L11a212

L11a211

L11a213

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a212 at Knotilus!

Link Presentations

[edit Notes on L11a212's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X20,17,21,18 X16,5,17,6 X18,11,19,12 X12,19,13,20 X22,16,7,15 X4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -5}, {4, -1, 2, -3, 8, -9, 5, -4, 10, -7, 6, -8, 9, -6, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^4-6 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u-2 v^3+5 v^2-6 v+2}{u v^2}$ (db)
Jones polynomial	$-\frac{16}{q^{9/2}}+\frac{11}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{4}{q^{21/2}}+\frac{8}{q^{19/2}}-\frac{13}{q^{17/2}}+\frac{17}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{18}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-2 a^9 z^3+a^7 z^7+3 a^7 z^5+2 a^7 z^3+a^7 z+a^7 z^{-1} +a^5 z^7+3 a^5 z^5+a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z$ (db)
Kauffman polynomial	$-z^4 a^{14}-4 z^5 a^{13}+2 z^3 a^{13}-8 z^6 a^{12}+7 z^4 a^{12}-z^2 a^{12}-11 z^7 a^{11}+14 z^5 a^{11}-5 z^3 a^{11}-11 z^8 a^{10}+17 z^6 a^{10}-7 z^4 a^{10}-7 z^9 a^9+6 z^7 a^9+9 z^5 a^9-8 z^3 a^9+2 z a^9-2 z^{10} a^8-11 z^8 a^8+39 z^6 a^8-31 z^4 a^8+7 z^2 a^8-11 z^9 a^7+30 z^7 a^7-22 z^5 a^7+7 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^{10} a^6-3 z^8 a^6+25 z^6 a^6-28 z^4 a^6+10 z^2 a^6-a^6-4 z^9 a^5+12 z^7 a^5-9 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -3 z^8 a^4+11 z^6 a^4-12 z^4 a^4+4 z^2 a^4-z^7 a^3+4 z^5 a^3-5 z^3 a^3+2 z a^3$ (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 1 2 χ 0                       1 1 -2                     2   -2 -4                   5 1   4 -6                 7 3     -4 -8               9 4       5 -10             9 7         -2 -12           10 9           1 -14         7 9             2 -16       6 10               -4 -18     3 8                 5 -20   1 5                   -4 -22   3                     3 -24 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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