L11a541

L11a540

L11a542

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a541 at Knotilus!

Link Presentations

[edit Notes on L11a541's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X8,16,5,15 X14,8,15,7 X16,19,17,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {9, -3, 5, -4}, {11, -2, 3, -6, 4, -8, 7, -9, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)(t(4)-1)^{2}(-t(2)t(1)+t(2)t(4)t(1)-t(4)t(1)+2t(1)+t(2)-2t(2)t(4)+t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-13q^{3/2}+18{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {25}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {15}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+5a^{5}z^{-1}+a^{5}z^{-3}-3z^{5}a^{3}-8z^{3}a^{3}-12za^{3}-10a^{3}z^{-1}-3a^{3}z^{-3}+z^{7}a+3z^{5}a+6z^{3}a+9za+9az^{-1}+3az^{-3}-z^{5}a^{-1}-z^{3}a^{-1}-2za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-13a^{3}z^{9}-9az^{9}-4a^{6}z^{8}-12a^{4}z^{8}-24a^{2}z^{8}-16z^{8}-3a^{7}z^{7}-4a^{5}z^{7}+5a^{3}z^{7}-7az^{7}-13z^{7}a^{-1}-a^{8}z^{6}+3a^{6}z^{6}+24a^{4}z^{6}+48a^{2}z^{6}-5z^{6}a^{-2}+23z^{6}+8a^{7}z^{5}+28a^{5}z^{5}+44a^{3}z^{5}+43az^{5}+18z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+9a^{6}z^{4}+2a^{4}z^{4}-11a^{2}z^{4}+2z^{4}a^{-2}-5z^{4}-9a^{7}z^{3}-43a^{5}z^{3}-68a^{3}z^{3}-43az^{3}-9z^{3}a^{-1}-3a^{8}z^{2}-14a^{6}z^{2}-30a^{4}z^{2}-24a^{2}z^{2}-5z^{2}+6a^{7}z+31a^{5}z+48a^{3}z+30az+7za^{-1}+a^{8}+6a^{6}+18a^{4}+21a^{2}+9-2a^{7}z^{-1}-11a^{5}z^{-1}-18a^{3}z^{-1}-14az^{-1}-5a^{-1}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   9 1   8 2                 9 4     -5 0               16 9       7 -2             15 15         0 -4           12 10           2 -6         8 15             7 -8       7 12               -5 -10     2 10                 8 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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