L11a504

L11a503

L11a505

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a504 at Knotilus!

Link Presentations

[edit Notes on L11a504's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X22,17,13,18 X20,15,21,16 X16,21,17,22 X12,19,7,20 X18,11,19,12 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, {11, -4, 6, -7, 5, -9, 8, -6, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+2u^{2}v^{2}w-u^{2}v^{2}-u^{2}vw^{3}+u^{2}vw^{2}-u^{2}vw+u^{2}v-uv^{2}w^{3}+uv^{2}w^{2}-uv^{2}w+uv^{2}+2uvw^{3}-2uvw^{2}+2uvw-2uv-uw^{3}+uw^{2}-uw+u-vw^{3}+vw^{2}-vw+v+w^{3}-2w^{2}+2w-2}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-4}-q^{-5}+5q^{-6}-6q^{-7}+10q^{-8}-11q^{-9}+12q^{-10}-11q^{-11}+9q^{-12}-6q^{-13}+3q^{-14}-q^{-15}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{14}-a^{12}z^{6}-3a^{12}z^{4}+2a^{12}z^{2}+4a^{12}z^{-2}+9a^{12}+a^{10}z^{8}+4a^{10}z^{6}-a^{10}z^{4}-18a^{10}z^{2}-5a^{10}z^{-2}-19a^{10}+a^{8}z^{8}+7a^{8}z^{6}+18a^{8}z^{4}+21a^{8}z^{2}+2a^{8}z^{-2}+11a^{8}$ (db)
Kauffman polynomial	$z^{3}a^{19}+3z^{4}a^{18}+6z^{5}a^{17}-4z^{3}a^{17}+za^{17}+9z^{6}a^{16}-13z^{4}a^{16}+5z^{2}a^{16}-a^{16}+10z^{7}a^{15}-20z^{5}a^{15}+9z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+8z^{8}a^{14}-17z^{6}a^{14}+3z^{4}a^{14}+3z^{2}a^{14}-a^{14}z^{-2}+4z^{9}a^{13}-3z^{7}a^{13}-23z^{5}a^{13}+33z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+z^{10}a^{12}+6z^{8}a^{12}-32z^{6}a^{12}+37z^{4}a^{12}-20z^{2}a^{12}-4a^{12}z^{-2}+13a^{12}+5z^{9}a^{11}-16z^{7}a^{11}-z^{5}a^{11}+39z^{3}a^{11}-35za^{11}+9a^{11}z^{-1}+z^{10}a^{10}-z^{8}a^{10}-13z^{6}a^{10}+36z^{4}a^{10}-39z^{2}a^{10}-5a^{10}z^{-2}+22a^{10}+z^{9}a^{9}-3z^{7}a^{9}-4z^{5}a^{9}+20z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-7z^{6}a^{8}+18z^{4}a^{8}-21z^{2}a^{8}-2a^{8}z^{-2}+11a^{8}$ (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 -7                       1 1 -9                     1 1 0 -11                   4     4 -13                 2 1     -1 -15               8 4       4 -17             4 3         -1 -19           8 7           1 -21         5 6             1 -23       4 6               -2 -25     2 5                 3 -27   1 4                   -3 -29   2                     2 -31 1                       -1

Integral Khovanov Homology (db, data source)

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

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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