From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a190 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X18,8,19,7 X16,10,17,9 X2,12,3,11 X22,13,1,14 X20,15,21,16 X8,18,9,17 X6,20,7,19 X10,21,11,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -11, 6, -2, 7, -3, 8, -5, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 12 14 18 16 2 22 20 8 6 10
A Braid Representative
A Morse Link Presentation K11a190 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a190/ThurstonBennequinNumber
Hyperbolic Volume 12.2349
A-Polynomial See Data:K11a190/A-polynomial

[edit Notes for K11a190's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a190's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+9 t^2-19 t+25-19 t^{-1} +9 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, 0 }
Jones polynomial -q^7+3 q^6-5 q^5+8 q^4-11 q^3+13 q^2-13 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +2 z^4 a^{-4} -3 z^4+2 a^2 z^2-4 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2-2 a^{-2} +3 a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +6 z^8 a^{-2} +4 z^8 a^{-4} +3 z^8 a^{-6} +5 z^8+6 a z^7-17 z^7 a^{-3} -10 z^7 a^{-5} +z^7 a^{-7} +5 a^2 z^6-24 z^6 a^{-2} -26 z^6 a^{-4} -13 z^6 a^{-6} -6 z^6+3 a^3 z^5-9 a z^5-11 z^5 a^{-1} +11 z^5 a^{-3} +6 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-6 a^2 z^4+27 z^4 a^{-2} +35 z^4 a^{-4} +16 z^4 a^{-6} +z^4-3 a^3 z^3+7 a z^3+11 z^3 a^{-1} +3 z^3 a^{-5} +4 z^3 a^{-7} -a^4 z^2+3 a^2 z^2-14 z^2 a^{-2} -18 z^2 a^{-4} -6 z^2 a^{-6} +2 z^2-2 a z-3 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} -a^2+2 a^{-2} +3 a^{-4} + a^{-6}
The A2 invariant Data:K11a190/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a190/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_86,}

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

Vassiliev invariants

V2 and V3: (-1, 1)
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
-4 8 8 \frac{178}{3} \frac{86}{3} -32 \frac{80}{3} -\frac{160}{3} 72 -\frac{32}{3} 32 -\frac{712}{3} -\frac{344}{3} -\frac{4831}{30} \frac{982}{15} -\frac{12902}{45} \frac{1663}{18} -\frac{2911}{30}

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=0 is the signature of K11a190. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15           1-1
13          2 2
11         31 -2
9        52  3
7       63   -3
5      75    2
3     66     0
1    67      -1
-1   47       3
-3  25        -3
-5 14         3
-7 2          -2
-91           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.