L11n257

L11n256

L11n258

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Link Presentations

[edit Notes on L11n257's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, -3, 6, -5, -9, 8, 3, -7, 4, -6, 5, -4, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{2}-uvw-3uw^{2}+4uw-2u+2vw^{3}-4vw^{2}+3vw+w^{2}-w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$2q^{-1}-2q^{-2}+6q^{-3}-6q^{-4}+8q^{-5}-6q^{-6}+6q^{-7}-5q^{-8}+2q^{-9}-q^{-10}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+3a^{8}z^{-2}+5a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}z^{-2}-2a^{4}+2z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+9a^{11}z^{3}-7a^{11}z+2a^{11}z^{-1}+2a^{10}z^{8}-8a^{10}z^{6}+8a^{10}z^{4}-2a^{10}z^{2}-a^{10}z^{-2}+a^{10}+a^{9}z^{9}+2a^{9}z^{7}-24a^{9}z^{5}+40a^{9}z^{3}-27a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-23a^{8}z^{6}+22a^{8}z^{4}-7a^{8}z^{2}-3a^{8}z^{-2}+5a^{8}+a^{7}z^{9}+5a^{7}z^{7}-33a^{7}z^{5}+49a^{7}z^{3}-34a^{7}z+10a^{7}z^{-1}+4a^{6}z^{8}-13a^{6}z^{6}+11a^{6}z^{4}-4a^{6}z^{2}-2a^{6}z^{-2}+4a^{6}+4a^{5}z^{7}-13a^{5}z^{5}+17a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+2a^{4}z^{6}-3a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}-3a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 3 0 -5               3   1 4 -7             4 4     0 -9           5 3       2 -11         2 5 1       2 -13       4 4           0 -15     1 2             1 -17   1 4               -3 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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