L11a424

L11a423

L11a425

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a424 at Knotilus!

Link Presentations

[edit Notes on L11a424's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+3t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-3t(2)^{2}-3t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+3t(1)t(3)^{2}-3t(3)^{2}+2t(1)-5t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+7q^{-7}-11q^{-6}+18q^{-5}-20q^{-4}+22q^{-3}-q^{2}-20q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+a^{6}z^{-2}+5a^{6}-4a^{4}z^{4}-5a^{4}z^{2}-2a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+2a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z-2a^{9}z^{-1}+3a^{8}z^{8}-3a^{8}z^{6}-5a^{8}z^{4}+9a^{8}z^{2}+2a^{8}z^{-2}-6a^{8}+3a^{7}z^{9}-7a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z^{-1}+a^{6}z^{10}+11a^{6}z^{8}-26a^{6}z^{6}+19a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+8a^{5}z^{9}-a^{5}z^{7}-24a^{5}z^{5}+24a^{5}z^{3}-6a^{5}z+a^{4}z^{10}+18a^{4}z^{8}-37a^{4}z^{6}+21a^{4}z^{4}-2a^{4}z^{2}-a^{4}+5a^{3}z^{9}+11a^{3}z^{7}-37a^{3}z^{5}+24a^{3}z^{3}-2a^{3}z+10a^{2}z^{8}-10a^{2}z^{6}-4a^{2}z^{4}+3a^{2}z^{2}-a^{2}+10az^{7}-15az^{5}+z^{5}a^{-1}+5az^{3}+5z^{6}-5z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 9 4     5 -3               12 7       -5 -5             10 8         2 -7           10 12           2 -9         8 10             -2 -11       4 11               7 -13     3 7                 -4 -15   1 6                   5 -17   1                     -1 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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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