L10a31

L10a30

L10a32

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a31 at Knotilus!

Link Presentations

[edit Notes on L10a31's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+7q^{3/2}-9{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+2z^{3}a+2za+az^{-1}+z^{5}a^{-1}+2z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-a^{3}z^{7}-6az^{7}-6z^{7}a^{-1}+9a^{4}z^{6}+22a^{2}z^{6}-7z^{6}a^{-2}+6z^{6}+5a^{5}z^{5}+18a^{3}z^{5}+27az^{5}+8z^{5}a^{-1}-6z^{5}a^{-3}-12a^{4}z^{4}-24a^{2}z^{4}+9z^{4}a^{-2}-3z^{4}a^{-4}-8a^{5}z^{3}-30a^{3}z^{3}-27az^{3}+2z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+5a^{4}z^{2}+11a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}+5a^{5}z+15a^{3}z+10az-3za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                     1 1 8                   2   -2 6                 4 1   3 4               3 2     -1 2             6 4       2 0           5 5         0 -2         3 4           -1 -4       3 5             2 -6     1 3               -2 -8   1 3                 2 -10   1                   -1 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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