From Knot Atlas
Jump to: navigation, search






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

Visit K11n1 at Knotilus!

Knot K11n1.
A graph, knot K11n1.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X11,16,12,17 X13,20,14,21 X6,15,7,16 X17,22,18,1 X19,12,20,13 X21,18,22,19
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, -4, 8, 6, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 10 14 2 -16 -20 6 -22 -12 -18
A Braid Representative
A Morse Link Presentation K11n1 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n1/ThurstonBennequinNumber
Hyperbolic Volume 9.70048
A-Polynomial See Data:K11n1/A-polynomial

[edit Notes for K11n1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n1's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+7 t-11+7 t^{-1} - t^{-2}
Conway polynomial -z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, -2 }
Jones polynomial  q^{-1} - q^{-2} +3 q^{-3} -4 q^{-4} +4 q^{-5} -4 q^{-6} +4 q^{-7} -3 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8+2 a^8-z^4 a^6-2 z^2 a^6-2 a^6+2 z^2 a^4+a^4+z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-10 z^6 a^{10}+14 z^4 a^{10}-7 z^2 a^{10}+a^{10}+z^9 a^9-2 z^7 a^9-7 z^5 a^9+13 z^3 a^9-4 z a^9+4 z^8 a^8-19 z^6 a^8+25 z^4 a^8-12 z^2 a^8+2 a^8+z^9 a^7-2 z^7 a^7-5 z^5 a^7+8 z^3 a^7-2 z a^7+2 z^8 a^6-9 z^6 a^6+13 z^4 a^6-9 z^2 a^6+2 a^6+z^7 a^5-3 z^5 a^5+3 z^3 a^5-z a^5+2 z^4 a^4-3 z^2 a^4+a^4+z^3 a^3+z^2 a^2-a^2
The A2 invariant Data:K11n1/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n1/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_48,}

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

Vassiliev invariants

V2 and V3: (3, -7)
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
12 -56 72 286 42 -672 -\frac{4688}{3} -\frac{800}{3} -216 288 1568 3432 504 \frac{87151}{10} \frac{1694}{15} \frac{52822}{15} \frac{529}{6} \frac{4431}{10}

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 K11n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1         11
-3        110
-5       2  2
-7      21  -1
-9     22   0
-11    22    0
-13   22     0
-15  12      1
-17 12       -1
-19 1        1
-211         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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.