L11a466

L11a465

L11a467

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a466 at Knotilus!

Link Presentations

[edit Notes on L11a466's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)\left(2t(2)t(3)^{2}-t(3)^{2}-4t(2)t(3)+4t(3)+t(2)-2\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+7q^{-7}-12q^{-6}+16q^{-5}-17q^{-4}+19q^{-3}-q^{2}-15q^{-2}+3q+12q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-a^{6}+z^{6}a^{4}+z^{4}a^{4}-2z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{6}a^{2}+3z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}+5a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+4z^{8}a^{8}-6z^{6}a^{8}-3z^{4}a^{8}+6z^{2}a^{8}-a^{8}+3z^{9}a^{7}+2z^{7}a^{7}-17z^{5}a^{7}+15z^{3}a^{7}-6za^{7}+z^{10}a^{6}+8z^{8}a^{6}-15z^{6}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-3a^{6}+6z^{9}a^{5}-z^{7}a^{5}-15z^{5}a^{5}+12z^{3}a^{5}-za^{5}-2a^{5}z^{-1}+z^{10}a^{4}+8z^{8}a^{4}-9z^{6}a^{4}-9z^{4}a^{4}+16z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+3z^{9}a^{3}+4z^{7}a^{3}-10z^{5}a^{3}+3z^{3}a^{3}+5za^{3}-2a^{3}z^{-1}+4z^{8}a^{2}+2z^{6}a^{2}-15z^{4}a^{2}+18z^{2}a^{2}+a^{2}z^{-2}-8a^{2}+4z^{7}a-3z^{5}a-3z^{3}a+3za+3z^{6}-6z^{4}+5z^{2}-2+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     2   2 1                   4 1   -3 -1                 8 2     6 -3               9 6       -3 -5             10 6         4 -7           7 9           2 -9         9 10             -1 -11       5 9               4 -13     2 7                 -5 -15   1 5                   4 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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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