L10a36

L10a35

L10a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a36 at Knotilus!

Link Presentations

[edit Notes on L10a36's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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