L11a286

L11a285

L11a287

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a286 at Knotilus!

Link Presentations

[edit Notes on L11a286's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}-t(1)^{2}t(2)^{2}-t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+5q^{3/2}-9{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}-az^{7}+a^{5}z^{5}-4a^{3}z^{5}-4az^{5}+z^{5}a^{-1}+3a^{5}z^{3}-4a^{3}z^{3}-4az^{3}+3z^{3}a^{-1}+a^{5}z-a^{3}z-az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}+3a^{8}z^{4}-a^{8}z^{2}+5a^{7}z^{5}-3a^{7}z^{3}+7a^{6}z^{6}-8a^{6}z^{4}+9a^{5}z^{7}-19a^{5}z^{5}+9a^{5}z^{3}-2a^{5}z+9a^{4}z^{8}-25a^{4}z^{6}+17a^{4}z^{4}-4a^{4}z^{2}+6a^{3}z^{9}-15a^{3}z^{7}+9a^{3}z^{3}-2a^{3}z+2a^{2}z^{10}+3a^{2}z^{8}+z^{8}a^{-2}-35a^{2}z^{6}-5z^{6}a^{-2}+44a^{2}z^{4}+8z^{4}a^{-2}-12a^{2}z^{2}-4z^{2}a^{-2}+9az^{9}+3z^{9}a^{-1}-40az^{7}-16z^{7}a^{-1}+52az^{5}+28z^{5}a^{-1}-22az^{3}-18z^{3}a^{-1}+2az+2za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}-5z^{8}-8z^{6}+24z^{4}-11z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     2   2 4                   3 1   -2 2                 6 2     4 0               4 3       -1 -2             8 6         2 -4           6 6           0 -6         5 6             -1 -8       3 6               3 -10     2 5                 -3 -12   1 3                   2 -14   2                     -2 -16 1                       1

Integral Khovanov Homology (db, data source)

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

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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