L11a380

L11a379

L11a381

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a380 at Knotilus!

Link Presentations

[edit Notes on L11a380's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{2}-u^{4}v-u^{3}v^{4}+4u^{3}v^{3}-5u^{3}v^{2}+3u^{3}v-u^{3}+2u^{2}v^{4}-6u^{2}v^{3}+7u^{2}v^{2}-6u^{2}v+2u^{2}-uv^{4}+3uv^{3}-5uv^{2}+4uv-u-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-7{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {15}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+5z^{3}a^{5}+2za^{5}-z^{7}a^{3}-3z^{5}a^{3}-2z^{3}a^{3}+a^{3}z^{-1}+2z^{5}a+5z^{3}a+za-az^{-1}-z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-2a^{9}z^{3}+6a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}+9a^{7}z^{7}-15a^{7}z^{5}+14a^{7}z^{3}-5a^{7}z+9a^{6}z^{8}-15a^{6}z^{6}+9a^{6}z^{4}-a^{6}z^{2}+6a^{5}z^{9}-5a^{5}z^{7}-11a^{5}z^{5}+14a^{5}z^{3}-5a^{5}z+2a^{4}z^{10}+8a^{4}z^{8}-31a^{4}z^{6}+25a^{4}z^{4}-7a^{4}z^{2}+10a^{3}z^{9}-27a^{3}z^{7}+19a^{3}z^{5}-8a^{3}z^{3}+4a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}+2a^{2}z^{8}-21a^{2}z^{6}+21a^{2}z^{4}-6a^{2}z^{2}+a^{2}+4az^{9}-12az^{7}+z^{7}a^{-1}+8az^{5}-4z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}+2az-az^{-1}-2za^{-1}+3z^{8}-11z^{6}+12z^{4}-4z^{2}$ (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 6                       1 1 4                     2   -2 2                   5 1   4 0                 6 2     -4 -2               9 5       4 -4             9 7         -2 -6           9 8           1 -8         7 10             3 -10       5 8               -3 -12     2 7                 5 -14   1 5                   -4 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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