L10a142

L10a141

L10a143

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a142 at Knotilus!

Link Presentations

[edit Notes on L10a142's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w^{2}+uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{2}-2uvw+uv+uw-u+v^{3}w^{3}-v^{3}w^{2}-v^{2}w^{3}+2v^{2}w^{2}-v^{2}w-vw^{2}+2vw-v-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} -2 q^{-4} +4 q^{-5} -4 q^{-6} +7 q^{-7} -7 q^{-8} +7 q^{-9} -5 q^{-10} +4 q^{-11} -2 q^{-12} + q^{-13} } (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-3a^{10}z^{4}-12a^{10}z^{2}-2a^{10}z^{-2}-11a^{10}+2a^{8}z^{6}+10a^{8}z^{4}+15a^{8}z^{2}+a^{8}z^{-2}+8a^{8}+a^{6}z^{6}+4a^{6}z^{4}+3a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{16}z^{4}-2a^{16}z^{2}+a^{16}+2a^{15}z^{5}-3a^{15}z^{3}+2a^{14}z^{6}-a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{7}-2a^{13}z^{5}+a^{13}z^{3}+2a^{12}z^{8}-5a^{12}z^{6}+9a^{12}z^{4}-8a^{12}z^{2}-a^{12}z^{-2}+5a^{12}+a^{11}z^{9}-8a^{11}z^{5}+18a^{11}z^{3}-11a^{11}z+2a^{11}z^{-1}+5a^{10}z^{8}-22a^{10}z^{6}+38a^{10}z^{4}-31a^{10}z^{2}-2a^{10}z^{-2}+13a^{10}+a^{9}z^{9}-11a^{9}z^{5}+18a^{9}z^{3}-11a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-14a^{8}z^{6}+23a^{8}z^{4}-20a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+4a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+3a^{6}z^{2}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 2     2 -11               2 2     0 -13             5 2       3 -15           3 3         0 -17         4 4           0 -19       1 3             2 -21     3 4               -1 -23   1 3                 2 -25   1                   -1 -27 1                     1

Integral Khovanov Homology (db, data source)

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

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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