L11a520

L11a519

L11a521

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a520 at Knotilus!

Link Presentations

[edit Notes on L11a520's Link Presentations]

Planar diagram presentation	X8192 X18,8,19,7 X16,6,17,5 X10,16,11,15 X14,3,15,4 X4,11,5,12 X2,20,3,19 X20,9,21,10 X12,13,7,14 X22,18,13,17 X6,21,1,22
Gauss code	{1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(uvw^{2}-uvw+uv-uw^{2}+2uw-u-vw^{2}+2vw-v+w^{2}-w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+5q^{4}-13q^{3}+23q^{2}-31q+37-36q^{-1}+33q^{-2}-23q^{-3}+15q^{-4}-6q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+a^{4}z^{-2}-2a^{2}z^{6}-z^{6}a^{-2}-4a^{2}z^{4}-2z^{4}a^{-2}-2a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}+z^{8}+4z^{6}+7z^{4}+4z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$7a^{2}z^{10}+7z^{10}+17a^{3}z^{9}+36az^{9}+19z^{9}a^{-1}+15a^{4}z^{8}+22a^{2}z^{8}+21z^{8}a^{-2}+28z^{8}+6a^{5}z^{7}-27a^{3}z^{7}-67az^{7}-21z^{7}a^{-1}+13z^{7}a^{-3}+a^{6}z^{6}-27a^{4}z^{6}-72a^{2}z^{6}-32z^{6}a^{-2}+5z^{6}a^{-4}-81z^{6}-6a^{5}z^{5}+6a^{3}z^{5}+26az^{5}-13z^{5}a^{-3}+z^{5}a^{-5}+11a^{4}z^{4}+41a^{2}z^{4}+21z^{4}a^{-2}-2z^{4}a^{-4}+53z^{4}+a^{3}z^{3}-az^{3}+3z^{3}a^{-1}+5z^{3}a^{-3}+2a^{4}z^{2}-2a^{2}z^{2}-6z^{2}a^{-2}-10z^{2}-a^{3}z-az+a^{4}+a^{2}+1+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   9 1   -8 5                 14 4     10 3               17 9       -8 1             20 14         6 -1           18 19           1 -3         15 18             -3 -5       10 20               10 -7     5 13                 -8 -9   1 10                   9 -11   5                     -5 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{20}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=-1$ ${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{18}$ ${\mathbb {Z} }^{18}$ $r=0$ ${\mathbb {Z} }^{19}\oplus {\mathbb {Z} }_{2}^{18}$ ${\mathbb {Z} }^{20}$ $r=1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{17}$ ${\mathbb {Z} }^{17}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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