L10a139

L10a138

L10a140

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a139 at Knotilus!

Link Presentations

[edit Notes on L10a139's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	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 \frac{(t(2)-1) \left(t(3) t(2)^2-2 t(2)^2-3 t(1) t(2)+2 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+t(1)-2 t(1) t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} \sqrt{t(3)}}} (db)
Jones polynomial	$q^{-6}-3q^{-5}+q^{4}+6q^{-4}-2q^{3}-8q^{-3}+5q^{2}+10q^{-2}-7q-10q^{-1}+11$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+2z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{4}-2z^{2}-2z^{-2}-3-2z^{2}a^{-2}+a^{-2}z^{-2}+a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}-5a^{4}z^{4}+z^{4}a^{-4}+6a^{4}z^{2}-2z^{2}a^{-4}-a^{4}+a^{-4}+a^{3}z^{9}+6a^{3}z^{7}-24a^{3}z^{5}+2z^{5}a^{-3}+21a^{3}z^{3}-2z^{3}a^{-3}-6a^{3}z+6a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}-a^{2}z^{4}-3z^{4}a^{-2}+8a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-3a^{-2}+az^{9}+6az^{7}+3z^{7}a^{-1}-17az^{5}+11az^{3}-5z^{3}a^{-1}+2az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-2z^{6}-3z^{4}+10z^{2}+2z^{-2}-7$ (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums $\chi$ (fixed 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 j} , alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   2 1 -1 5                 3     3 3               4 2     -2 1             7 3       4 -1           5 6         1 -3         5 5           0 -5       3 5             2 -7     3 5               -2 -9   1 4                 3 -11   2                   -2 -13 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ 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=-5} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ 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}} $r=-4$ 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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{3}$ 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=-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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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=-1} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}^{2}$ 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}^{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=4} ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L10a138 L10a140