L11n25

L11n24

L11n26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n25 at Knotilus!

Link Presentations

[edit Notes on L11n25's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X11,14,12,15 X5,13,6,12 X13,5,14,22 X10,19,11,20 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, 4, -10, -7, 8, -9, 7, 11, -2, 3, -6, 10, -4, 6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-2a^{3}z^{-1}-a^{-3}z^{-1}-az^{5}-z^{5}a^{-1}-z^{3}a^{-1}+2az+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-2z^{4}a^{-4}-3a^{4}z^{2}+a^{4}+4a^{3}z^{7}+4z^{7}a^{-3}-7a^{3}z^{5}-12z^{5}a^{-3}+14a^{3}z^{3}+8z^{3}a^{-3}-11a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}-12a^{2}z^{6}-14z^{6}a^{-2}+19a^{2}z^{4}+10z^{4}a^{-2}-13a^{2}z^{2}-4z^{2}a^{-2}+3a^{2}+a^{-2}+2az^{9}+2z^{9}a^{-1}+4az^{7}+4z^{7}a^{-1}-19az^{5}-24z^{5}a^{-1}+22az^{3}+19z^{3}a^{-1}-10az-za^{-1}+az^{-1}-a^{-1}z^{-1}+10z^{8}-28z^{6}+27z^{4}-14z^{2}+2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               3 1   -2 4             7 3     4 2           5 3       -2 0         7 7         0 -2       6 7           1 -4     3 5             -2 -6   2 6               4 -8   3                 -3 -10 2                   2

Integral Khovanov Homology (db, data source)

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

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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