L11a429

L11a428

L11a430

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a429 at Knotilus!

Link Presentations

[edit Notes on L11a429's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X22,20,11,19 X8,22,9,21 X20,8,21,7 X2,9,3,10 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {5, -1, 9, -8, 10, -6}, {3, -2, 4, -5, 6, -4, 11, -3, 7, -9, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-5t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-4t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+9t(1)t(3)t(2)-9t(3)t(2)+4t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-4t(1)t(3)+5t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-14q^{6}+20q^{5}-22q^{4}+23q^{3}-19q^{2}-q^{-2}+15q+4q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+4z^{2}a^{-2}-4z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-2}-6a^{-4}+2a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-2z^{4}a^{-10}+z^{2}a^{-10}+4z^{7}a^{-9}-9z^{5}a^{-9}+5z^{3}a^{-9}+7z^{8}a^{-8}-16z^{6}a^{-8}+11z^{4}a^{-8}-4z^{2}a^{-8}+a^{-8}+6z^{9}a^{-7}-7z^{7}a^{-7}-4z^{5}a^{-7}+3z^{3}a^{-7}+2z^{10}a^{-6}+12z^{8}a^{-6}-34z^{6}a^{-6}+23z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+12z^{9}a^{-5}-18z^{7}a^{-5}+6z^{5}a^{-5}-5z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+13z^{8}a^{-4}-28z^{6}a^{-4}+12z^{4}a^{-4}+8z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+6z^{9}a^{-3}-9z^{5}a^{-3}+z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+8z^{8}a^{-2}-7z^{6}a^{-2}-4z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}+7z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-6z^{4}+3z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 9 4     -5 11               11 5       6 9             11 9         -2 7           12 11           1 5         8 12             4 3       7 11               -4 1     3 10                 7 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ 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=-2} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ 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}^{10}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ 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} ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ 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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}} ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=8} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a428 L11a430