L11n212

L11n211

L11n213

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n212 at Knotilus!

Link Presentations

[edit Notes on L11n212's Link Presentations]

Planar diagram presentation	X10,1,11,2 X13,21,14,20 X3,12,4,13 X2,19,3,20 X14,5,15,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X22,18,9,17 X21,4,22,5
Gauss code	{1, -4, -3, 11, 5, -9, 6, -7}, {7, -1, 8, 3, -2, -5, 9, -6, 10, -8, 4, 2, -11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n211 L11n213