L11a498

L11a497

L11a499

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a498 at Knotilus!

Link Presentations

[edit Notes on L11a498's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(2)t(3)^{4}+2t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+2t(3)^{3}+4t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-4t(3)^{2}-3t(1)t(3)+2t(1)t(2)t(3)+2t(3)+t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+10q^{2}-14q+19-20q^{-1}+21q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+3a^{4}z^{4}+9a^{4}z^{2}+4a^{4}z^{-2}+10a^{4}-2a^{2}z^{6}-8a^{2}z^{4}+z^{4}a^{-2}-15a^{2}z^{2}-5a^{2}z^{-2}+z^{2}a^{-2}-13a^{2}+a^{-2}-z^{6}-z^{4}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+9a^{3}z^{9}+6az^{9}+3a^{6}z^{8}+12a^{4}z^{8}+22a^{2}z^{8}+13z^{8}+a^{7}z^{7}-3a^{5}z^{7}-8a^{3}z^{7}+10az^{7}+14z^{7}a^{-1}-11a^{6}z^{6}-49a^{4}z^{6}-66a^{2}z^{6}+10z^{6}a^{-2}-18z^{6}-4a^{7}z^{5}-17a^{5}z^{5}-39a^{3}z^{5}-48az^{5}-18z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+60a^{4}z^{4}+66a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}+9z^{4}+6a^{7}z^{3}+34a^{5}z^{3}+66a^{3}z^{3}+44az^{3}+6z^{3}a^{-1}-8a^{6}z^{2}-35a^{4}z^{2}-43a^{2}z^{2}+6z^{2}a^{-2}-10z^{2}-4a^{7}z-21a^{5}z-39a^{3}z-22az+3a^{6}+16a^{4}+21a^{2}-2a^{-2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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                     4 1 -3 5                   6     6 3                 8 4     -4 1               11 6       5 -1             11 10         -1 -3           10 9           1 -5         6 11             5 -7       6 10               -4 -9     2 7                 5 -11   1 5                   -4 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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