L11a383

L11a382

L11a384

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a383 at Knotilus!

Link Presentations

[edit Notes on L11a383's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+4u^{3}v^{3}-3u^{3}v^{2}-3u^{2}v^{3}+7u^{2}v^{2}-3u^{2}v-3uv^{2}+4uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {2}{q^{7/2}}}-9q^{5/2}+{\frac {4}{q^{5/2}}}+10q^{3/2}-{\frac {7}{q^{3/2}}}-q^{13/2}+2q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-6z^{5}a^{-3}-12z^{3}a^{-3}-8za^{-3}-a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+8z^{7}a^{-1}-6az^{5}+24z^{5}a^{-1}-12az^{3}+32z^{3}a^{-1}-8az+17za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+2z^{6}a^{-6}-5z^{4}a^{-6}+z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}+a^{4}z^{6}-15z^{6}a^{-4}-4a^{4}z^{4}+22z^{4}a^{-4}+4a^{4}z^{2}-10z^{2}a^{-4}+3z^{9}a^{-3}+2a^{3}z^{7}-11z^{7}a^{-3}-7a^{3}z^{5}+18z^{5}a^{-3}+6a^{3}z^{3}-11z^{3}a^{-3}-a^{3}z+7za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+2a^{2}z^{8}+z^{8}a^{-2}-5a^{2}z^{6}-14z^{6}a^{-2}+2a^{2}z^{4}+31z^{4}a^{-2}-a^{2}z^{2}-16z^{2}a^{-2}+a^{-2}+2az^{9}+5z^{9}a^{-1}-7az^{7}-23z^{7}a^{-1}+15az^{5}+49z^{5}a^{-1}-22az^{3}-47z^{3}a^{-1}+9az+19za^{-1}-a^{-1}z^{-1}+z^{10}-z^{8}-3z^{6}+10z^{4}-10z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     1   -1 10                   3 1   2 8                 4 1     -3 6               5 3       2 4             6 5         -1 2           5 4           1 0         4 7             3 -2       3 4               -1 -4     1 4                 3 -6   1 3                   -2 -8   1                     1 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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