L10a112

L10a111

L10a113

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Link Presentations

[edit Notes on L10a112's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,6,15,5 X18,7,19,8 X16,10,17,9 X10,11,1,12 X6,14,7,13 X4,19,5,20 X20,15,11,16 X2,18,3,17
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-v+1\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-\frac{4}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+8 q^{5/2}-\frac{14}{q^{5/2}}-13 q^{3/2}+\frac{16}{q^{3/2}}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -2 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -2 a^3 z+5 a z-4 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^6 z^4+4 a^5 z^5-a^5 z^3+9 a^4 z^6+z^6 a^{-4} -8 a^4 z^4-2 z^4 a^{-4} +3 a^4 z^2+z^2 a^{-4} +13 a^3 z^7+4 z^7 a^{-3} -20 a^3 z^5-10 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -4 a^3 z-2 z a^{-3} +10 a^2 z^8+6 z^8 a^{-2} -10 a^2 z^6-13 z^6 a^{-2} -3 a^2 z^4+6 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +3 a z^9+3 z^9 a^{-1} +16 a z^7+7 z^7 a^{-1} -47 a z^5-33 z^5 a^{-1} +34 a z^3+28 z^3 a^{-1} -10 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +16 z^8-33 z^6+14 z^4-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   3   3 6                 5 1   -4 4               8 3     5 2             8 5       -3 0           10 8         2 -2         8 10           2 -4       6 8             -2 -6     3 8               5 -8   1 6                 -5 -10   3                   3 -12 1                     -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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