L11a403

L11a402

L11a404

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a403 at Knotilus!

Link Presentations

[edit Notes on L11a403's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,9,17,10 X14,8,15,7 X18,12,19,11 X20,15,21,16 X22,18,11,17 X10,19,5,20 X8,22,9,21 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {5, -2, 11, -4, 6, -3, 7, -5, 8, -6, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-2uvw^{4}+5uvw^{3}-5uvw^{2}+4uvw-uv+uw^{4}-3uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+vw^{4}-4vw^{3}+5vw^{2}-5vw+2v+w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-15q^{4}+20q^{3}-22q^{2}+23q-18+15q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}+a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-10z^{4}a^{-2}-2a^{2}z^{2}-9z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+2z^{6}+7z^{4}+7z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+14z^{9}a^{-1}+9z^{9}a^{-3}+4a^{2}z^{8}+23z^{8}a^{-2}+15z^{8}a^{-4}+12z^{8}+a^{3}z^{7}-11az^{7}-28z^{7}a^{-1}-2z^{7}a^{-3}+14z^{7}a^{-5}-14a^{2}z^{6}-77z^{6}a^{-2}-25z^{6}a^{-4}+9z^{6}a^{-6}-57z^{6}-3a^{3}z^{5}-2az^{5}-6z^{5}a^{-1}-30z^{5}a^{-3}-19z^{5}a^{-5}+4z^{5}a^{-7}+17a^{2}z^{4}+73z^{4}a^{-2}+15z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+67z^{4}+3a^{3}z^{3}+13az^{3}+26z^{3}a^{-1}+27z^{3}a^{-3}+10z^{3}a^{-5}-z^{3}a^{-7}-8a^{2}z^{2}-30z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-29z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 9 3     -6 7               11 6       5 5             12 10         -2 3           11 10           1 1         9 14             5 -1       6 9               -3 -3     3 10                 7 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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