K11n80

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n80 at Knotilus!

Knot presentations

 Planar diagram presentation X4251 X8394 X14,6,15,5 X10,8,11,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,12,20,13 X21,16,22,17 Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -4, -6, 10, 7, -3, -8, 11, -9, 6, -10, 8, -11, 9 Dowker-Thistlethwaite code 4 8 14 10 2 -18 6 -20 -22 -12 -16
A Braid Representative

Three dimensional invariants

 Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n80/ThurstonBennequinNumber Hyperbolic Volume 10.8343 A-Polynomial See Data:K11n80/A-polynomial

Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

Polynomial invariants

 Alexander polynomial $-t^3+t^2+5 t-9+5 t^{-1} + t^{-2} - t^{-3}$ Conway polynomial $-z^6-5 z^4+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 15, -2 } Jones polynomial $q^2-q+1-2 q^{-2} +3 q^{-3} -3 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8}$ HOMFLY-PT polynomial (db, data sources) $-a^8+2 z^2 a^6+a^6+3 z^2 a^4+3 a^4-z^6 a^2-6 z^4 a^2-9 z^2 a^2-5 a^2+z^4+4 z^2+3$ Kauffman polynomial (db, data sources) $z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+z^7 a^7-6 z^3 a^7+3 z a^7+3 z^6 a^6-7 z^4 a^6+3 z^2 a^6-a^6+3 z^5 a^5-6 z^3 a^5+3 z a^5+z^8 a^4-8 z^6 a^4+20 z^4 a^4-15 z^2 a^4+3 a^4+z^9 a^3-8 z^7 a^3+18 z^5 a^3-13 z^3 a^3+5 z a^3+2 z^8 a^2-16 z^6 a^2+36 z^4 a^2-26 z^2 a^2+5 a^2+z^9 a-7 z^7 a+14 z^5 a-10 z^3 a+3 z a+z^8-7 z^6+15 z^4-12 z^2+3$ The A2 invariant Data:K11n80/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n80/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Vassiliev invariants

 V2 and V3: (0, -3)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $-24$ $0$ $128$ $40$ $0$ $-304$ $-64$ $-56$ $0$ $288$ $0$ $0$ $1008$ $56$ $\frac{920}{3}$ $\frac{224}{3}$ $16$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of K11n80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-101234χ
5           11
3            0
1         11 0
-1       21   1
-3      211   -2
-5     221    1
-7    22      0
-9   221      1
-11  12        1
-13 12         -1
-15 1          1
-171           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.