L11a377

L11a376

L11a378

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a377 at Knotilus!

Link Presentations

[edit Notes on L11a377's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-2t(2)^{4}t(1)^{3}+5t(2)^{3}t(1)^{3}-4t(2)^{2}t(1)^{3}+t(2)t(1)^{3}+t(2)^{4}t(1)^{2}-5t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-4t(2)^{2}t(1)+5t(2)t(1)-2t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-10q^{11/2}+13q^{9/2}-15q^{7/2}+14q^{5/2}-13q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+5z^{5}a^{-1}-18z^{5}a^{-3}+5z^{5}a^{-5}+8z^{3}a^{-1}-20z^{3}a^{-3}+8z^{3}a^{-5}+5za^{-1}-8za^{-3}+4za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-3z^{3}a^{-9}+za^{-9}+5z^{6}a^{-8}-5z^{4}a^{-8}+2z^{2}a^{-8}+6z^{7}a^{-7}-6z^{5}a^{-7}+z^{3}a^{-7}+za^{-7}+6z^{8}a^{-6}-8z^{6}a^{-6}+3z^{4}a^{-6}+5z^{9}a^{-5}-10z^{7}a^{-5}+11z^{5}a^{-5}-11z^{3}a^{-5}+4za^{-5}+2z^{10}a^{-4}+2z^{8}a^{-4}-16z^{6}a^{-4}+16z^{4}a^{-4}-6z^{2}a^{-4}+9z^{9}a^{-3}-32z^{7}a^{-3}+40z^{5}a^{-3}-27z^{3}a^{-3}+9za^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}-z^{8}a^{-2}-15z^{6}a^{-2}+20z^{4}a^{-2}-7z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+az^{7}-15z^{7}a^{-1}-4az^{5}+16z^{5}a^{-1}+4az^{3}-8z^{3}a^{-1}-az+4za^{-1}-a^{-1}z^{-1}+3z^{8}-12z^{6}+13z^{4}-4z^{2}$ (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 18                       1 -1 16                     2   2 14                   4 1   -3 12                 6 2     4 10               7 4       -3 8             8 6         2 6           7 8           1 4         6 7             -1 2       4 8               4 0     2 5                 -3 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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