L11a273

L11a272

L11a274

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a273 at Knotilus!

Link Presentations

[edit Notes on L11a273's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2u^{3}v^{4}+2u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+3u^{2}v^{4}-5u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-5uv^{2}+3uv-u-v^{3}+2v^{2}-2v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {11}{q^{17/2}}}-{\frac {11}{q^{19/2}}}+{\frac {10}{q^{21/2}}}-{\frac {8}{q^{23/2}}}+{\frac {5}{q^{25/2}}}-{\frac {3}{q^{27/2}}}+{\frac {1}{q^{29/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+3a^{11}z^{5}+13a^{11}z^{3}+14a^{11}z+3a^{11}z^{-1}-2a^{9}z^{7}-11a^{9}z^{5}-19a^{9}z^{3}-12a^{9}z-2a^{9}z^{-1}-a^{7}z^{7}-5a^{7}z^{5}-7a^{7}z^{3}-3a^{7}z$ (db)
Kauffman polynomial	$-z^{4}a^{18}+z^{2}a^{18}-3z^{5}a^{17}+4z^{3}a^{17}-za^{17}-4z^{6}a^{16}+4z^{4}a^{16}-4z^{7}a^{15}+3z^{5}a^{15}+2z^{3}a^{15}-2za^{15}-4z^{8}a^{14}+7z^{6}a^{14}-7z^{4}a^{14}+4z^{2}a^{14}-a^{14}-3z^{9}a^{13}+6z^{7}a^{13}-5z^{5}a^{13}+3z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{10}a^{12}-4z^{8}a^{12}+25z^{6}a^{12}-40z^{4}a^{12}+24z^{2}a^{12}-3a^{12}-6z^{9}a^{11}+25z^{7}a^{11}-39z^{5}a^{11}+35z^{3}a^{11}-18za^{11}+3a^{11}z^{-1}-z^{10}a^{10}-2z^{8}a^{10}+22z^{6}a^{10}-35z^{4}a^{10}+20z^{2}a^{10}-3a^{10}-3z^{9}a^{9}+14z^{7}a^{9}-23z^{5}a^{9}+23z^{3}a^{9}-13za^{9}+2a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}-z^{7}a^{7}+5z^{5}a^{7}-7z^{3}a^{7}+3za^{7}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   3     3 -12                 3 2     -1 -14               6 3       3 -16             5 3         -2 -18           6 6           0 -20         4 5             1 -22       4 6               -2 -24     2 5                 3 -26   1 3                   -2 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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