L10a88

L10a87

L10a89

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a88 at Knotilus!

Link Presentations

[edit Notes on L10a88's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,16,19,15 X20,13,9,14 X6,17,7,18 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 2, -8, 3, -7, 4, -10}, {9, -1, 10, -2, 6, -3, 5, -4, 7, -5, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{2}-2u^{3}v+u^{3}+u^{2}v^{3}-5u^{2}v^{2}+7u^{2}v-2u^{2}-2uv^{3}+7uv^{2}-5uv+u+v^{3}-2v^{2}+v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {13}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-z^{3}a^{3}-2za^{3}-a^{3}z^{-1}-z^{5}a-z^{3}a-za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-7a^{5}z^{7}-11a^{3}z^{7}-7az^{7}-a^{8}z^{6}+6a^{6}z^{6}+11a^{4}z^{6}-3a^{2}z^{6}-7z^{6}+10a^{7}z^{5}+26a^{5}z^{5}+27a^{3}z^{5}+7az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}+a^{4}z^{4}+13a^{2}z^{4}-z^{4}a^{-2}+8z^{4}-10a^{7}z^{3}-26a^{5}z^{3}-20a^{3}z^{3}-az^{3}+3z^{3}a^{-1}-2a^{8}z^{2}-3a^{6}z^{2}-5a^{4}z^{2}-6a^{2}z^{2}-2z^{2}+3a^{7}z+10a^{5}z+7a^{3}z+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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 5       -2 -4           6 5         1 -6         5 7           2 -8       4 6             -2 -10     2 6               4 -12   1 3                 -2 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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