L10a16

L10a15

L10a17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a16 at Knotilus!

Link Presentations

[edit Notes on L10a16's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X20,11,5,12 X18,13,19,14 X14,17,15,18 X12,19,13,20 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 5, -6, 8, -3, 6, -5, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(2)^{3}+3t(1)t(2)^{2}-6t(2)^{2}-6t(1)t(2)+3t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {7}{q^{5/2}}}-q^{3/2}+{\frac {5}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-3a^{7}z-2a^{7}z^{-1}+2a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}+2a^{3}z^{3}+a^{3}z+a^{3}z^{-1}+az^{3}-az-az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+9z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-a^{8}-z^{9}a^{7}+15z^{5}a^{7}-26z^{3}a^{7}+13za^{7}-2a^{7}z^{-1}-5z^{8}a^{6}+20z^{6}a^{6}-23z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-z^{9}a^{5}-2z^{7}a^{5}+16z^{5}a^{5}-18z^{3}a^{5}+8za^{5}-a^{5}z^{-1}-3z^{8}a^{4}+8z^{6}a^{4}-8z^{4}a^{4}+7z^{2}a^{4}-2a^{4}-3z^{7}a^{3}+3z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-3z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-a^{2}-3z^{5}a+3z^{3}a-3za+az^{-1}-2z^{4}+z^{2}-z^{3}a^{-1}+za^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                     1 1 2                   1   -1 0                 3 1   2 -2               3 2     -1 -4             4 2       2 -6           3 3         0 -8         3 4           -1 -10       3 4             1 -12     1 2               -1 -14   1 3                 2 -16   1                   -1 -18 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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