L11n299

L11n298

L11n300

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n299 at Knotilus!

Link Presentations

[edit Notes on L11n299's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, -7, 9}, {11, -2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{2}t(3)^{4}+t(2)t(3)^{4}+2t(2)^{2}t(3)^{3}-t(2)t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}+t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-2t(1)t(3)+t(1)t(2)t(3)+t(1)-t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-3q^{-5}+4q^{-4}+q^{3}-2q^{-3}-2q^{2}+3q^{-2}+q$ (db)
Signature	0 (db)
HOMFLY-PT 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 -z^2 a^6+a^6 z^{-2} -a^6+2 z^4 a^4+6 z^2 a^4-2 a^4 z^{-2} +a^4-z^6 a^2-5 z^4 a^2-6 z^2 a^2+a^2 z^{-2} -a^2+z^2+1+z^2 a^{-2} } (db)
Kauffman polynomial	$2a^{5}z^{9}+2a^{3}z^{9}+3a^{6}z^{8}+6a^{4}z^{8}+4a^{2}z^{8}+z^{8}+a^{7}z^{7}-8a^{5}z^{7}-9a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}-15a^{6}z^{6}-35a^{4}z^{6}-27a^{2}z^{6}+z^{6}a^{-2}-6z^{6}-4a^{7}z^{5}+a^{5}z^{5}-15az^{5}-10z^{5}a^{-1}+19a^{6}z^{4}+55a^{4}z^{4}+48a^{2}z^{4}-4z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}+10a^{5}z^{3}+18a^{3}z^{3}+21az^{3}+10z^{3}a^{-1}-8a^{6}z^{2}-29a^{4}z^{2}-30a^{2}z^{2}+2z^{2}a^{-2}-7z^{2}-a^{7}z-3a^{5}z-7a^{3}z-7az-2za^{-1}+3a^{4}+4a^{2}+2-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     1   -1 3                 1 1 1   -1 1               2 2 1     1 -1             2 3 1       0 -3           3 3 3         3 -5         3 5 1           1 -7       2 2 2             2 -9     2 4 1               1 -11   1 1                   0 -13   2                     2 -15 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=-3} $i=-1$ 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 i=1} $r=-7$ 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}} $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ 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}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ 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} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=-2} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ 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}^{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=-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}\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}^{3}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ 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=0} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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=1} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ 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} ${\mathbb {Z} }$ 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=3} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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