L11a211

L11a210

L11a212

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a211 at Knotilus!

Link Presentations

[edit Notes on L11a211's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-2t(2)^{3}+6t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+6t(2)^{2}-2t(1)^{2}t(2)+8t(1)t(2)-5t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {17}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}+a^{7}z^{-1}-z^{7}a^{5}-4z^{5}a^{5}-7z^{3}a^{5}-6za^{5}-a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-2z^{3}a^{3}+z^{5}a+2z^{3}a$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-3z^{3}a^{9}+za^{9}-6z^{8}a^{8}+8z^{6}a^{8}-5z^{4}a^{8}+2z^{2}a^{8}-5z^{9}a^{7}+5z^{7}a^{7}-4z^{5}a^{7}+5z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{10}a^{6}-7z^{8}a^{6}+21z^{6}a^{6}-22z^{4}a^{6}+9z^{2}a^{6}-a^{6}-11z^{9}a^{5}+26z^{7}a^{5}-27z^{5}a^{5}+16z^{3}a^{5}-6za^{5}+a^{5}z^{-1}-2z^{10}a^{4}-8z^{8}a^{4}+30z^{6}a^{4}-28z^{4}a^{4}+8z^{2}a^{4}-6z^{9}a^{3}+12z^{7}a^{3}-5z^{5}a^{3}+z^{3}a^{3}-7z^{8}a^{2}+19z^{6}a^{2}-14z^{4}a^{2}+3z^{2}a^{2}-4z^{7}a+10z^{5}a-5z^{3}a-z^{6}+2z^{4}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 8 3     5 -4               10 6       -4 -6             10 7         3 -8           9 10           1 -10         8 10             -2 -12       4 9               5 -14     3 8                 -5 -16   1 5                   4 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a210 L11a212