L11a278

L11a277

L11a279

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a278 at Knotilus!

Link Presentations

[edit Notes on L11a278's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X8,9,1,10 X4,13,5,14 X20,17,21,18 X18,8,19,7 X6,20,7,19 X22,15,9,16 X16,21,17,22
Gauss code	{1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 10, -11, 7, -8, 9, -7, 11, -10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)-2t(2)+2}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {5}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-3a^{9}z+a^{7}z^{7}+5a^{7}z^{5}+7a^{7}z^{3}+3a^{7}z+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-4a^{13}z^{3}+a^{13}z+2a^{12}z^{6}-3a^{12}z^{4}+a^{12}z^{2}+2a^{11}z^{7}-4a^{11}z^{5}+4a^{11}z^{3}+2a^{10}z^{8}-6a^{10}z^{6}+8a^{10}z^{4}-a^{10}z^{2}+2a^{9}z^{9}-9a^{9}z^{7}+17a^{9}z^{5}-11a^{9}z^{3}+2a^{9}z+a^{8}z^{10}-4a^{8}z^{8}+7a^{8}z^{6}-6a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{9}-15a^{7}z^{7}+27a^{7}z^{5}-22a^{7}z^{3}+5a^{7}z+a^{6}z^{10}-5a^{6}z^{8}+11a^{6}z^{6}-16a^{6}z^{4}+8a^{6}z^{2}+a^{5}z^{9}-3a^{5}z^{7}-2a^{5}z^{5}+8a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+3a^{4}z^{2}-a^{4}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                         0 -4                   3 1   2 -6                 2 1     -1 -8               3 2       1 -10             3 2         -1 -12           4 3           1 -14         3 4             1 -16       2 3               -1 -18     1 3                 2 -20   1 2                   -1 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a277 L11a279