L11a381

L11a380

L11a382

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a381 at Knotilus!

Link Presentations

[edit Notes on L11a381's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,13,21,14 X10,21,1,22 X14,4,15,3 X18,6,19,5 X6,11,7,12 X22,7,11,8 X4,16,5,15 X8,18,9,17 X16,10,17,9 X2,20,3,19
Gauss code	{1, -11, 4, -8, 5, -6, 7, -9, 10, -3}, {6, -1, 2, -4, 8, -10, 9, -5, 11, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}-t(2)^{4}t(1)^{3}+5t(2)^{3}t(1)^{3}-8t(2)^{2}t(1)^{3}+5t(2)t(1)^{3}-t(1)^{3}+2t(2)^{4}t(1)^{2}-9t(2)^{3}t(1)^{2}+15t(2)^{2}t(1)^{2}-9t(2)t(1)^{2}+2t(1)^{2}-t(2)^{4}t(1)+5t(2)^{3}t(1)-8t(2)^{2}t(1)+5t(2)t(1)-t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {10}{q^{5/2}}}+26q^{3/2}-{\frac {17}{q^{3/2}}}-q^{13/2}+4q^{11/2}-27{\sqrt {q}}+{\frac {22}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-2z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3az^{3}+3z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+3za^{-1}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+12z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-2a^{4}z^{4}+12z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+9z^{9}a^{-3}+4a^{3}z^{7}-3z^{7}a^{-3}-8a^{3}z^{5}-15z^{5}a^{-3}+5a^{3}z^{3}+13z^{3}a^{-3}-a^{3}z-za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+8a^{2}z^{8}+18z^{8}a^{-2}-17a^{2}z^{6}-46z^{6}a^{-2}+12a^{2}z^{4}+33z^{4}a^{-2}-4a^{2}z^{2}-9z^{2}a^{-2}+a^{-2}+8az^{9}+17z^{9}a^{-1}-11az^{7}-27z^{7}a^{-1}+7z^{5}a^{-1}-2z^{3}a^{-1}+2az+5za^{-1}-a^{-1}z^{-1}+3z^{10}+14z^{8}-42z^{6}+31z^{4}-9z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   7 1   6 8                 10 3     -7 6               13 7       6 4             14 11         -3 2           13 12           1 0         10 15             5 -2       7 12               -5 -4     3 10                 7 -6   1 7                   -6 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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