L11n373

L11n372

L11n374

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n373 at Knotilus!

Link Presentations

[edit Notes on L11n373's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,19,8,18 X21,15,22,14 X9,20,10,21 X13,9,14,8 X15,17,16,22 X17,5,18,16 X19,12,20,13 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {-8, 3, -9, 5, -4, 7}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 4, -7, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(uv^{2}w^{3}-uv^{2}w^{2}-uvw^{3}+2uvw^{2}-2uvw-uw^{2}+uw+v^{2}w^{2}-v^{2}w-2vw^{2}+2vw-v-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+3q^{-4}-6q^{3}-5q^{-3}+10q^{2}+9q^{-2}-11q-11q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{6}-z^{6}a^{-2}-5a^{2}z^{4}-4z^{4}a^{-2}-10a^{2}z^{2}-5z^{2}a^{-2}-2a^{2}z^{-2}-8a^{2}-a^{-2}+z^{8}+6z^{6}+13z^{4}+12z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+4z^{8}a^{-2}+9z^{8}+3a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+4z^{7}a^{-3}-18a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}-6a^{3}z^{5}-10az^{5}-10z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+37a^{2}z^{4}-6z^{4}a^{-4}+37z^{4}+7a^{3}z^{3}+18az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-5}-14a^{4}z^{2}-36a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-24z^{2}-7a^{3}z-10az-3za^{-1}+za^{-3}+za^{-5}+7a^{4}+14a^{2}-a^{-2}-a^{-4}+8+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               4 1   -3 5             6 2     4 3           6 5       -1 1         7 5         2 -1       5 7           2 -3     4 6             -2 -5   2 6               4 -7 1 3                 -2 -9 3                   3

Integral Khovanov Homology (db, data source)

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

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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