L11a525

L11a524

L11a526

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a525 at Knotilus!

Link Presentations

[edit Notes on L11a525's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,15,7,16 X22,19,13,20 X16,9,17,10 X10,22,11,21 X20,12,21,11 X18,5,19,6 X2738 X4,13,5,14 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 7, -3}, {10, -2, 3, -5, 11, -8, 4, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-u^{2}v^{2}w+u^{2}v^{2}-2u^{2}vw^{2}+5u^{2}vw-2u^{2}v+2u^{2}w^{2}-3u^{2}w+u^{2}-2uv^{2}w^{2}+3uv^{2}w-uv^{2}-uvw^{3}+6uvw^{2}-6uvw+uv+uw^{3}-3uw^{2}+2uw-v^{2}w^{3}+3v^{2}w^{2}-2v^{2}w+2vw^{3}-5vw^{2}+2vw-w^{3}+w^{2}}{uvw^{3/2}}}$ (db)
Jones polynomial	$q-3+7q^{-1}-12q^{-2}+17q^{-3}-18q^{-4}+20q^{-5}-16q^{-6}+13q^{-7}-8q^{-8}+4q^{-9}-q^{-10}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+4z^{2}a^{8}+a^{8}z^{-2}+3a^{8}-3z^{4}a^{6}-4z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{4}a^{2}+2z^{2}a^{2}+a^{2}+z^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-3a^{11}z^{5}+3a^{11}z^{3}-a^{11}z+4a^{10}z^{8}-14a^{10}z^{6}+15a^{10}z^{4}-6a^{10}z^{2}+2a^{10}+5a^{9}z^{9}-13a^{9}z^{7}+2a^{9}z^{5}+10a^{9}z^{3}-3a^{9}z+2a^{8}z^{10}+8a^{8}z^{8}-46a^{8}z^{6}+59a^{8}z^{4}-34a^{8}z^{2}-a^{8}z^{-2}+10a^{8}+12a^{7}z^{9}-30a^{7}z^{7}+12a^{7}z^{5}+8a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+14a^{6}z^{8}-56a^{6}z^{6}+66a^{6}z^{4}-41a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+7a^{5}z^{9}-7a^{5}z^{7}-9a^{5}z^{5}+12a^{5}z^{3}-6a^{5}z+2a^{5}z^{-1}+10a^{4}z^{8}-18a^{4}z^{6}+15a^{4}z^{4}-8a^{4}z^{2}-a^{4}z^{-2}+4a^{4}+9a^{3}z^{7}-13a^{3}z^{5}+9a^{3}z^{3}+6a^{2}z^{6}-6a^{2}z^{4}+4a^{2}z^{2}-a^{2}+3az^{5}-2az^{3}+z^{4}-z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 8 3     -5 -5               9 4       5 -7             10 9         -1 -9           10 8           2 -11         7 11             4 -13       6 9               -3 -15     3 8                 5 -17   1 5                   -4 -19   3                     3 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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