L11a308

L11a307

L11a309

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a308 at Knotilus!

Link Presentations

[edit Notes on L11a308's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)+t(2)-1)(t(1)t(2)+1)(t(2)t(1)-t(1)-t(2))\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {15}{q^{9/2}}}+{\frac {17}{q^{11/2}}}-{\frac {17}{q^{13/2}}}+{\frac {15}{q^{15/2}}}-{\frac {12}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-3a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}+a^{7}z^{7}+4a^{7}z^{5}+7a^{7}z^{3}+8a^{7}z+3a^{7}z^{-1}+a^{5}z^{7}+3a^{5}z^{5}-5a^{5}z-2a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-2a^{13}z^{3}+6a^{12}z^{6}-6a^{12}z^{4}+3a^{12}z^{2}+9a^{11}z^{7}-15a^{11}z^{5}+13a^{11}z^{3}-3a^{11}z+9a^{10}z^{8}-16a^{10}z^{6}+12a^{10}z^{4}-4a^{10}z^{2}+a^{10}+6a^{9}z^{9}-7a^{9}z^{7}-3a^{9}z^{5}+4a^{9}z^{3}-a^{9}z^{-1}+2a^{8}z^{10}+7a^{8}z^{8}-29a^{8}z^{6}+29a^{8}z^{4}-17a^{8}z^{2}+3a^{8}+10a^{7}z^{9}-30a^{7}z^{7}+31a^{7}z^{5}-23a^{7}z^{3}+12a^{7}z-3a^{7}z^{-1}+2a^{6}z^{10}+a^{6}z^{8}-18a^{6}z^{6}+21a^{6}z^{4}-11a^{6}z^{2}+3a^{6}+4a^{5}z^{9}-13a^{5}z^{7}+12a^{5}z^{5}-7a^{5}z^{3}+7a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+11a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z$ (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                     2   -2 -4                   5 1   4 -6                 6 3     -3 -8               9 4       5 -10             8 6         -2 -12           9 9           0 -14         7 9             2 -16       5 8               -3 -18     2 7                 5 -20   1 5                   -4 -22   2                     2 -24 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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