L11a543

L11a542

L11a544

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a543 at Knotilus!

Link Presentations

[edit Notes on L11a543's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{2}wx-uv^{2}w-uv^{2}x+uv^{2}+2uvw^{2}x-uvw^{2}-5uvwx+4uvw+2uvx-3uv-uw^{2}x+uw^{2}+2uwx-3uw-ux+2u+2v^{2}w^{2}x+v^{2}\left(-w^{2}\right)-3v^{2}wx+2v^{2}w+v^{2}x-v^{2}-3vw^{2}x+2vw^{2}+4vwx-5vw-vx+2v+w^{2}x-w^{2}-wx+2w}{{\sqrt {u}}vw{\sqrt {x}}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {14}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {22}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}+a^{13}(-z)+3a^{11}z^{3}-3a^{11}z^{-3}+2a^{11}z-5a^{11}z^{-1}-2a^{9}z^{5}-a^{9}z^{3}+3a^{9}z^{-3}+8a^{9}z+10a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-a^{7}z^{-3}-8a^{7}z-5a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+8a^{15}z^{3}-3a^{15}z+6a^{14}z^{8}-12a^{14}z^{6}+4a^{14}z^{4}+a^{14}z^{2}+4a^{13}z^{9}+4a^{13}z^{7}-31a^{13}z^{5}+33a^{13}z^{3}-a^{13}z^{-3}-15a^{13}z+5a^{13}z^{-1}+a^{12}z^{10}+15a^{12}z^{8}-38a^{12}z^{6}+23a^{12}z^{4}+4a^{12}z^{2}+3a^{12}z^{-2}-10a^{12}+8a^{11}z^{9}+a^{11}z^{7}-38a^{11}z^{5}+49a^{11}z^{3}-3a^{11}z^{-3}-29a^{11}z+12a^{11}z^{-1}+a^{10}z^{10}+15a^{10}z^{8}-31a^{10}z^{6}+12a^{10}z^{4}+18a^{10}z^{2}+6a^{10}z^{-2}-19a^{10}+4a^{9}z^{9}+7a^{9}z^{7}-29a^{9}z^{5}+39a^{9}z^{3}-3a^{9}z^{-3}-29a^{9}z+12a^{9}z^{-1}+6a^{8}z^{8}-3a^{8}z^{6}-9a^{8}z^{4}+15a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+13a^{7}z^{3}-a^{7}z^{-3}-11a^{7}z+5a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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 -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               12 5       7 -14             9 9         0 -16           13 10           3 -18         8 13             5 -20       6 9               -3 -22     3 9                 6 -24   1 5                   -4 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a542 L11a544