L11a271

L11a270

L11a272

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a271 at Knotilus!

Link Presentations

[edit Notes on L11a271's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11a270 L11a272