L10a107

L10a106

L10a108

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a107 at Knotilus!

Link Presentations

[edit Notes on L10a107's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L10a106 L10a108