L11a252

L11a251

L11a253

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a252 at Knotilus!

Link Presentations

[edit Notes on L11a252's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{5}-3u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+5u^{2}v^{4}-9u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+6uv^{3}-9uv^{2}+5uv-u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {17}{q^{9/2}}}-{\frac {20}{q^{7/2}}}-q^{5/2}+{\frac {19}{q^{5/2}}}+4q^{3/2}-{\frac {17}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {13}{q^{11/2}}}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+6a^{3}z^{7}-az^{7}-4a^{5}z^{5}+12a^{3}z^{5}-4az^{5}-4a^{5}z^{3}+8a^{3}z^{3}-4az^{3}-a^{3}z-az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-7a^{8}z^{4}+a^{8}z^{2}+11a^{7}z^{7}-14a^{7}z^{5}+5a^{7}z^{3}+11a^{6}z^{8}-16a^{6}z^{6}+5a^{6}z^{4}+8a^{5}z^{9}-10a^{5}z^{7}-4a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+7a^{4}z^{8}-34a^{4}z^{6}+28a^{4}z^{4}-5a^{4}z^{2}-a^{4}+14a^{3}z^{9}-43a^{3}z^{7}+38a^{3}z^{5}-10a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}-24a^{2}z^{6}+28a^{2}z^{4}-7a^{2}z^{2}+6az^{9}-21az^{7}+z^{7}a^{-1}+21az^{5}-3z^{5}a^{-1}-7az^{3}+2z^{3}a^{-1}+az+4z^{8}-14z^{6}+13z^{4}-3z^{2}$ (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 6                       1 1 4                     3   -3 2                   5 1   4 0                 7 3     -4 -2               10 5       5 -4             10 8         -2 -6           10 9           1 -8         7 10             3 -10       6 10               -4 -12     3 8                 5 -14   1 5                   -4 -16   3                     3 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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