L11a426

L11a425

L11a427

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a426 at Knotilus!

Link Presentations

[edit Notes on L11a426's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, -9, 8, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{4}-3uv^{2}w^{3}+4uv^{2}w^{2}-2uv^{2}w-uvw^{4}+5uvw^{3}-8uvw^{2}+6uvw-uv-uw^{3}+3uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-3v^{2}w^{2}+v^{2}w+vw^{4}-6vw^{3}+8vw^{2}-5vw+v+2w^{3}-4w^{2}+3w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-8}+4q^{-7}-10q^{-6}+16q^{-5}-22q^{-4}+q^{3}+26q^{-3}-4q^{2}-24q^{-2}+10q+23q^{-1}-15$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-11a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+10a^{4}z^{2}-13a^{2}z^{2}+4z^{2}-2a^{6}+8a^{4}-10a^{2}+4-a^{6}z^{-2}+4a^{4}z^{-2}-5a^{2}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-a^{9}z^{3}+4a^{8}z^{6}-4a^{8}z^{4}+a^{8}z^{2}+9a^{7}z^{7}-14a^{7}z^{5}+11a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+11a^{6}z^{8}-15a^{6}z^{6}+9a^{6}z^{4}-5a^{6}z^{2}-a^{6}z^{-2}+3a^{6}+7a^{5}z^{9}+7a^{5}z^{7}-37a^{5}z^{5}+38a^{5}z^{3}-18a^{5}z+5a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-56a^{4}z^{6}+51a^{4}z^{4}-28a^{4}z^{2}-4a^{4}z^{-2}+12a^{4}+14a^{3}z^{9}-13a^{3}z^{7}-23a^{3}z^{5}+36a^{3}z^{3}-24a^{3}z+9a^{3}z^{-1}+2a^{2}z^{10}+19a^{2}z^{8}-57a^{2}z^{6}+z^{6}a^{-2}+58a^{2}z^{4}-2z^{4}a^{-2}-35a^{2}z^{2}+z^{2}a^{-2}-5a^{2}z^{-2}+15a^{2}+7az^{9}-7az^{7}+4z^{7}a^{-1}-9az^{5}-8z^{5}a^{-1}+14az^{3}+4z^{3}a^{-1}-11az+5az^{-1}+8z^{8}-19z^{6}+18z^{4}-12z^{2}-2z^{-2}+7$ (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 7                       1 1 5                     3   -3 3                   7 1   6 1                 9 4     -5 -1               14 6       8 -3             12 11         -1 -5           14 12           2 -7         9 13             4 -9       7 13               -6 -11     3 9                 6 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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