L11a74

L11a73

L11a75

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a74 at Knotilus!

Link Presentations

[edit Notes on L11a74's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X22,13,5,14 X20,18,21,17 X18,9,19,10 X8,19,9,20 X14,21,15,22 X10,16,11,15 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 6, -9, 11, -2, 4, -8, 9, -3, 5, -6, 7, -5, 8, -4}

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} , 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 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)^5+2 t(1) t(2)^4-6 t(2)^4-8 t(1) t(2)^3+12 t(2)^3+12 t(1) t(2)^2-8 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}} (db)
Jones polynomial	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 q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}} (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4za^{7}-3a^{7}z^{-1}+5z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-4z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}-z^{5}a+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+13a^{8}z^{4}-6a^{8}z^{2}+a^{8}+3a^{7}z^{9}-4a^{7}z^{7}-14a^{7}z^{5}+28a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-42a^{6}z^{6}+46a^{6}z^{4}-18a^{6}z^{2}+3a^{6}+8a^{5}z^{9}-9a^{5}z^{7}-30a^{5}z^{5}+50a^{5}z^{3}-25a^{5}z+4a^{5}z^{-1}+a^{4}z^{10}+17a^{4}z^{8}-52a^{4}z^{6}+45a^{4}z^{4}-15a^{4}z^{2}+2a^{4}+5a^{3}z^{9}+7a^{3}z^{7}-39a^{3}z^{5}+38a^{3}z^{3}-15a^{3}z+2a^{3}z^{-1}+10a^{2}z^{8}-13a^{2}z^{6}+4a^{2}z^{4}+z^{4}a^{-2}-2a^{2}z^{2}+a^{2}+11az^{7}-15az^{5}+4z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}-2az+8z^{6}-7z^{4}+z^{2}$ (db)

Khovanov Homology

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

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 8 3     5 -2               9 6       -3 -4             10 7         3 -6           8 9           1 -8         7 10             -3 -10       5 9               4 -12     2 6                 -4 -14   1 5                   4 -16   2                     -2 -18 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=-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 i=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 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 r=-7} 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}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-6$ 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^{2}} ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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=-4} ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ 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}^{7}} $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ 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}^{8}} $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ 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}} 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} 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}^{7}\oplus{\mathbb Z}_2^{9}} ${\mathbb {Z} }^{9}$ $r=0$ 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}\oplus{\mathbb Z}_2^{7}} 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}^{8}} $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=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}_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}}

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.

L11a73 L11a75