L11n46

L11n45

L11n47

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n46 at Knotilus!

Link Presentations

[edit Notes on L11n46's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{7}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-4 a^9 z^3-5 a^9 z-3 a^9 z^{-1} +a^7 z^7+6 a^7 z^5+14 a^7 z^3+16 a^7 z+7 a^7 z^{-1} -2 a^5 z^5-9 a^5 z^3-11 a^5 z-4 a^5 z^{-1}$ (db)
Kauffman polynomial	$a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-3 a^{13} z^3+2 a^{12} z^6-a^{12} z^4-2 a^{12} z^2+2 a^{11} z^7-2 a^{11} z^5+a^{11} z^3+2 a^{10} z^8-5 a^{10} z^6+7 a^{10} z^4-a^{10} z^2+a^9 z^9-2 a^9 z^7+3 a^9 z^5-3 a^9 z^3+6 a^9 z-3 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+18 a^8 z^4-16 a^8 z^2+7 a^8+a^7 z^9-4 a^7 z^7+10 a^7 z^5-19 a^7 z^3+18 a^7 z-7 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+9 a^6 z^4-15 a^6 z^2+7 a^6+3 a^5 z^5-12 a^5 z^3+12 a^5 z-4 a^5 z^{-1}$ (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 -4                   2 2 -6                 2 2 0 -8               3     3 -10             2 2     0 -12           5 3       2 -14         2 2         0 -16       3 5           -2 -18     1 2             1 -20   1 3               -2 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n45 L11n47