L10a128

L10a127

L10a129

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a128 at Knotilus!

Link Presentations

[edit Notes on L10a128's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(1)-3t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+6q^{-4}-4q^{3}-8q^{-3}+7q^{2}+12q^{-2}-10q-12q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-5a^{4}+3a^{2}z^{4}+z^{4}a^{-2}+7a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+5a^{2}-z^{6}-3z^{4}-4z^{2}-1$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}+3a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-13a^{4}z^{4}+z^{4}a^{-4}+18a^{4}z^{2}+2a^{4}z^{-2}-9a^{4}+a^{3}z^{9}+5a^{3}z^{7}-14a^{3}z^{5}+4z^{5}a^{-3}+4a^{3}z^{3}-3z^{3}a^{-3}+5a^{3}z-2a^{3}z^{-1}+6a^{2}z^{8}-4a^{2}z^{6}+7z^{6}a^{-2}-16a^{2}z^{4}-8z^{4}a^{-2}+19a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}-8a^{2}+az^{9}+10az^{7}+7z^{7}a^{-1}-19az^{5}-6z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+3az+za^{-1}+4z^{8}+4z^{6}-16z^{4}+10z^{2}-2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   3   -3 5                 4 1   3 3               6 3     -3 1             7 4       3 -1           6 7         1 -3         6 6           0 -5       4 8             4 -7     2 4               -2 -9   1 5                 4 -11   1                   -1 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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