L10a19

L10a18

L10a20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a19 at Knotilus!

Link Presentations

[edit Notes on L10a19's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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