L11a307

L11a306

L11a308

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a307 at Knotilus!

Link Presentations

[edit Notes on L11a307's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-3u^{2}v^{3}+9u^{2}v^{2}-8u^{2}v+3u^{2}+3uv^{3}-8uv^{2}+9uv-3u+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {17}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-5za^{7}-a^{7}z^{-1}+3z^{5}a^{5}+8z^{3}a^{5}+5za^{5}-z^{7}a^{3}-4z^{5}a^{3}-7z^{3}a^{3}-5za^{3}+z^{5}a+2z^{3}a$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-11a^{9}z^{5}+11a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+6a^{8}z^{8}-5a^{8}z^{6}-3a^{8}z^{4}+4a^{8}z^{2}-a^{8}+4a^{7}z^{9}+3a^{7}z^{7}-14a^{7}z^{5}+12a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+a^{6}z^{10}+13a^{6}z^{8}-31a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+8a^{5}z^{9}-8a^{5}z^{7}-8a^{5}z^{5}+10a^{5}z^{3}-a^{5}z+a^{4}z^{10}+13a^{4}z^{8}-40a^{4}z^{6}+41a^{4}z^{4}-16a^{4}z^{2}+4a^{3}z^{9}-a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+6a^{2}z^{8}-16a^{2}z^{6}+13a^{2}z^{4}-4a^{2}z^{2}+4az^{7}-11az^{5}+7az^{3}+z^{6}-2z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   4 1   -3 -2                 8 3     5 -4               10 5       -5 -6             9 7         2 -8           10 10           0 -10         7 9             -2 -12       5 10               5 -14     3 7                 -4 -16     5                   5 -18 1 3                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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