L11a479

L11a478

L11a480

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a479 at Knotilus!

Link Presentations

[edit Notes on L11a479's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}(t(2)+t(3))}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-14q+19-20q^{-1}+21q^{-2}-16q^{-3}+13q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{6}a^{2}-z^{4}a^{2}-2a^{2}z^{-2}-a^{2}-z^{6}-2z^{4}-3z^{2}+z^{-2}-1+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+3a^{6}z^{8}-11a^{6}z^{6}+14a^{6}z^{4}-9a^{6}z^{2}+3a^{6}+4a^{5}z^{9}-10a^{5}z^{7}+a^{5}z^{5}+11a^{5}z^{3}-7a^{5}z+2a^{4}z^{10}+6a^{4}z^{8}-37a^{4}z^{6}+48a^{4}z^{4}+z^{4}a^{-4}-27a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+12a^{3}z^{9}-28a^{3}z^{7}+7a^{3}z^{5}+4z^{5}a^{-3}+15a^{3}z^{3}-z^{3}a^{-3}-10a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+16a^{2}z^{8}-54a^{2}z^{6}+9z^{6}a^{-2}+48a^{2}z^{4}-8z^{4}a^{-2}-20a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}-a^{-2}+8az^{9}-4az^{7}+13z^{7}a^{-1}-19az^{5}-17z^{5}a^{-1}+17az^{3}+7z^{3}a^{-1}-6az-za^{-1}+2az^{-1}+13z^{8}-19z^{6}+5z^{4}+z^{2}-z^{-2}+1$ (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 9                       1 1 7                     3   -3 5                   6 1   5 3                 8 3     -5 1               11 6       5 -1             11 10         -1 -3           10 9           1 -5         8 13             5 -7       5 8               -3 -9     2 8                 6 -11   1 5                   -4 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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