L11a529

L11a528

L11a530

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a529 at Knotilus!

Link Presentations

[edit Notes on L11a529's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(3)^{3}+t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}-3t(1)^{2}t(3)^{2}+3t(1)t(2)^{2}t(3)^{2}-3t(2)^{2}t(3)^{2}+5t(1)t(3)^{2}+3t(1)^{2}t(2)t(3)^{2}-7t(1)t(2)t(3)^{2}+5t(2)t(3)^{2}-2t(3)^{2}+3t(1)^{2}t(3)+2t(1)^{2}t(2)^{2}t(3)-5t(1)t(2)^{2}t(3)+3t(2)^{2}t(3)-3t(1)t(3)-5t(1)^{2}t(2)t(3)+7t(1)t(2)t(3)-3t(2)t(3)-t(1)^{2}-t(1)^{2}t(2)^{2}+2t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-2t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+5q^{-12}-10q^{-11}+17q^{-10}-22q^{-9}+26q^{-8}-25q^{-7}+23q^{-6}-16q^{-5}+10q^{-4}-4q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}\left(-z^{2}\right)+a^{12}z^{-2}+2a^{12}+a^{10}z^{4}-6a^{10}z^{2}-2a^{10}z^{-2}-9a^{10}+6a^{8}z^{4}+11a^{8}z^{2}+a^{8}z^{-2}+7a^{8}+4a^{6}z^{4}+4a^{6}z^{2}+a^{4}z^{4}$ (db)
Kauffman polynomial	$z^{7}a^{15}-2z^{5}a^{15}+z^{3}a^{15}+5z^{8}a^{14}-15z^{6}a^{14}+13z^{4}a^{14}-z^{2}a^{14}-2a^{14}+8z^{9}a^{13}-23z^{7}a^{13}+17z^{5}a^{13}-z^{3}a^{13}+4z^{10}a^{12}+6z^{8}a^{12}-46z^{6}a^{12}+46z^{4}a^{12}-11z^{2}a^{12}-a^{12}z^{-2}+3a^{12}+21z^{9}a^{11}-53z^{7}a^{11}+26z^{5}a^{11}+9z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}+4z^{10}a^{10}+20z^{8}a^{10}-76z^{6}a^{10}+70z^{4}a^{10}-33z^{2}a^{10}-2a^{10}z^{-2}+11a^{10}+13z^{9}a^{9}-13z^{7}a^{9}-17z^{5}a^{9}+19z^{3}a^{9}-9za^{9}+2a^{9}z^{-1}+19z^{8}a^{8}-35z^{6}a^{8}+28z^{4}a^{8}-19z^{2}a^{8}-a^{8}z^{-2}+7a^{8}+16z^{7}a^{7}-20z^{5}a^{7}+8z^{3}a^{7}+10z^{6}a^{6}-8z^{4}a^{6}+4z^{2}a^{6}+4z^{5}a^{5}+z^{4}a^{4}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 10 4     -6 -11               13 6       7 -13             12 10         -2 -15           14 13           1 -17         10 14             4 -19       7 12               -5 -21     4 11                 7 -23   1 6                   -5 -25   4                     4 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-8$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-7$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-6$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=-5$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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