L10a169

L10a168

L10a170

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Japanese family symbol

Link Presentations

[edit Notes on L10a169's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X14,9,11,10 X20,13,15,14 X10,19,5,20 X18,12,19,11 X4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 6, -8}, {4, -5, 10, -9, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1)^2 (x-1)^2}{\sqrt{u} \sqrt{v} w x}$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-3 a^3 z^3+3 a z^3-z^3 a^{-1} -a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3}$ (db)
Kauffman polynomial	$a^7 z^5+5 a^6 z^6-4 a^6 z^4+11 a^5 z^7-17 a^5 z^5+8 a^5 z^3-a^5 z^{-3} +a^5 z^{-1} +11 a^4 z^8-12 a^4 z^6+3 a^4 z^{-2} -2 a^4+4 a^3 z^9+19 a^3 z^7-48 a^3 z^5+z^5 a^{-3} +24 a^3 z^3-3 a^3 z^{-3} +a^3 z+22 a^2 z^8-34 a^2 z^6+5 z^6 a^{-2} +8 a^2 z^4-4 z^4 a^{-2} +6 a^2 z^{-2} -3 a^2+4 a z^9+19 a z^7+11 z^7 a^{-1} -48 a z^5-17 z^5 a^{-1} +24 a z^3+8 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} +a z+ a^{-1} z^{-1} +11 z^8-12 z^6+3 z^{-2} -2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   4   -4 4                 7 1   6 2               8 4     -4 0             14 7       7 -2           10 12         2 -4         12 10           2 -6       7 14             7 -8     4 8               -4 -10   1 7                 6 -12   4                   -4 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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