L11a373

L11a372

L11a374

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a373 at Knotilus!

Link Presentations

[edit Notes on L11a373's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-4u^{3}v^{3}+6u^{3}v^{2}-4u^{3}v+u^{3}-2u^{2}v^{4}+7u^{2}v^{3}-11u^{2}v^{2}+7u^{2}v-2u^{2}+uv^{4}-4uv^{3}+6uv^{2}-4uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {21}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-3a^{5}z^{3}-3a^{5}z+a^{3}z^{7}+4a^{3}z^{5}+7a^{3}z^{3}+5a^{3}z+a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+2az^{3}-2z^{3}a^{-1}-2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-5a^{5}z^{9}-11a^{3}z^{9}-6az^{9}-5a^{6}z^{8}-7a^{4}z^{8}-11a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+9a^{5}z^{7}+21a^{3}z^{7}+az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+24a^{4}z^{6}+28a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}-5a^{5}z^{5}-16a^{3}z^{5}+11az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-9a^{6}z^{4}-23a^{4}z^{4}-22a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-6a^{7}z^{3}+2a^{5}z^{3}+9a^{3}z^{3}-7az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}+3a^{6}z^{2}+8a^{4}z^{2}+6a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z-5a^{3}z-az+2za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               11 6       5 -2             11 9         -2 -4           10 10           0 -6         8 12             4 -8       5 9               -4 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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