L11a540

L11a539

L11a541

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a540 at Knotilus!

Link Presentations

[edit Notes on L11a540's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(x-1)\left(uvwx^{2}-2uvwx+2uvx-uv-uwx^{2}+3uwx-2ux+2u-2vwx^{2}+2vwx-3vx+v+wx^{2}-2wx+2x-1\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z+2a^{9}z^{-1}-3a^{7}z^{3}-3a^{7}z^{-3}-9a^{7}z-9a^{7}z^{-1}+3a^{5}z^{5}+11a^{5}z^{3}+3a^{5}z^{-3}+17a^{5}z+12a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-8a^{3}z^{3}-a^{3}z^{-3}-10a^{3}z-5a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-3za^{11}+a^{11}z^{-1}-2z^{6}a^{10}+3z^{4}a^{10}-a^{10}-3z^{7}a^{9}+3z^{5}a^{9}-z^{3}a^{9}+3za^{9}-3a^{9}z^{-1}+a^{9}z^{-3}-3z^{8}a^{8}-2z^{6}a^{8}+12z^{4}a^{8}-16z^{2}a^{8}-3a^{8}z^{-2}+11a^{8}-3z^{9}a^{7}-z^{7}a^{7}+9z^{5}a^{7}-20z^{3}a^{7}+20za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-z^{10}a^{6}-9z^{8}a^{6}+19z^{6}a^{6}-3z^{4}a^{6}-28z^{2}a^{6}-6a^{6}z^{-2}+24a^{6}-7z^{9}a^{5}+4z^{7}a^{5}+23z^{5}a^{5}-39z^{3}a^{5}+29za^{5}-14a^{5}z^{-1}+3a^{5}z^{-3}-z^{10}a^{4}-12z^{8}a^{4}+32z^{6}a^{4}-16z^{4}a^{4}-12z^{2}a^{4}-3a^{4}z^{-2}+13a^{4}-4z^{9}a^{3}-2z^{7}a^{3}+28z^{5}a^{3}-31z^{3}a^{3}+18za^{3}-6a^{3}z^{-1}+a^{3}z^{-3}-6z^{8}a^{2}+12z^{6}a^{2}-2z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+10z^{5}a-8z^{3}a+3za-z^{6}+2z^{4}-z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               9 8       -1 -6             10 6         4 -8           7 9           2 -10         9 10             -1 -12       4 11               7 -14     2 5                 -3 -16     4                   4 -18 1 2                     -1 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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