L11a392

L11a391

L11a393

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a392 at Knotilus!

Link Presentations

[edit Notes on L11a392's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(3)^{3}+2t(2)t(3)^{3}-2t(3)^{3}-7t(1)t(3)^{2}+4t(1)t(2)t(3)^{2}-7t(2)t(3)^{2}+5t(3)^{2}+7t(1)t(3)-5t(1)t(2)t(3)+7t(2)t(3)-4t(3)-2t(1)+2t(1)t(2)-2t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-4q^{-3}+10q^{-4}-13q^{-5}+18q^{-6}-18q^{-7}+19q^{-8}-14q^{-9}+10q^{-10}-6q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+3a^{12}z^{-2}+4a^{12}-6z^{2}a^{10}-2a^{10}z^{-2}-7a^{10}+3z^{4}a^{8}+2z^{2}a^{8}-a^{8}z^{-2}-a^{8}+4z^{4}a^{6}+7z^{2}a^{6}+a^{6}z^{-2}+4a^{6}+z^{4}a^{4}$ (db)
Kauffman polynomial	$z^{7}a^{15}-5z^{5}a^{15}+9z^{3}a^{15}-7za^{15}+2a^{15}z^{-1}+2z^{8}a^{14}-7z^{6}a^{14}+7z^{4}a^{14}-2z^{2}a^{14}-a^{14}z^{-2}+a^{14}+2z^{9}a^{13}-z^{7}a^{13}-17z^{5}a^{13}+35z^{3}a^{13}-27za^{13}+8a^{13}z^{-1}+z^{10}a^{12}+6z^{8}a^{12}-25z^{6}a^{12}+24z^{4}a^{12}-7z^{2}a^{12}-3a^{12}z^{-2}+5a^{12}+7z^{9}a^{11}-7z^{7}a^{11}-27z^{5}a^{11}+50z^{3}a^{11}-34za^{11}+10a^{11}z^{-1}+z^{10}a^{10}+15z^{8}a^{10}-40z^{6}a^{10}+27z^{4}a^{10}-8z^{2}a^{10}-2a^{10}z^{-2}+4a^{10}+5z^{9}a^{9}+8z^{7}a^{9}-35z^{5}a^{9}+26z^{3}a^{9}-10za^{9}+2a^{9}z^{-1}+11z^{8}a^{8}-12z^{6}a^{8}-2z^{4}a^{8}+4z^{2}a^{8}+a^{8}z^{-2}-3a^{8}+13z^{7}a^{7}-16z^{5}a^{7}+2z^{3}a^{7}+4za^{7}-2a^{7}z^{-1}+10z^{6}a^{6}-11z^{4}a^{6}+7z^{2}a^{6}+a^{6}z^{-2}-4a^{6}+4z^{5}a^{5}+z^{4}a^{4}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 7 4     -3 -11               11 6       5 -13             10 10         0 -15           9 8           1 -17         5 10             5 -19       5 9               -4 -21     1 5                 4 -23   1 5                   -4 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

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

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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