L10a5

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a5 at Knotilus!

Link Presentations

[edit Notes on L10a5's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X12,5,13,6 X20,13,5,14 X16,12,17,11 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, 3, -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	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {3}{q^{7/2}}}-12q^{5/2}+{\frac {6}{q^{5/2}}}+14q^{3/2}-{\frac {11}{q^{3/2}}}+q^{11/2}-15{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}-2a^{3}z+5az-4za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-4a^{2}z^{8}-7z^{8}a^{-2}-11z^{8}-3a^{3}z^{7}-5az^{7}-12z^{7}a^{-1}-10z^{7}a^{-3}-a^{4}z^{6}+9a^{2}z^{6}+5z^{6}a^{-2}-8z^{6}a^{-4}+23z^{6}+9a^{3}z^{5}+25az^{5}+34z^{5}a^{-1}+14z^{5}a^{-3}-4z^{5}a^{-5}+3a^{4}z^{4}-4a^{2}z^{4}+5z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}-11z^{4}-9a^{3}z^{3}-26az^{3}-26z^{3}a^{-1}-7z^{3}a^{-3}+2z^{3}a^{-5}-2a^{4}z^{2}-a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+4a^{3}z+10az+8za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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                 5 1   -4 6               7 3     4 4             7 5       -2 2           8 7         1 0         7 9           2 -2       4 6             -2 -4     2 7               5 -6   1 4                 -3 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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