L11a214

L11a213

L11a215

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a214 at Knotilus!

Link Presentations

[edit Notes on L11a214's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,20,7,19 X16,5,17,6 X18,10,19,9 X4,15,5,16 X10,18,11,17 X14,22,15,21 X20,14,21,13 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -6, 4, -11}, {10, -1, 5, -7, 11, -2, 9, -8, 6, -4, 7, -5, 3, -9, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-4t(2)-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+3az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-4a^{3}z-az-za^{-1}+za^{-3}+a^{5}z^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-12az^{9}-6z^{9}a^{-1}-7a^{4}z^{8}-12a^{2}z^{8}-7z^{8}a^{-2}-12z^{8}-6a^{5}z^{7}+5a^{3}z^{7}+22az^{7}+7z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+8a^{4}z^{6}+34a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+40z^{6}-a^{7}z^{5}+10a^{5}z^{5}-a^{3}z^{5}-14az^{5}+7z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-2a^{4}z^{4}-35a^{2}z^{4}-10z^{4}a^{-2}+2z^{4}a^{-4}-41z^{4}+2a^{7}z^{3}-9a^{5}z^{3}+9az^{3}-7z^{3}a^{-1}-5z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}+16a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}+22z^{2}-a^{7}z+5a^{5}z-7az+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 8 3     5 2               11 6       -5 0             11 8         3 -2           11 12           1 -4         8 10             -2 -6       5 11               6 -8     3 8                 -5 -10     5                   5 -12 1 3                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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