L11a383

L11a382

L11a384

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a383 at Knotilus!

Link Presentations

[edit Notes on L11a383's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X10,11,1,12 X22,15,11,16 X16,8,17,7 X6,22,7,21 X20,6,21,5 X4,20,5,19 X18,10,19,9 X8,18,9,17
Gauss code	{1, -2, 3, -9, 8, -7, 6, -11, 10, -4}, {4, -1, 2, -3, 5, -6, 11, -10, 9, -8, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+4u^{3}v^{3}-3u^{3}v^{2}-3u^{2}v^{3}+7u^{2}v^{2}-3u^{2}v-3uv^{2}+4uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {2}{q^{7/2}}}-9q^{5/2}+{\frac {4}{q^{5/2}}}+10q^{3/2}-{\frac {7}{q^{3/2}}}-q^{13/2}+2q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-6z^{5}a^{-3}-12z^{3}a^{-3}-8za^{-3}-a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+8z^{7}a^{-1}-6az^{5}+24z^{5}a^{-1}-12az^{3}+32z^{3}a^{-1}-8az+17za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman 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^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -5 z^4 a^{-6} +z^2 a^{-6} +3 z^7 a^{-5} -8 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} +a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+22 z^4 a^{-4} +4 a^4 z^2-10 z^2 a^{-4} +3 z^9 a^{-3} +2 a^3 z^7-11 z^7 a^{-3} -7 a^3 z^5+18 z^5 a^{-3} +6 a^3 z^3-11 z^3 a^{-3} -a^3 z+7 z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +2 a^2 z^8+z^8 a^{-2} -5 a^2 z^6-14 z^6 a^{-2} +2 a^2 z^4+31 z^4 a^{-2} -a^2 z^2-16 z^2 a^{-2} + a^{-2} +2 a z^9+5 z^9 a^{-1} -7 a z^7-23 z^7 a^{-1} +15 a z^5+49 z^5 a^{-1} -22 a z^3-47 z^3 a^{-1} +9 a z+19 z a^{-1} - a^{-1} z^{-1} +z^{10}-z^8-3 z^6+10 z^4-10 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     1   -1 10                   3 1   2 8                 4 1     -3 6               5 3       2 4             6 5         -1 2           5 4           1 0         4 7             3 -2       3 4               -1 -4     1 4                 3 -6   1 3                   -2 -8   1                     1 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a382 L11a384