L11a486

L11a485

L11a487

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a486 at Knotilus!

Link Presentations

[edit Notes on L11a486's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X18,12,5,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {4, -1, 2, -5, 7, -8, 10, -4, 6, -9, 11, -2, 3, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)^{3}+2t(2)t(3)^{2}-2t(3)^{2}-2t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+10q^{7}-17q^{6}+23q^{5}-25q^{4}+27q^{3}-22q^{2}+17q-9+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}+a^{-8}-3z^{4}a^{-6}-5z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+2z^{6}a^{-4}+5z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}z^{-2}-a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+8z^{9}a^{-3}+15z^{9}a^{-5}+7z^{9}a^{-7}+11z^{8}a^{-2}+25z^{8}a^{-4}+22z^{8}a^{-6}+8z^{8}a^{-8}+8z^{7}a^{-1}+z^{7}a^{-3}-14z^{7}a^{-5}-3z^{7}a^{-7}+4z^{7}a^{-9}-16z^{6}a^{-2}-67z^{6}a^{-4}-65z^{6}a^{-6}-17z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-22z^{5}a^{-3}-23z^{5}a^{-5}-20z^{5}a^{-7}-8z^{5}a^{-9}+12z^{4}a^{-2}+63z^{4}a^{-4}+62z^{4}a^{-6}+14z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+5z^{3}a^{-1}+19z^{3}a^{-3}+27z^{3}a^{-5}+20z^{3}a^{-7}+6z^{3}a^{-9}-4z^{2}a^{-2}-27z^{2}a^{-4}-30z^{2}a^{-6}-8z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-za^{-1}-za^{-3}-5za^{-5}-7za^{-7}-2za^{-9}-a^{-2}+2a^{-4}+5a^{-6}+3a^{-8}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   7 1   6 13                 10 3     -7 11               13 7       6 9             14 12         -2 7           13 11           2 5         9 14             5 3       8 13               -5 1     3 11                 8 -1   1 6                   -5 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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