L10a4

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L10a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a4 at Knotilus!

Link Presentations

[edit Notes on L10a4's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X20,18,5,17 X18,11,19,12 X10,19,11,20 X2,14,3,13
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 6, -9, 8, -4, 10, -2, 3, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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