L10a21

L10a20

L10a22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a21 at Knotilus!

Link Presentations

[edit Notes on L10a21's Link Presentations]

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

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	$-8q^{9/2}+12q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {4}{q^{5/2}}}+16q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+4q^{11/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}+2az^{3}-6z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+az-3za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-12z^{8}a^{-2}-6z^{8}a^{-4}-6z^{8}-7az^{7}-12z^{7}a^{-1}-12z^{7}a^{-3}-7z^{7}a^{-5}-4a^{2}z^{6}+19z^{6}a^{-2}+5z^{6}a^{-4}-4z^{6}a^{-6}+6z^{6}-a^{3}z^{5}+13az^{5}+32z^{5}a^{-1}+31z^{5}a^{-3}+12z^{5}a^{-5}-z^{5}a^{-7}+6a^{2}z^{4}-7z^{4}a^{-2}+4z^{4}a^{-4}+6z^{4}a^{-6}+z^{4}+a^{3}z^{3}-8az^{3}-25z^{3}a^{-1}-23z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}-a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}a^{-6}-z^{2}+2az+6za^{-1}+6za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 5 1   4 8               7 3     -4 6             8 5       3 4           8 7         -1 2         8 8           0 0       5 10             5 -2     3 6               -3 -4   1 5                 4 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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