L11n374

L11n373

L11n375

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n374 at Knotilus!

Link Presentations

[edit Notes on L11n374's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X13,20,14,21 X7,19,8,18 X17,9,18,8 X19,22,20,17 X21,14,22,15
Gauss code	{1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(2uw^{2}-uw+w-2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-9}+3q^{-8}-5q^{-7}+8q^{-6}-8q^{-5}+8q^{-4}-6q^{-3}+6q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+3a^{8}-z^{6}a^{6}-4z^{4}a^{6}-5z^{2}a^{6}+a^{6}z^{-2}-2a^{6}-z^{6}a^{4}-3z^{4}a^{4}-2z^{2}a^{4}-2a^{4}z^{-2}-3a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$z^{3}a^{11}-2za^{11}+3z^{4}a^{10}-4z^{2}a^{10}+a^{10}+2z^{7}a^{9}-7z^{5}a^{9}+14z^{3}a^{9}-6za^{9}+3z^{8}a^{8}-12z^{6}a^{8}+22z^{4}a^{8}-12z^{2}a^{8}+3a^{8}+z^{9}a^{7}+2z^{7}a^{7}-14z^{5}a^{7}+20z^{3}a^{7}-6za^{7}+5z^{8}a^{6}-15z^{6}a^{6}+14z^{4}a^{6}-5z^{2}a^{6}+a^{6}z^{-2}-a^{6}+z^{9}a^{5}+2z^{7}a^{5}-12z^{5}a^{5}+7z^{3}a^{5}+2za^{5}-2a^{5}z^{-1}+2z^{8}a^{4}-2z^{6}a^{4}-9z^{4}a^{4}+9z^{2}a^{4}+2a^{4}z^{-2}-6a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+4za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-4z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               5 1   4 -5             3 3     0 -7           5 3       2 -9       1 4 3         0 -11       5 5           0 -13     2 5             3 -15   1 3               -2 -17   2                 2 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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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