L10n113

L10n112

L11a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n113 at Knotilus!

Link Presentations

[edit Notes on L10n113's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X20,15,17,16 X16,19,13,20 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, 8, -9}, {-10, 3, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, $w$, ...)	${\frac {-uvw+uvxy-uvy+uv+uwy-uxy+vw-vx-wxy+wx-w+xy}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}{\sqrt {y}}}}$ (db)
Jones polynomial	$q^{6}-q^{5}+5q^{4}-q^{3}+q^{-3}+5q^{2}+q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{-6}z^{-4}+2a^{-6}z^{-2}+a^{-6}-4a^{-4}z^{-4}-3z^{2}a^{-4}-9a^{-4}z^{-2}-8a^{-4}+2z^{4}a^{-2}+a^{2}z^{-4}+6a^{-2}z^{-4}+a^{2}z^{2}+9z^{2}a^{-2}+3a^{2}z^{-2}+15a^{-2}z^{-2}+3a^{2}+16a^{-2}-z^{4}-4z^{-4}-7z^{2}-11z^{-2}-12$ (db)
Kauffman polynomial	$z^{6}a^{-6}-5z^{4}a^{-6}-a^{-6}z^{-4}+10z^{2}a^{-6}+5a^{-6}z^{-2}-10a^{-6}+z^{7}a^{-5}-z^{5}a^{-5}-10z^{3}a^{-5}+4a^{-5}z^{-3}+20za^{-5}-15a^{-5}z^{-1}+6z^{6}a^{-4}-25z^{4}a^{-4}-4a^{-4}z^{-4}+30z^{2}a^{-4}+14a^{-4}z^{-2}-25a^{-4}+z^{7}a^{-3}+3z^{5}a^{-3}-30z^{3}a^{-3}+12a^{-3}z^{-3}+55za^{-3}-41a^{-3}z^{-1}+a^{2}z^{6}+5z^{6}a^{-2}-6a^{2}z^{4}-25z^{4}a^{-2}-a^{2}z^{-4}-6a^{-2}z^{-4}+10a^{2}z^{2}+40z^{2}a^{-2}+5a^{2}z^{-2}+18a^{-2}z^{-2}-10a^{2}-31a^{-2}+az^{5}+5z^{5}a^{-1}-10az^{3}-30z^{3}a^{-1}+4az^{-3}+12a^{-1}z^{-3}+20az+55za^{-1}-15az^{-1}-41a^{-1}z^{-1}+z^{6}-11z^{4}-4z^{-4}+30z^{2}+14z^{-2}-25$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                     1 1 11                       0 9                 5 1   4 7               1 5     4 5             4         4 3         4   1         5 1       1 10 4           5 -1         6             6 -3     1                 1 -5 1                     1 -7 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $i=3$ 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=-4} ${\mathbb {Z} }$ 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}} $r=-3$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ 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}^{6}} ${\mathbb {Z} }^{10}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{4}$ $r=2$ 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}\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} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=4} ${\mathbb {Z} }^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L10n112 L11a1