K11n140

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n140 at Knotilus!

Knot presentations

 Planar diagram presentation X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X20,9,21,10 X18,11,19,12 X2,13,3,14 X15,6,16,7 X22,18,1,17 X10,19,11,20 X8,21,9,22 Gauss code 1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -6, 10, -5, 11, -9 Dowker-Thistlethwaite code 4 12 -16 -14 20 18 2 -6 22 10 8
A Braid Representative

Three dimensional invariants

 Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n140/ThurstonBennequinNumber Hyperbolic Volume 11.4116 A-Polynomial See Data:K11n140/A-polynomial

Four dimensional invariants

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

Polynomial invariants

 Alexander polynomial $-2 t^2+13 t-21+13 t^{-1} -2 t^{-2}$ Conway polynomial $-2 z^4+5 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 51, -2 } Jones polynomial $q-3+5 q^{-1} -7 q^{-2} +9 q^{-3} -8 q^{-4} +8 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8}$ HOMFLY-PT polynomial (db, data sources) $-2 a^8+3 z^2 a^6+2 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2+z^2$ Kauffman polynomial (db, data sources) $3 z^5 a^9-10 z^3 a^9+7 z a^9+z^8 a^8-2 z^6 a^8+z^2 a^8-2 a^8+z^9 a^7-3 z^7 a^7+9 z^5 a^7-17 z^3 a^7+9 z a^7+4 z^8 a^6-12 z^6 a^6+15 z^4 a^6-6 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-3 z^5 a^5+z^3 a^5+2 z a^5+3 z^8 a^4-6 z^6 a^4+9 z^4 a^4-5 z^2 a^4+a^4+4 z^7 a^3-6 z^5 a^3+4 z^3 a^3+4 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-4 z^3 a+z^4-z^2$ The A2 invariant Data:K11n140/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n140/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: (5, -11)
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 $20$ $-88$ $200$ $\frac{1606}{3}$ $\frac{338}{3}$ $-1760$ $-\frac{10288}{3}$ $-\frac{1696}{3}$ $-696$ $\frac{4000}{3}$ $3872$ $\frac{32120}{3}$ $\frac{6760}{3}$ $\frac{131935}{6}$ $-\frac{5030}{3}$ $\frac{106310}{9}$ $\frac{4837}{18}$ $\frac{11071}{6}$

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 K11n140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-1012χ
3         11
1        2 -2
-1       31 2
-3      53  -2
-5     42   2
-7    45    1
-9   44     0
-11  14      3
-13 24       -2
-15 1        1
-172         -2
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

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