L11n294

L11n293

L11n295

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n294 at Knotilus!

Link Presentations

[edit Notes on L11n294's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,22,5,21 X11,19,12,18 X9,17,10,16 X17,11,18,22 X19,9,20,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {-6, -2, 11, -4, 3, 7, -8, 6, -9, -3, 5, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}+t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)t(3)+t(1)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+q^{7}-q^{6}+3q^{5}-q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-21z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-15a^{-6}+8a^{-8}-a^{-10}+2a^{-4}z^{-2}-5a^{-6}z^{-2}+4a^{-8}z^{-2}-a^{-10}z^{-2}$ (db)
Kauffman polynomial	$-2za^{-11}+a^{-11}z^{-1}-3z^{2}a^{-10}-a^{-10}z^{-2}+2a^{-10}+2z^{7}a^{-9}-12z^{5}a^{-9}+19z^{3}a^{-9}-13za^{-9}+5a^{-9}z^{-1}+3z^{8}a^{-8}-19z^{6}a^{-8}+36z^{4}a^{-8}-29z^{2}a^{-8}-4a^{-8}z^{-2}+13a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-9z^{5}a^{-7}+31z^{3}a^{-7}-27za^{-7}+9a^{-7}z^{-1}+4z^{8}a^{-6}-26z^{6}a^{-6}+53z^{4}a^{-6}-45z^{2}a^{-6}-5a^{-6}z^{-2}+20a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+3z^{5}a^{-5}+12z^{3}a^{-5}-16za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-19z^{2}a^{-4}-2a^{-4}z^{-2}+10a^{-4}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 17               2 1 -1 15             1 2 1 0 13           2 2     0 11         1 1 2     2 9       1 3         2 7     2 1 1         2 5   1 3             2 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

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

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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