial	$-z^6 a^{12}+4 z^4 a^{12}-4 z^2 a^{12}-2 z^7 a^{11}+7 z^5 a^{11}-7 z^3 a^{11}+2 z a^{11}-2 z^8 a^{10}+5 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}-a^{10}-z^9 a^9+2 z^5 a^9+3 z^3 a^9-2 z a^9+a^9 z^{-1} -4 z^8 a^8+9 z^6 a^8-10 z^4 a^8+11 z^2 a^8-3 a^8-z^9 a^7-5 z^5 a^7+15 z^3 a^7-12 z a^7+3 a^7 z^{-1} -2 z^8 a^6+2 z^6 a^6-5 z^4 a^6+6 z^2 a^6-3 a^6-2 z^7 a^5+2 z^3 a^5-5 z a^5+2 a^5 z^{-1} -z^6 a^4-2 z^4 a^4+z^2 a^4-3 z^3 a^3+3 z a^3$ (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 -1 0 χ -2                   2 2 -4                 3 1 -2 -6               4 1   3 -8             4 3     -1 -10           5 4       1 -12         4 5         1 -14       3 4           -1 -16     1 4             3 -18   1 3               -2 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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