L11a527

L11a526

L11a528

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a527 at Knotilus!

Link Presentations

[edit Notes on L11a527's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{2}-u^{2}v^{2}w+u^{2}vw^{3}-4u^{2}vw^{2}+4u^{2}vw-u^{2}v+2u^{2}w^{2}-3u^{2}w+u^{2}+uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-2uvw^{3}+7uvw^{2}-7uvw+2uv-3uw^{2}+3uw-u-v^{2}w^{3}+3v^{2}w^{2}-2v^{2}w+vw^{3}-4vw^{2}+4vw-v+w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+8q^{-7}-13q^{-6}+19q^{-5}-21q^{-4}+23q^{-3}-q^{2}-19q^{-2}+4q+15q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-3z^{4}a^{6}-8z^{2}a^{6}-2a^{6}z^{-2}-8a^{6}+2z^{6}a^{4}+7z^{4}a^{4}+11z^{2}a^{4}+a^{4}z^{-2}+7a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-3a^{10}z^{4}+3a^{10}z^{2}-a^{10}+3a^{9}z^{7}-7a^{9}z^{5}+4a^{9}z^{3}+5a^{8}z^{8}-10a^{8}z^{6}+6a^{8}z^{4}-5a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+5a^{7}z^{9}-6a^{7}z^{7}-3a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+10a^{6}z^{8}-35a^{6}z^{6}+43a^{6}z^{4}-31a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+12a^{5}z^{9}-21a^{5}z^{7}+8a^{5}z^{5}+7a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-45a^{4}z^{6}+52a^{4}z^{4}-31a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+7a^{3}z^{9}-4a^{3}z^{7}-9a^{3}z^{5}+11a^{3}z^{3}-3a^{3}z+10a^{2}z^{8}-17a^{2}z^{6}+13a^{2}z^{4}-7a^{2}z^{2}+2a^{2}+8az^{7}-12az^{5}+z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}-az+4z^{6}-5z^{4}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   6 1   -5 -1                 9 3     6 -3               11 7       -4 -5             12 8         4 -7           10 12           2 -9         9 11             -2 -11       5 11               6 -13     3 8                 -5 -15   1 6                   5 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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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