L11n361

L11n360

L11n362

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Link Presentations

[edit Notes on L11n361's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X20,9,21,10 X10,19,11,20 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, 7, -6, -10, 9}, {-4, -1, 2, -5, 6, -7, 11, -2, 3, 8, -9, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}\left(v^{2}+vw+w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-4q^{-8}+7q^{-7}-7q^{-6}+8q^{-5}-7q^{-4}+6q^{-3}-3q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{4}-5a^{6}z^{2}-2a^{6}z^{-2}-6a^{6}-a^{4}z^{4}+a^{4}z^{-2}+a^{4}+2a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+4z^{3}a^{11}-za^{11}+3z^{8}a^{10}-14z^{6}a^{10}+19z^{4}a^{10}-8z^{2}a^{10}+2z^{9}a^{9}-5z^{7}a^{9}-5z^{5}a^{9}+12z^{3}a^{9}-2za^{9}+8z^{8}a^{8}-35z^{6}a^{8}+47z^{4}a^{8}-27z^{2}a^{8}-a^{8}z^{-2}+7a^{8}+2z^{9}a^{7}-2z^{7}a^{7}-12z^{5}a^{7}+17z^{3}a^{7}-9za^{7}+2a^{7}z^{-1}+5z^{8}a^{6}-19z^{6}a^{6}+28z^{4}a^{6}-25z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+4z^{7}a^{5}-10z^{5}a^{5}+10z^{3}a^{5}-7za^{5}+2a^{5}z^{-1}+2z^{6}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+z^{5}a^{3}+z^{3}a^{3}+za^{3}+3z^{2}a^{2}-2a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 2 -1 -5               3     3 -7             4 3     -1 -9           4 3       1 -11         3 4         1 -13       4 4           0 -15     2 5             3 -17   1 2               -1 -19   2                 2 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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