L11a279

L11a278

L11a280

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a279 at Knotilus!

Link Presentations

[edit Notes on L11a279's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X14,6,15,5 X4,20,5,19 X18,8,19,7 X6,16,7,15 X22,17,9,18 X16,21,17,22
Gauss code	{1, -2, 3, -7, 6, -9, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -11, 10, -8, 7, -5, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{4}-t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {12}{q^{9/2}}}-{\frac {14}{q^{7/2}}}-q^{5/2}+{\frac {13}{q^{5/2}}}+3q^{3/2}-{\frac {12}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-5a^{5}z-a^{5}z^{-1}+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+21a^{3}z^{3}+11a^{3}z+3a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-6az-2az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+6z^{4}a^{8}-2z^{2}a^{8}-6z^{7}a^{7}+9z^{5}a^{7}-6z^{3}a^{7}+za^{7}-6z^{8}a^{6}+12z^{6}a^{6}-12z^{4}a^{6}+6z^{2}a^{6}-a^{6}-5z^{9}a^{5}+13z^{7}a^{5}-17z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+a^{5}z^{-1}-2z^{10}a^{4}-z^{8}a^{4}+18z^{6}a^{4}-27z^{4}a^{4}+16z^{2}a^{4}-3a^{4}-9z^{9}a^{3}+36z^{7}a^{3}-52z^{5}a^{3}+39z^{3}a^{3}-15za^{3}+3a^{3}z^{-1}-2z^{10}a^{2}+2z^{8}a^{2}+13z^{6}a^{2}-20z^{4}a^{2}+10z^{2}a^{2}-3a^{2}-4z^{9}a+16z^{7}a-19z^{5}a+12z^{3}a-7za+2az^{-1}-3z^{8}+12z^{6}-12z^{4}+3z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-4z^{3}a^{-1}+za^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   4 1   3 0                 4 2     -2 -2               8 4       4 -4             7 6         -1 -6           7 6           1 -8         5 7             2 -10       4 7               -3 -12     2 5                 3 -14   1 4                   -3 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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