L10a20

L10a19

L10a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a20 at Knotilus!

Link Presentations

[edit Notes on L10a20's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-13q^{5/2}+15q^{3/2}-17{\sqrt {q}}+{\frac {14}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z+2za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+6az^{3}-7z^{3}a^{-1}+7az-7za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+4z^{5}a^{-5}-z^{3}a^{-5}+a^{4}z^{6}+9z^{6}a^{-4}-3a^{4}z^{4}-9z^{4}a^{-4}+3a^{4}z^{2}+4z^{2}a^{-4}-a^{4}-a^{-4}+3a^{3}z^{7}+12z^{7}a^{-3}-8a^{3}z^{5}-18z^{5}a^{-3}+8a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+4a^{2}z^{8}+8z^{8}a^{-2}-6a^{2}z^{6}-3z^{6}a^{-2}-4a^{2}z^{4}-14z^{4}a^{-2}+10a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+2az^{9}+2z^{9}a^{-1}+8az^{7}+17z^{7}a^{-1}-31az^{5}-45z^{5}a^{-1}+29az^{3}+33z^{3}a^{-1}-13az-13za^{-1}+4az^{-1}+4a^{-1}z^{-1}+12z^{8}-19z^{6}-5z^{4}+15z^{2}-7$ (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                 6 1   -5 6               7 3     4 4             8 6       -2 2           9 7         2 0         7 10           3 -2       5 7             -2 -4     2 7               5 -6   1 5                 -4 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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