L11a538

L11a537

L11a539

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a538 at Knotilus!

Link Presentations

[edit Notes on L11a538's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)(t(4)-1)\left(t(1)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}-3t(1)t(4)+2t(1)t(2)t(4)-3t(2)t(4)+2t(4)+t(1)-t(1)t(2)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	${\frac {19}{q^{9/2}}}-{\frac {20}{q^{7/2}}}+{\frac {13}{q^{5/2}}}-{\frac {8}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {14}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {22}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}+a^{9}z^{-3}-6z^{3}a^{7}-11za^{7}-9a^{7}z^{-1}-3a^{7}z^{-3}+3z^{5}a^{5}+8z^{3}a^{5}+12za^{5}+10a^{5}z^{-1}+3a^{5}z^{-3}+z^{5}a^{3}-z^{3}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+9a^{11}z^{3}-6a^{11}z+2a^{11}z^{-1}+4a^{10}z^{8}-4a^{10}z^{6}-8a^{10}z^{4}+14a^{10}z^{2}-6a^{10}+3a^{9}z^{9}+7a^{9}z^{7}-34a^{9}z^{5}+43a^{9}z^{3}-a^{9}z^{-3}-30a^{9}z+11a^{9}z^{-1}+a^{8}z^{10}+12a^{8}z^{8}-21a^{8}z^{6}-8a^{8}z^{4}+27a^{8}z^{2}+3a^{8}z^{-2}-18a^{8}+7a^{7}z^{9}+9a^{7}z^{7}-59a^{7}z^{5}+79a^{7}z^{3}-3a^{7}z^{-3}-52a^{7}z+18a^{7}z^{-1}+a^{6}z^{10}+15a^{6}z^{8}-26a^{6}z^{6}-a^{6}z^{4}+27a^{6}z^{2}+6a^{6}z^{-2}-21a^{6}+4a^{5}z^{9}+11a^{5}z^{7}-44a^{5}z^{5}+59a^{5}z^{3}-3a^{5}z^{-3}-40a^{5}z+14a^{5}z^{-1}+7a^{4}z^{8}-7a^{4}z^{6}-2a^{4}z^{4}+12a^{4}z^{2}+3a^{4}z^{-2}-9a^{4}+6a^{3}z^{7}-10a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}+3a^{2}z^{6}-4a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   6 1   5 -4                 9 4     -5 -6               11 4       7 -8             8 9         1 -10           14 11           3 -12         9 14             5 -14       5 8               -3 -16     2 9                 7 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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