L10a15

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L10a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a15 at Knotilus!

Link Presentations

[edit Notes on L10a15's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-5t(1)t(2)^{3}+5t(2)^{3}+5t(1)t(2)^{2}-5t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-6a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+7a^{5}z+4a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}-az^{3}-2az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-3z^{2}a^{10}+a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-3z^{8}a^{8}+10z^{4}a^{8}-9z^{2}a^{8}+3a^{8}-z^{9}a^{7}-7z^{7}a^{7}+21z^{5}a^{7}-25z^{3}a^{7}+14za^{7}-3a^{7}z^{-1}-6z^{8}a^{6}+10z^{6}a^{6}-4z^{4}a^{6}-6z^{2}a^{6}+3a^{6}-z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-31z^{3}a^{5}+19za^{5}-4a^{5}z^{-1}-3z^{8}a^{4}+5z^{6}a^{4}-4z^{4}a^{4}-z^{2}a^{4}+2a^{4}-3z^{7}a^{3}+6z^{5}a^{3}-6z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}-2z^{6}a^{2}+4z^{4}a^{2}-z^{2}a^{2}-z^{5}a+3z^{3}a-2za$ (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 χ 2                     1 1 0                   1   -1 -2                 4 1   3 -4               5 2     -3 -6             6 3       3 -8           5 5         0 -10         6 6           0 -12       4 6             2 -14     2 5               -3 -16   1 4                 3 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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