L10a24

L10a23

L10a25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a24 at Knotilus!

Link Presentations

[edit Notes on L10a24's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,12,15,11 X10,4,11,3 X12,5,13,6 X20,13,5,14 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code	{1, -9, 5, -3}, {6, -1, 2, -10, 8, -5, 4, -6, 7, -4, 9, -8, 10, -2, 3, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+2a^{5}z^{3}+a^{5}z-a^{5}z^{-1}-a^{3}z^{5}+2a^{3}z+3a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}-2az-2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+4a^{6}z^{8}-11a^{6}z^{6}+9a^{6}z^{4}-4a^{6}z^{2}+a^{6}+2a^{5}z^{9}+2a^{5}z^{7}-16a^{5}z^{5}+14a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+10a^{4}z^{8}-26a^{4}z^{6}+24a^{4}z^{4}-11a^{4}z^{2}+3a^{4}+2a^{3}z^{9}+7a^{3}z^{7}-21a^{3}z^{5}+12a^{3}z^{3}+3a^{3}z-3a^{3}z^{-1}+6a^{2}z^{8}-7a^{2}z^{6}+4a^{2}z^{4}+z^{4}a^{-2}-5a^{2}z^{2}+3a^{2}+8az^{7}-10az^{5}+4z^{5}a^{-1}+2az^{3}-3z^{3}a^{-1}+3az-2az^{-1}+7z^{6}-7z^{4}$ (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                   3   3 2                 4 1   -3 0               7 3     4 -2             7 6       -1 -4           7 5         2 -6         5 7           2 -8       4 7             -3 -10     2 5               3 -12   1 4                 -3 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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