L11a270

L11a269

L11a271

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a270 at Knotilus!

Link Presentations

[edit Notes on L11a270'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 X20,15,21,16 X22,17,9,18 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, -7, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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