L11a428

L11a427

L11a429

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a428 at Knotilus!

Link Presentations

[edit Notes on L11a428's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-2uv^{2}w+uv^{2}-2uvw^{4}+5uvw^{3}-5uvw^{2}+4uvw-2uv-2uw^{3}+2uw^{2}-2uw+u-v^{2}w^{4}+2v^{2}w^{3}-2v^{2}w^{2}+2v^{2}w+2vw^{4}-4vw^{3}+5vw^{2}-5vw+2v-w^{4}+2w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-9q^{3}-12q^{-3}+14q^{2}+19q^{-2}-19q-20q^{-1}+22$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}+3a^{4}-10a^{2}-2a^{-2}+9+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}+6a^{4}z^{8}-18a^{4}z^{6}+4z^{6}a^{-4}+24a^{4}z^{4}-5z^{4}a^{-4}-22a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}+5a^{3}z^{9}-6a^{3}z^{7}+8z^{7}a^{-3}-9a^{3}z^{5}-13z^{5}a^{-3}+20a^{3}z^{3}+6z^{3}a^{-3}-16a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}+10z^{8}a^{-2}-41a^{2}z^{6}-19z^{6}a^{-2}+59a^{2}z^{4}+16z^{4}a^{-2}-46a^{2}z^{2}-8z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+20a^{2}+2a^{-2}+12az^{9}+7z^{9}a^{-1}-23az^{7}-6z^{7}a^{-1}+5az^{5}-7z^{5}a^{-1}+26az^{3}+16z^{3}a^{-1}-27az-13za^{-1}+9az^{-1}+5a^{-1}z^{-1}+2z^{10}+14z^{8}-45z^{6}+53z^{4}-30z^{2}-4z^{-2}+13$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 8 3     5 3               11 6       -5 1             11 8         3 -1           11 13           2 -3         8 9             -1 -5       5 12               7 -7     3 7                 -4 -9     5                   5 -11 1 3                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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