L11a378

L11a377

L11a379

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a378 at Knotilus!

Link Presentations

[edit Notes on L11a378's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}+2t(2)^{3}t(1)^{3}-4t(2)^{2}t(1)^{3}+3t(2)t(1)^{3}-t(1)^{3}+t(2)^{4}t(1)^{2}-4t(2)^{3}t(1)^{2}+5t(2)^{2}t(1)^{2}-4t(2)t(1)^{2}+t(1)^{2}-t(2)^{4}t(1)+3t(2)^{3}t(1)-4t(2)^{2}t(1)+2t(2)t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {1}{q^{9/2}}}-8q^{7/2}+{\frac {3}{q^{7/2}}}+9q^{5/2}-{\frac {6}{q^{5/2}}}-12q^{3/2}+{\frac {9}{q^{3/2}}}+q^{13/2}-2q^{11/2}+12{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{5}-2z^{5}a^{-1}-z^{5}a^{-3}+a^{3}z^{3}-az^{3}-5z^{3}a^{-1}-2z^{3}a^{-3}+z^{3}a^{-5}+a^{3}z+az-2za^{-1}-za^{-3}+2za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-6z^{6}a^{-6}+12z^{4}a^{-6}-8z^{2}a^{-6}+2z^{9}a^{-5}-11z^{7}a^{-5}+21z^{5}a^{-5}+a^{5}z^{3}-17z^{3}a^{-5}+6za^{-5}+z^{10}a^{-4}-16z^{6}a^{-4}+3a^{4}z^{4}+31z^{4}a^{-4}-15z^{2}a^{-4}+6z^{9}a^{-3}-24z^{7}a^{-3}+6a^{3}z^{5}+28z^{5}a^{-3}-4a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+8za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+7z^{8}a^{-2}+9a^{2}z^{6}-36z^{6}a^{-2}-13a^{2}z^{4}+40z^{4}a^{-2}+5a^{2}z^{2}-13z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+10az^{7}-3z^{7}a^{-1}-20az^{5}-19z^{5}a^{-1}+10az^{3}+18z^{3}a^{-1}-3az-2za^{-1}-a^{-1}z^{-1}+8z^{8}-17z^{6}+5z^{4}-z^{2}$ (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 6 7 χ 14                       1 -1 12                     1   1 10                   3 1   -2 8                 5 1     4 6               4 3       -1 4             8 5         3 2           5 5           0 0         6 7             -1 -2       4 6               2 -4     2 5                 -3 -6   1 4                   3 -8   2                     -2 -10 1                       1

Integral Khovanov Homology (db, data source)

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

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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