L11a421

L11a420

L11a422

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a421 at Knotilus!

Link Presentations

[edit Notes on L11a421's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3uv^{2}w^{2}-3uv^{2}w+uv^{2}+3uvw^{3}-7uvw^{2}+5uvw-uv+uw^{4}-4uw^{3}+5uw^{2}-2uw+2v^{2}w^{3}-5v^{2}w^{2}+4v^{2}w-v^{2}+vw^{4}-5vw^{3}+7vw^{2}-3vw-w^{4}+3w^{3}-3w^{2}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-2}-4q^{-3}+10q^{-4}-15q^{-5}+21q^{-6}-22q^{-7}+23q^{-8}-18q^{-9}+14q^{-10}-8q^{-11}+3q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-9z^{2}a^{10}-5a^{10}z^{-2}-13a^{10}+5z^{4}a^{8}+9z^{2}a^{8}+2a^{8}z^{-2}+7a^{8}+4z^{4}a^{6}+5z^{2}a^{6}+a^{6}+z^{4}a^{4}$ (db)
Kauffman polynomial	$z^{7}a^{15}-4z^{5}a^{15}+6z^{3}a^{15}-4za^{15}+a^{15}z^{-1}+3z^{8}a^{14}-10z^{6}a^{14}+12z^{4}a^{14}-7z^{2}a^{14}-a^{14}z^{-2}+3a^{14}+4z^{9}a^{13}-7z^{7}a^{13}-9z^{5}a^{13}+26z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+2z^{10}a^{12}+9z^{8}a^{12}-44z^{6}a^{12}+52z^{4}a^{12}-29z^{2}a^{12}-4a^{12}z^{-2}+15a^{12}+13z^{9}a^{11}-23z^{7}a^{11}-17z^{5}a^{11}+48z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+2z^{10}a^{10}+22z^{8}a^{10}-71z^{6}a^{10}+69z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+9z^{9}a^{9}-35z^{5}a^{9}+35z^{3}a^{9}-18za^{9}+5a^{9}z^{-1}+16z^{8}a^{8}-27z^{6}a^{8}+19z^{4}a^{8}-15z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+15z^{7}a^{7}-19z^{5}a^{7}+7z^{3}a^{7}+10z^{6}a^{6}-9z^{4}a^{6}+5z^{2}a^{6}-a^{6}+4z^{5}a^{5}+z^{4}a^{4}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 9 4     -5 -11               12 6       6 -13             11 10         -1 -15           12 11           1 -17         8 13             5 -19       6 10               -4 -21     2 8                 6 -23   1 6                   -5 -25   2                     2 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-7$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-6$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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