L11a441

L11a440

L11a442

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a441 at Knotilus!

Link Presentations

[edit Notes on L11a441's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X12,15,5,16 X22,17,13,18 X16,7,17,8 X8,22,9,21 X20,12,21,11 X18,10,19,9 X10,20,11,19 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 8, -9, 7, -3}, {11, -2, 3, -5, 4, -8, 9, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-t(3)t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-6t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-t(3)^{3}t(2)-4t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-2t(1)t(2)+6t(1)t(3)t(2)-4t(3)t(2)+t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-6q^{-5}+q^{4}+11q^{-4}-4q^{3}-14q^{-3}+9q^{2}+19q^{-2}-13q-18q^{-1}+17$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-2a^{6}+3a^{4}z^{4}+9a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-2a^{2}z^{6}-8a^{2}z^{4}+z^{4}a^{-2}-13a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-9a^{2}-z^{6}-z^{4}+3z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+8a^{3}z^{9}+5az^{9}+3a^{6}z^{8}+9a^{4}z^{8}+16a^{2}z^{8}+10z^{8}+a^{7}z^{7}-6a^{5}z^{7}-13a^{3}z^{7}+6az^{7}+12z^{7}a^{-1}-12a^{6}z^{6}-41a^{4}z^{6}-48a^{2}z^{6}+9z^{6}a^{-2}-10z^{6}-4a^{7}z^{5}-6a^{5}z^{5}-11a^{3}z^{5}-29az^{5}-16z^{5}a^{-1}+4z^{5}a^{-3}+16a^{6}z^{4}+55a^{4}z^{4}+47a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+5a^{7}z^{3}+17a^{5}z^{3}+24a^{3}z^{3}+20az^{3}+7z^{3}a^{-1}-z^{3}a^{-3}-9a^{6}z^{2}-34a^{4}z^{2}-29a^{2}z^{2}+4z^{2}a^{-2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   6 1   5 3                 7 3     -4 1               10 6       4 -1             11 10         -1 -3           8 7           1 -5         6 11             5 -7       5 8               -3 -9     2 7                 5 -11   1 4                   -3 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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