L11a429

L11a428

L11a430

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Link Presentations

[edit Notes on L11a429's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-2 t(2)^2-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-5 t(1) t(2)+9 t(1) t(3) t(2)-9 t(3) t(2)+4 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-4 t(1) t(3)+5 t(3)-2}{\sqrt{t(1)} t(2) t(3)}} (db)
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-14q^{6}+20q^{5}-22q^{4}+23q^{3}-19q^{2}-q^{-2}+15q+4q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+4z^{2}a^{-2}-4z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-2}-6a^{-4}+2a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-10} -2 z^4 a^{-10} +z^2 a^{-10} +4 z^7 a^{-9} -9 z^5 a^{-9} +5 z^3 a^{-9} +7 z^8 a^{-8} -16 z^6 a^{-8} +11 z^4 a^{-8} -4 z^2 a^{-8} + a^{-8} +6 z^9 a^{-7} -7 z^7 a^{-7} -4 z^5 a^{-7} +3 z^3 a^{-7} +2 z^{10} a^{-6} +12 z^8 a^{-6} -34 z^6 a^{-6} +23 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +12 z^9 a^{-5} -18 z^7 a^{-5} +6 z^5 a^{-5} -5 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +13 z^8 a^{-4} -28 z^6 a^{-4} +12 z^4 a^{-4} +8 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +6 z^9 a^{-3} -9 z^5 a^{-3} +z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +8 z^8 a^{-2} -7 z^6 a^{-2} -4 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2} +7 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-6 z^4+3 z^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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 9 4     -5 11               11 5       6 9             11 9         -2 7           12 11           1 5         8 12             4 3       7 11               -4 1     3 10                 7 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}} ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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